text stringlengths 1 2.12k | source dict |
|---|---|
$M \sim N$) if there is a one-to-one correspondence between the elements of M and the elements of N. The concept of equivalence is applicable both to finite and infinite sets. Two finite sets are equivalent if and only if they have the same number of elements. We can now define a countable set as a set equivalent to the set $\mathscr{Z^{+}}$ of all positive integers. It is clear that two sets are equivalent to a third set are equivalent to each other, and in particular that any two countable sets are equivalent. $\bf{Example1}$ The sets of points in any two closed intervals$[a,b]$and$[c,d]\$ are equivalent; you can “see’ a one-to-one correspondence by drawing the following diagram: Step 1: draw cd as a base of a triangle. Let the third vertex of the triangle be O. Draw a line segment “ab” above the base of the triangle; where “a” lies on one side of the triangle and “b” lies on the third side of the third triangle. Note that two points p and q correspond to each other if and only if they lie on the same ray emanating from the point O in which the extensions of the line segments ac and bd intersect. | {
"domain": "madhavamathcompetition.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9883127433812521,
"lm_q1q2_score": 0.8783716472247783,
"lm_q2_score": 0.8887587993853654,
"openwebmath_perplexity": 294.34835768822796,
"openwebmath_score": 0.9011602997779846,
"tags": null,
"url": "https://madhavamathcompetition.com/category/chennai-mathematical-institute-entrance/"
} |
$\bf{Example2}$
The set of all points z in the complex plane is equivalent to the set of all points z on a sphere. In fact, a one-to-one correspondence $z \leftrightarrow \alpha$ can be established by using stereographic projection. The origin is the North Pole of the sphere.
$\bf{Example3}$
The set of all points x in the open unit interval $(0,1)$ is equivalent to the set of all points y on the whole real line. For example, the formula $y=\frac{1}{\pi}\arctan{x}+\frac{1}{2}$ establishes a one-to-one correspondence between these two sets. $\bf{QED}$.
The last example and the examples in Section 2 show that an infinite set is sometimes equivalent to one of its proper subsets. For example, there are “as many” positive integers as integers of arbitrary sign, there are “as many” points in the interval $(0,1)$ as on the whole real line, and so on. This fact is characteristic of all infinite sets (and can be used to define such sets) as shown by:
$\bf{Theorem4}$
$\bf{Every \hspace{0.1in} infinite \hspace{0.1in} set \hspace{0.1in}is \hspace{0.1in} equivalent \hspace{0.1in} to \hspace{0.1in}one \hspace{0.1in}of \hspace{0.1in}its \hspace{0.1in}proper \hspace{0.1in}subsets.}$
$\bf{Proof}$
According to Theorem 3, every infinite set M contains a countable subset. Let this subset be $A=\{a_{1}, a_{2}, a_{3}, \ldots, a_{n}, \ldots \}$ and partition A into two countable subsets $A_{1}=\{a_{1}, a_{3}, a_{5}, \ldots \}$ and $A_{2}=\{a_{2}, a_{4}, a_{6}, \ldots \}$.
Obviously, we can establish a one-to-one correspondence between the countable subsets A and $A_{1}$ (merely let $a_{n} \leftrightarrow a_{2n-1}$). This correspondence can be extended to a one-to-one correspondence between the sets $A \bigcup (M-A)=M$ and $A_{1} \bigcup (M-A)=M-A_{2}$ by simply assigning x itself to each element $x \in M-A$. But $M-A_{2}$ is a proper subset of M. $\bf{QED}$.
More later, to be continued,
Regards,
Nalin Pithwa
A fifth degree equation in two variables: a clever solution
Question: | {
"domain": "madhavamathcompetition.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9883127433812521,
"lm_q1q2_score": 0.8783716472247783,
"lm_q2_score": 0.8887587993853654,
"openwebmath_perplexity": 294.34835768822796,
"openwebmath_score": 0.9011602997779846,
"tags": null,
"url": "https://madhavamathcompetition.com/category/chennai-mathematical-institute-entrance/"
} |
Regards,
Nalin Pithwa
A fifth degree equation in two variables: a clever solution
Question:
Verify the identity: $(2xy+(x^{2}-2y^{2}))^{5}+(2xy-(x^{2}-2y^{2}))^{5}=(2xy+(x^{2}+2y^{2})i)^{5}+(2xy-(x^{2}+2y^{2})i)^{5}$
let us observe first that each of the fifth degree expression is just a quadratic in two variables x and y. Let us say the above identity to be verified is:
$P_{1}+P_{2}=P_{3}+P_{4}$
Method I:
Use binomial expansion. It is a very longish tedious method.
Method II:
Factorize each of the quadratic expressions $P_{1}, P_{2}, P_{3}, P_{4}$ using quadratic formula method (what is known in India as Sridhar Acharya’s method):
Now fill in the above details.
You will conclude very happily that :
The above identity is transformed to :
$P_{1}=(x+y+\sqrt{3}y)^{5}(x+y-\sqrt{3}y)^{5}$
$P_{2}=(-1)^{5}(x-y-\sqrt{3}y)^{5}(x-y+\sqrt{3}y)^{5}$
$P_{3}=(i^{2}(x-y-\sqrt{3}y)(x-y+\sqrt{3}y))^{5}$
$P_{4}=((-i^{2})(x+y+\sqrt{3}y)(x-y-\sqrt{3}y))^{5}$
You will find that $P_{1}=P_{4}$ and $P_{2}=P_{4}$
Hence, it is verified that the given identity $P_{1}+P_{2}=P_{3}+P_{4}$. QED.
Regards,
Nalin Pithwa.
Set Theory, Relations, Functions: preliminaries: part 10: more tutorial problems for practice
Problem 1:
Prove that a function f is 1-1 iff $f^{-1}(f(A))=A$ for all $A \subset X$. Given that $f: X \longrightarrow Y$.
Problem 2:
Prove that a function if is onto iff $f(f^{-1}(C))=C$ for all $C \subset Y$. Given that $f: X \longrightarrow Y$.
Problem 3:
(a) How many functions are there from a non-empty set S into $\phi$\?
(b) How many functions are there from $\phi$ into an arbitrary set $S$?
(c) Show that the notation $\{ X_{i} \}_{i \in I}$ implicitly involves the notion of a function.
Problem 4:
Let $f: X \longrightarrow Y$ be a function, let $A \subset X$, $B \subset X$, $C \subset Y$ and $D \subset Y$. Prove that
i) $f(A \bigcap B) \subset f(A) \bigcap f(B)$
ii) $f^{-1}(f(A)) \supset A$
iii) $f(f^{-1}(C)) \subset C$
Problem 5: | {
"domain": "madhavamathcompetition.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9883127433812521,
"lm_q1q2_score": 0.8783716472247783,
"lm_q2_score": 0.8887587993853654,
"openwebmath_perplexity": 294.34835768822796,
"openwebmath_score": 0.9011602997779846,
"tags": null,
"url": "https://madhavamathcompetition.com/category/chennai-mathematical-institute-entrance/"
} |
ii) $f^{-1}(f(A)) \supset A$
iii) $f(f^{-1}(C)) \subset C$
Problem 5:
Let I be a non-empty set and for each $i \in I$, let $X_{i}$ be a set. Prove that
(a) for any set B, we have $B \bigcap \bigcup_{i \in I}X_{i}=\bigcup_{i \in I}(B \bigcap X_{i})$
(b) if each $X_{i}$ is a subset of a given set S, then $(\bigcup_{i \in I}X_{i})^{'}=\bigcap_{i \in I}(X_{i})^{'}$ where the prime indicates complement.
Problem 6:
Let A, B, C be subsets of a set S. Prove the following statements:
(i) $A- (B-C)=(A-B)\bigcup(A \bigcap B \bigcap C)$
(ii) $(A-B) \times C=(A \times C)-(B \times C)$
🙂 🙂 🙂
Nalin Pithwa
Set Theory, Relations, Functions: Preliminaries: Part IX: (tutorial problems)
Reference: Introductory Real Analysis, Kolmogorov and Fomin, Dover Publications.
Problem 1:
Prove that if $A \bigcup B=A$ and $A \bigcap B=A$, then $A=B$.
Problem 2:
Show that in general $(A-B)\bigcup B \neq A$.
Problem 3:
Let $A = \{ 2,4, \ldots, 2n, \ldots\}$ and $B= \{ 3,6,\ldots, 3n, \ldots\}$. Find $A \bigcap B$ and $A - B$.
Problem 4:
Prove that (a) $(A-B)\bigcap (C)=(A \bigcap C)-(B \bigcap C)$
Prove that (b) $A \Delta B = (A \bigcup B)-(A \bigcap B)$
Problem 5:
Prove that $\bigcup_{a}A_{\alpha}-\bigcup_{a}B_{\alpha}=\bigcup_{\alpha}(A_{\alpha}-B_{\alpha})$
Problem 6:
Let $A_{n}$ be the set of all positive integers divisible by $n$. Find the sets (i) $\bigcup_{n=2}^{\infty}A_{n}$ (ii) $\bigcap_{n=2}^{\infty}A_{n}$.
Problem 7:
Find (i) $\bigcup_{n=1}^{\infty}[n+\frac{1}{n}, n - \frac{1}{n}]$ (ii) $\bigcap_{n=1}^{\infty}(a-\frac{1}{n},b+\frac{1}{n})$
Problem 8:
Let $A_{\alpha}$ be the set of points lying on the curve $y=\frac{1}{x^{\alpha}}$ where $(0. What is $\bigcap_{\alpha \geq 1}A_{\alpha}$?
Problem 9: | {
"domain": "madhavamathcompetition.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9883127433812521,
"lm_q1q2_score": 0.8783716472247783,
"lm_q2_score": 0.8887587993853654,
"openwebmath_perplexity": 294.34835768822796,
"openwebmath_score": 0.9011602997779846,
"tags": null,
"url": "https://madhavamathcompetition.com/category/chennai-mathematical-institute-entrance/"
} |
Problem 9:
Let $y=f(x) = $ for all real x, where $$ is the fractional part of x. Prove that every closed interval of length 1 has the same image under f. What is the image? Is f one-to-one? What is the pre-image of the interval $\frac{1}{4} \leq y \leq \frac{3}{4}$? Partition the real line into classes of points with the same image.
Problem 10:
Given a set M, let $\mathscr{R}$ be the set of all ordered pairs on the form $(a,a)$ with $a \in M$, and let $aRb$ if and only if $(a,b) \in \mathscr{R}$. Interpret the relation R.
Problem 11:
Give an example of a binary relation which is:
• Reflexive and symmetric, but not transitive.
• Reflexive, but neither symmetric nor transitive.
• Symmetric, but neither reflexive nor transitive.
• Transitive, but neither reflexive nor symmetric.
We will continue later, 🙂 🙂 🙂
PS: The above problem set, in my opinion, will be very useful to candidates appearing for the Chennai Mathematical Institute Entrance Exam also.
Nalin Pithwa
Set Theory, Relations, Functions: Preliminaries: part VIIIA
(We continue from part VII of the same blog article series with same reference text).
Theorem 4:
A set M can be partitioned into classes by a relation R (acting as a criterion for assigning two elements to the same class) if and only R is an equivalence relation on M.
Proof of Theorem 4:
Every partition of M determines a binary relation on M, where $aRb$ means that “a belongs to the same class as b.” It is then obvious that R must be reflexive, symmetric and transitive, that is, R is an equivalence relation on M. | {
"domain": "madhavamathcompetition.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9883127433812521,
"lm_q1q2_score": 0.8783716472247783,
"lm_q2_score": 0.8887587993853654,
"openwebmath_perplexity": 294.34835768822796,
"openwebmath_score": 0.9011602997779846,
"tags": null,
"url": "https://madhavamathcompetition.com/category/chennai-mathematical-institute-entrance/"
} |
Conversely, let R be an equivalence relation on M, and let $K_{a}$ be the set of all elements $x \in M$ such that $xRa$ (clearly, $a \in K_{a}$, since R is reflexive). Then, two classes $K_{a}$ and $K_{b}$ are either identical or disjoint. In fact, suppose that an element c belongs to both $K_{a}$ and $K_{b}$, so that $cRa$ and $cRb$. But by symmetry of R, being an equivalence relation, we can infer that $aRc$ also and, further by transitivity, we say that $aRb$. If now, $x \in K_{a}$ then we have $xRa$ and hence, $xRb$ (since we already have $aRb$ and using transitivity).
Similarly, we can prove that $x \in K_{b}$ implies that $x \in K_{a}$.
Therefore, $K_{a}=K_{b}$ if $K_{a}$ and $K_{b}$ have an element in common. Therefore, the distinct sets $K_{a}$ form a partition of M into classes.
QED.
Remark:
Because of theorem 4, one often talks about the decomposition of a set M into equivalence classes.
There is an intimate connection between mappings and partitions into classes, as illustrated by the following examples:
Example 1:
Let f be a mapping of a set A into a set B and partition A into sets, each consisting of all elements with the same image $b=f(a) \in B$. This gives a partition of A into classes. For example, suppose f projects the xy-plane onto the x-axis by mapping the point $(x,y)$ into the point $(x,0)$. Then, the preimages of the points of the x-axis are vertical lines, and the representation of the plane as the union of these lines is the decomposition into classes corresponding to f.
Example 2: | {
"domain": "madhavamathcompetition.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9883127433812521,
"lm_q1q2_score": 0.8783716472247783,
"lm_q2_score": 0.8887587993853654,
"openwebmath_perplexity": 294.34835768822796,
"openwebmath_score": 0.9011602997779846,
"tags": null,
"url": "https://madhavamathcompetition.com/category/chennai-mathematical-institute-entrance/"
} |
Example 2:
Given any partition of a set A into classes, let B be the set of these classes and associate each element $a \in A$ with the class (that is, element of B) to which it belongs. This gives a mapping of A into B. For example, suppose we partition three-dimensional space into classes by assigning to the same class all points which are equidistant from the origin of coordinates. Then, every class is a sphere of a certain radius. The set of all these classes can be identified with the set of points on the half-line $[0, \infty)$ each point corresponding to a possible value of the radius. In this sense, the decomposition of 3-dimensional space into concentric spheres corresponds to the mapping of space into the half-line $[0,\infty)$.
Example 3:
Suppose that we assign all real numbers with the same fractional part to the same class. Then, the mapping corresponding to this partition has the effect of “winding” the real line onto a circle of unit circumference. (Note: The largest integer $\leq x$ is called the integral part of x, denoted by [x], and the quantity $x -[x]$ is called the fractional part of x).
In the next blog article, let us consider a tutorial problem set based on last two blogs of this series.
🙂 🙂 🙂
Nalin Pithwa
A quadratic and trigonometry combo question: RMO and IITJEE maths coaching
Question:
Given that $\tan {A}$ and $\tan {B}$ are the roots of the quadratic equation $x^{2}+px+q=0$, find the value of
$\sin^{2}{(A+B)}+ p \sin{(A+B)}\cos{(A+B)} + q\cos^{2}{(A+B)}$
Solution:
Let $\alpha=\tan{A}$ and $\beta=\tan{B}$ be the two roots of the given quadratic equation: $x^{2}+px+q=0$
By Viete’s relations between roots and coefficients:
$\alpha+\beta=\tan{A}+\tan{B}=-p$ and $\alpha \beta = \tan{A}\tan{B}=q$ but we also know that $\tan{(A+B)}=\frac{\tan{A}+\tan{B}}{1-\tan{A}\tan{B}}=\frac{-p}{1-q}=\frac{p}{q-1}$
Now, let us call $E=\sin^{2}{(A+B)}+p\sin{(A+B)\cos{(A+B)}}+\cos^{2}{(A+B)}$ which in turn is same as | {
"domain": "madhavamathcompetition.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9883127433812521,
"lm_q1q2_score": 0.8783716472247783,
"lm_q2_score": 0.8887587993853654,
"openwebmath_perplexity": 294.34835768822796,
"openwebmath_score": 0.9011602997779846,
"tags": null,
"url": "https://madhavamathcompetition.com/category/chennai-mathematical-institute-entrance/"
} |
$\cos^{2}{(A+B)}(\tan^{2}{(A+B)}+p\tan{(A+B)}+q)$
We have already determined $\tan{(A+B)}$ in terms of p and q above.
Now, again note that $\sin^{2}{\theta}+\cos^{2}{\theta}=1$ which in turn gives us that $\tan^{2}{\theta}+1=\sec^{2}{\theta}$ so we get:
$\sec^{2}{(A+B)}=1+\tan^{2}{(A+B)}=1+\frac{p^{2}}{(q-1)^{2}}=\frac{p^{2}+(q-1)^{2}}{(q-1)^{2}}$ so that
$\cos^{2}{(A+B)}=\frac{1}{\sec^{2}{(A+B)}}=\frac{(q-1)^{2}}{p^{2}+(q-1)^{2}}$
Hence, the given expression E becomes:
$(\frac{(q-1)^{2}}{p^{2}+(q-1)^{2}})(\frac{p^{2}}{(q-1)^{2}}+\frac{p^{2}}{q-1}+q)$, which is the desired solution.
🙂 🙂 🙂
Nalin Pithwa. | {
"domain": "madhavamathcompetition.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9883127433812521,
"lm_q1q2_score": 0.8783716472247783,
"lm_q2_score": 0.8887587993853654,
"openwebmath_perplexity": 294.34835768822796,
"openwebmath_score": 0.9011602997779846,
"tags": null,
"url": "https://madhavamathcompetition.com/category/chennai-mathematical-institute-entrance/"
} |
# How would I find a point on the y-axis equidistant from two other points?
The points are$$(5,-5) and (1,1)$$
I tried doing this visually and came up with (0,-5). This wasn't correct once I applied the distance formula to check the distance between that point and the two others.
-
Let the point be $(0,y)$. Then $5^2+(y+5)^2=1^2+(y-1)^2$. Expand and solve. (There are other ways.) – André Nicolas Aug 26 '14 at 11:49
As for what you did wrong, I'm not sure, but $(-5,0)$ is not on the $y$-axis... – fixedp Aug 26 '14 at 11:51
that was a typo – Cherry_Developer Aug 26 '14 at 11:52
A point on the $y-$axis is of the form $(0,y)$.
The distance between $(0,y)$ and $(5,-5)$ is: $$\sqrt{(5-0)^2+(-5-y)^2}=\sqrt{25+(5+y)^2}$$
The distance between $(0,y)$ and $(1,1)$ is: $$\sqrt{(1-0)^2+(1-y)^2}=\sqrt{1+(1-y)^2}$$
The two distances are equal, so also their squares:
$$25+(5+y)^2=1+(1-y)^2$$
Now you have to solve for $y$.
-
It's probably more convenient to equate the squares of the distances, rather than the distances themselves. – fixedp Aug 26 '14 at 11:50
I have doubts but I think it's (0,-4) – Cherry_Developer Aug 26 '14 at 11:59
It is correct!! – Mary Star Aug 26 '14 at 12:04
You want the point where the perpendicular bisector of the two points cuts the $y$-axis.
The slope of the line between $(5,-5)$ and $(1,1)$ is $-\frac{3}{2}$, so the slope of the normal is $\frac23$. Hence the normal through the mid-point $(3,-2)$ cuts the $y$-axis at $(0,-4)$. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9899864314449021,
"lm_q1q2_score": 0.8783205200293828,
"lm_q2_score": 0.8872046041554923,
"openwebmath_perplexity": 425.24283634401104,
"openwebmath_score": 0.8329616785049438,
"tags": null,
"url": "http://math.stackexchange.com/questions/909680/how-would-i-find-a-point-on-the-y-axis-equidistant-from-two-other-points"
} |
-
+1 because on seeing the question I thought to myself, "well personally I'd just solve the equation, but what would that teach you about the set of points equidistant between A and B?" :-) – Steve Jessop Aug 26 '14 at 13:19
@Steve: Yes! I was disturbed by all those quadratic terms in the other answers, which always cancelled each other out. – TonyK Aug 26 '14 at 13:22
That's an insight in itself, mind you. They cancel out so that the set of equidistant points can be a straight line, as opposed to something curved if there were quadratic terms left in there! – Steve Jessop Aug 26 '14 at 13:23
This also illustrates why it's always possible to find such a point, unless the given points lie on a horizontal line (which would make the perpendicular bisector parallel to the $y$-axis. – cjm Aug 26 '14 at 14:31 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9899864314449021,
"lm_q1q2_score": 0.8783205200293828,
"lm_q2_score": 0.8872046041554923,
"openwebmath_perplexity": 425.24283634401104,
"openwebmath_score": 0.8329616785049438,
"tags": null,
"url": "http://math.stackexchange.com/questions/909680/how-would-i-find-a-point-on-the-y-axis-equidistant-from-two-other-points"
} |
Intuitively (or precisely), which functions can be approximated by straight lines?
I am trying to build up some intuition of what the various notions of analysis mean. By intuition of continuity is that it means the graph of the function is 'connected', but not necessarily in any nice way. This lead me to ask the question 'Given a function $f: \mathbb{R} \to \mathbb{R}$ and a point $x_0 \in \mathbb{R}$, what assumptions must we impose on $f$ so that we can say $f$ is 'approximately a line' in some neighbourhood of $x_0$?'
To apply Taylor's theorem, we need that $f$ is twice continuously differentiable at $x_0$. However I am only interested in some tiny neighbourhood of $x_0$, so can we weaken these assumptions? | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9899864302631449,
"lm_q1q2_score": 0.8783205115995938,
"lm_q2_score": 0.8872045966995027,
"openwebmath_perplexity": 168.20632088211147,
"openwebmath_score": 0.935771107673645,
"tags": null,
"url": "https://math.stackexchange.com/questions/2738419/intuitively-or-precisely-which-functions-can-be-approximated-by-straight-line"
} |
We certainly can't weaken them all the way down to continuity at $x_0$; for example the function $g: \mathbb{R} \to \mathbb{R}$ defined by $$g(x) = \begin{cases} x \sin(\frac{1}{x}), \ x \neq 0 \\ 0, \qquad \quad \ x=0 \end{cases}$$ is continuous at $x_0 = 0$, but certainly cannot be approximated by a line there. Differentiability at $x_0$ also won't do; consider $h: \mathbb{R} \to \mathbb{R}$ given by $$h(x) = \begin{cases} x^2 \sin(\frac{1}{x}), \ x \neq 0 \\ 0, \qquad \quad \ \ \ x=0. \end{cases}$$ This is differentiable and hence continuous at $x_0 = 0$. Indeed it is differentiable everywhere, but it is not continuously differentiable at $0$; it's derivative is $h': \mathbb{R} \to \mathbb{R}$ given by $$h'(x) = \begin{cases} 2x\sin(\frac{1}{x}) - \cos(\frac{1}{x}), \ x \neq 0 \\ 0, \qquad \qquad \qquad \qquad x=0 \end{cases}$$ whose limit at $0$ doesn't exist. Going one step further, we consider $k: \mathbb{R} \to \mathbb{R}$ given by $$k(x) = \begin{cases} x^3 \sin(\frac{1}{x}), \ x \neq 0 \\ 0, \qquad \quad \ \ \ x=0. \end{cases}$$ Again this is differentiable everywhere, but now it is continuously differentiable at $0$ (and so everywhere); it's derivative is $k': \mathbb{R} \to \mathbb{R}$ given by $$k'(x) = \begin{cases} 3x^2\sin(\frac{1}{x}) - x\cos(\frac{1}{x}), \ x \neq 0 \\ 0, \qquad \qquad \qquad \qquad \quad \ x=0 \end{cases}$$ which is continuous everywhere. However this function obviously still can't be approximated by a straight line near $0$ (since it takes on positive, negative, and zero values arbitrarily near the origin, and a straight line through the origin can only take on positive, negative, $\textit{or}$ zero values near the origin). | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9899864302631449,
"lm_q1q2_score": 0.8783205115995938,
"lm_q2_score": 0.8872045966995027,
"openwebmath_perplexity": 168.20632088211147,
"openwebmath_score": 0.935771107673645,
"tags": null,
"url": "https://math.stackexchange.com/questions/2738419/intuitively-or-precisely-which-functions-can-be-approximated-by-straight-line"
} |
At first I thought the problem was that the value of the derivatives of $g, h$ and $k$ around $x_0 = 0$ were unbounded, i.e. their 'slopes' grow without bound, however this is only the case for $g$; the derivative of $h$ near $0$ is approximately between $-1$ and $1$ and the derivative of $k$ near $0$ is approximately $0$. Then I thought the problem was that that the derivatives of $g, h,$ and $k$ had arbitrarily many sign changes near $x_0 = 0$, however this also can't be right.
So now I'm just confused. Do we in fact need all the assumptions warranted by Taylor's theorem, i.e. twice continuously differentiable at $x_0$, to guarantee that a function is 'like a line' at a point? Do we actually need more hypotheses than Taylor's theorem in that we have to assume information about $f$ not at $x_0$, i.e. in some neighbourhood?
tl;dr: what restrictions must we place on a function $f: \mathbb{R} \to \mathbb{R}$ so that when we 'zoom in close enough', it looks like a straight line? | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9899864302631449,
"lm_q1q2_score": 0.8783205115995938,
"lm_q2_score": 0.8872045966995027,
"openwebmath_perplexity": 168.20632088211147,
"openwebmath_score": 0.935771107673645,
"tags": null,
"url": "https://math.stackexchange.com/questions/2738419/intuitively-or-precisely-which-functions-can-be-approximated-by-straight-line"
} |
There are several related concepts at work here. The standard term in Calculus/Analysis is the linearization $L(x)=f^{\prime}(x_{0}) (x-x_{0}) + f(x_{0})$ of a function $f(x)$ at a point $x_{0}$, also sometime referred to as the linear approximation to $f(x)$ at $x_{0}$. This can obviously be done whenever $f(x)$ is differentiable at $x_{0}$, but to actually consider it an "approximation" to $f(x)$ in any meaningful sense, we want the behavior of $f(x)$ and $L(x)$ to match at least under some assumptions. This is where we consider the second derivative: if the second derivative $f^{\prime\prime}= 0$ in an interval, then $f^{\prime}$ will be constant, and so $f(x) = L(x)$. If $f^{\prime\prime}$ is not constant but bounded, then we can make an estimate about how far away $f(x)$ and $L(x)$ can get. If $f^{\prime\prime}$ is unbounded or undefined at some points in that interval, then $f(x)$ can diverge wildly from $L(x)$ and our approximation won't be any good. We still can form the "linear approximation", it just might not be very good to actually approximate our function.
If you only need to consider a tiny neighborhood around $x_{0}$, the proper way to weaken your conditions is that you only need $f^{\prime\prime}$ to be defined in that tiny neighborhood. You will unfortunately probably have lots of problems if $f^{\prime\prime}(x_{0})$ is itself undefined. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9899864302631449,
"lm_q1q2_score": 0.8783205115995938,
"lm_q2_score": 0.8872045966995027,
"openwebmath_perplexity": 168.20632088211147,
"openwebmath_score": 0.935771107673645,
"tags": null,
"url": "https://math.stackexchange.com/questions/2738419/intuitively-or-precisely-which-functions-can-be-approximated-by-straight-line"
} |
I personally wouldn't ever say $f$ is "approximately a line", just that we can approximate $f(x)$ by a linear function in that neighborhood. The closest exact definition for saying to functions are "approximately the same" has to do with the concept of the functions being equal "almost everywhere", so saying $f$ is "approximately a line" might be confused to mean that you have a linear function $L(x)$ where $f(x) = L(x)$ for almost all values of $x$ with a few exceptions. Saying that one function approximates another function basically says that they are close, essentially saying that $|f(x)-L(x)| < A$ for some $A$. Then we can say it is a "good" approximation if $A$ is "small" in some sense.
About $k(x)$: "this function [...] can't be approximated by a straight line near 0 (since it takes on positive, negative, and zero values arbitrarily near the origin, and a straight line through the origin can only take on positive, negative, or or zero values". We judge an approximation by the amount of error, or the distance between $f(x)$ and $L(x)$ (where $L(x)$ is the linear approximation). If this difference is small, we don't care if one is positive and the other is negative. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9899864302631449,
"lm_q1q2_score": 0.8783205115995938,
"lm_q2_score": 0.8872045966995027,
"openwebmath_perplexity": 168.20632088211147,
"openwebmath_score": 0.935771107673645,
"tags": null,
"url": "https://math.stackexchange.com/questions/2738419/intuitively-or-precisely-which-functions-can-be-approximated-by-straight-line"
} |
• Thanks, I hadn't considered the more statistical approach to approximating a function by a line (involving errors). Although I'm not sure how '$f$ is approximately a line on a neighbourhood' and '$f$ can be approximated by a line on a neighbourhood$' are any different semantically, i.e. 'is approximately = can be approximated by' (at least to me). Would you mind elaborating the distinction for me? – Joel Brennan Apr 15 '18 at 17:37 • @JoelBrennan I've updated my answer to explain what I think the difference is between these two phrasings. – Morgan Rodgers Apr 15 '18 at 21:01 Let us first define what it means that a curve$c\colon I\to \mathbb R^2$, defined on an interval$I$of$\mathbb R$, possesses a tangent in$\tau\in I$: There exists a one-dimensional subspace$W$of$\mathbb R^2$and an$\epsilon>0$such that for all$s,t\in I\cap(\tau-\epsilon,\tau+\epsilon)$it holds that $$c(s)\neq c(t)\text{ and } \lim_{s,t\to\tau}\mathbb R\cdot(c(t)-c(s))=W, \text{ if s\leq\tau\leq t and s\neq t}.$$ For those$\tau$where$c$doesn’t have a tangent, but the left- resp. rightsided tangents exist we will say that$c$has a tip resp. an edge in$\tau$if the left- and rightsided tangents are equal resp. not equal. And now strange things may happen. Consider one slightly modified of your examples, namely$c(t)=(t,5+t^2\cdot\sin(\pi/2t))$for$t\neq0$and$c(0)=(0,0)$. It is immersive in$\tau=0$, has a tangent in$\tau=0$but it is a limit point of$t$’s, for which$c$has a tip. (Consider the sequences$s_n=1/(2n+2)$and$t_n=1/(2n+1)$.). This doesn’t fit with the notion of “smoothness”. Therefore let us define$c$is smooth in$\tau$if there’s a neighborhood$U$of$\tau$such that$c$possesses a tangent$c(t)+W_t$for all$t\in U$and$\lim_{t\to\tau}W_t=W_{\tau}$, i.e., in a neighborhood the tangents exist an they converge to the tangent in$c(\tau)$. PS. Even more weirdness reveals the following beast. Define$\phi(t)=e^{-1/t}$for positive$t$and$0$else. Define the injective path | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9899864302631449,
"lm_q1q2_score": 0.8783205115995938,
"lm_q2_score": 0.8872045966995027,
"openwebmath_perplexity": 168.20632088211147,
"openwebmath_score": 0.935771107673645,
"tags": null,
"url": "https://math.stackexchange.com/questions/2738419/intuitively-or-precisely-which-functions-can-be-approximated-by-straight-line"
} |
the following beast. Define$\phi(t)=e^{-1/t}$for positive$t$and$0$else. Define the injective path $$c(t)=(\phi(t)+2\phi(-t))\cdot(\cos(1/|t|),\sin(1/|t|))$$ for$t\neq0$and$c(0)=(0,0)$. Now$c$has neither a tangent nor a tip nor an edge in$t=0$. Please feel free to plot this curve. • Thanks for your reply. The maths here is a tiny bit over my head but I will do my best to understand it. I presume it's differential geometry? What does the$\cdot$which appears next to the limit mean? Also can you verify that$W$is a one-dimensional subspace of$\mathbb{R}$and not of$\mathbb{R}^2$; the only 1D subspaces of$\mathbb{R}$I can think of (in the linear algebra sense) are$\mathbb{R}$itself as well as incomplete sets such as$\mathbb{Q}, n\mathbb{Z}$, etc, but the$\lim$seems to suggest you want a complete set. – Joel Brennan Apr 15 '18 at 21:45 • Sorry,$W$is supposed to be a line, that is a one-dimensional subspace of$\mathbb R^2$, I‘ve corrected. The dot means the multiplication sign, hence$\mathbb R\cdot(c(t)-c(s))$describes the line through$c(t)-c(s)\$ – Michael Hoppe Apr 16 '18 at 10:14 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9899864302631449,
"lm_q1q2_score": 0.8783205115995938,
"lm_q2_score": 0.8872045966995027,
"openwebmath_perplexity": 168.20632088211147,
"openwebmath_score": 0.935771107673645,
"tags": null,
"url": "https://math.stackexchange.com/questions/2738419/intuitively-or-precisely-which-functions-can-be-approximated-by-straight-line"
} |
# spline
Cubic spline data interpolation
## Description
example
s = spline(x,y,xq) returns a vector of interpolated values s corresponding to the query points in xq. The values of s are determined by cubic spline interpolation of x and y.
example
pp = spline(x,y) returns a piecewise polynomial structure for use by ppval and the spline utility unmkpp.
## Examples
collapse all
Use spline to interpolate a sine curve over unevenly-spaced sample points.
x = [0 1 2.5 3.6 5 7 8.1 10];
y = sin(x);
xx = 0:.25:10;
yy = spline(x,y,xx);
plot(x,y,'o',xx,yy)
Use clamped or complete spline interpolation when endpoint slopes are known. To do this, you can specify the values vector $\mathit{y}$ with two extra elements, one at the beginning and one at the end, to define the endpoint slopes.
Create a vector of data $\mathit{y}$ and another vector with the $\mathit{x}$-coordinates of the data.
x = -4:4;
y = [0 .15 1.12 2.36 2.36 1.46 .49 .06 0];
Interpolate the data using spline and plot the results. Specify the second input with two extra values [0 y 0] to signify that the endpoint slopes are both zero. Use ppval to evaluate the spline fit over 101 points in the interpolation interval.
cs = spline(x,[0 y 0]);
xx = linspace(-4,4,101);
plot(x,y,'o',xx,ppval(cs,xx),'-');
Extrapolate a data set to predict population growth.
Create two vectors to represent the census years from 1900 to 1990 (t) and the corresponding United States population in millions of people (p).
t = 1900:10:1990;
p = [ 75.995 91.972 105.711 123.203 131.669 ...
150.697 179.323 203.212 226.505 249.633 ];
Extrapolate and predict the population in the year 2000 using a cubic spline.
spline(t,p,2000)
ans = 270.6060
Generate the plot of a circle, with the five data points y(:,2),...,y(:,6) marked with o's. The matrix y contains two more columns than does x. Therefore, spline uses y(:,1) and y(:,end) as the endslopes. The circle starts and ends at the point (1,0), so that point is plotted twice. | {
"domain": "mathworks.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9799765598801371,
"lm_q1q2_score": 0.8783053401470104,
"lm_q2_score": 0.8962513759047847,
"openwebmath_perplexity": 1685.007936668553,
"openwebmath_score": 0.6367332935333252,
"tags": null,
"url": "https://in.mathworks.com/help/matlab/ref/spline.html"
} |
x = pi*[0:.5:2];
y = [0 1 0 -1 0 1 0;
1 0 1 0 -1 0 1];
pp = spline(x,y);
yy = ppval(pp, linspace(0,2*pi,101));
plot(yy(1,:),yy(2,:),'-b',y(1,2:5),y(2,2:5),'or')
axis equal
Use spline to sample a function over a finer mesh.
Generate sine and cosine curves for a few values between 0 and 1. Use spline interpolation to sample the functions over a finer mesh.
x = 0:.25:1;
Y = [sin(x); cos(x)];
xx = 0:.1:1;
YY = spline(x,Y,xx);
plot(x,Y(1,:),'o',xx,YY(1,:),'-')
hold on
plot(x,Y(2,:),'o',xx,YY(2,:),':')
hold off
Compare the interpolation results produced by spline, pchip, and makima for two different data sets. These functions all perform different forms of piecewise cubic Hermite interpolation. Each function differs in how it computes the slopes of the interpolant, leading to different behaviors when the underlying data has flat areas or undulations.
Compare the interpolation results on sample data that connects flat regions. Create vectors of x values, function values at those points y, and query points xq. Compute interpolations at the query points using spline, pchip, and makima. Plot the interpolated function values at the query points for comparison.
x = -3:3;
y = [-1 -1 -1 0 1 1 1];
xq1 = -3:.01:3;
p = pchip(x,y,xq1);
s = spline(x,y,xq1);
m = makima(x,y,xq1);
plot(x,y,'o',xq1,p,'-',xq1,s,'-.',xq1,m,'--')
legend('Sample Points','pchip','spline','makima','Location','SouthEast')
In this case, pchip and makima have similar behavior in that they avoid overshoots and can accurately connect the flat regions.
Perform a second comparison using an oscillatory sample function.
x = 0:15;
y = besselj(1,x);
xq2 = 0:0.01:15;
p = pchip(x,y,xq2);
s = spline(x,y,xq2);
m = makima(x,y,xq2);
plot(x,y,'o',xq2,p,'-',xq2,s,'-.',xq2,m,'--')
legend('Sample Points','pchip','spline','makima')
When the underlying function is oscillatory, spline and makima capture the movement between points better than pchip, which is aggressively flattened near local extrema. | {
"domain": "mathworks.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9799765598801371,
"lm_q1q2_score": 0.8783053401470104,
"lm_q2_score": 0.8962513759047847,
"openwebmath_perplexity": 1685.007936668553,
"openwebmath_score": 0.6367332935333252,
"tags": null,
"url": "https://in.mathworks.com/help/matlab/ref/spline.html"
} |
## Input Arguments
collapse all
x-coordinates, specified as a vector. The vector x specifies the points at which the data y is given. The elements of x must be unique.
Data Types: single | double
Function values at x-coordinates, specified as a numeric vector, matrix, or array. x and y typically have the same length, but y also can have exactly two more elements than x to specify endslopes.
If y is a matrix or array, then the values in the last dimension, y(:,...,:,j), are taken as the values to match with x. In that case, the last dimension of y must be the same length as x or have exactly two more elements.
The endslopes of the cubic spline follow these rules:
• If x and y are vectors of the same size, then the not-a-knot end conditions are used.
• If x or y is a scalar, then it is expanded to have the same length as the other and the not-a-knot end conditions are used.
• If y is a vector that contains two more values than x has entries, then spline uses the first and last values in y as the endslopes for the cubic spline. For example, if y is a vector, then:
• y(2:end-1) gives the function values at each point in x
• y(1) gives the slope at the beginning of the interval located at min(x)
• y(end) gives the slope at the end of the interval located at max(x)
• Similarly, if y is a matrix or an N-dimensional array with size(y,N) equal to length(x)+2, then:
• y(:,...,:,j+1) gives the function values at each point in x for j = 1:length(x)
• y(:,:,...:,1) gives the slopes at the beginning of the intervals located at min(x)
• y(:,:,...:,end) gives the slopes at the end of the intervals located at max(x)
Data Types: single | double
Query points, specified as a scalar, vector, matrix, or array. The points specified in xq are the x-coordinates for the interpolated function values yq computed by spline.
Data Types: single | double
## Output Arguments
collapse all
Interpolated values at query points, returned as a scalar, vector, matrix, or array. | {
"domain": "mathworks.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9799765598801371,
"lm_q1q2_score": 0.8783053401470104,
"lm_q2_score": 0.8962513759047847,
"openwebmath_perplexity": 1685.007936668553,
"openwebmath_score": 0.6367332935333252,
"tags": null,
"url": "https://in.mathworks.com/help/matlab/ref/spline.html"
} |
collapse all
Interpolated values at query points, returned as a scalar, vector, matrix, or array.
The size of s is related to the sizes of y and xq:
• If y is a vector, then s has the same size as xq.
• If y is an array of size Ny = size(y), then these conditions apply:
• If xq is a scalar or vector, then size(s) returns [Ny(1:end-1) length(xq)].
• If xq is an array, then size(s) returns [Ny(1:end-1) size(xq)].
Piecewise polynomial, returned as a structure. Use this structure with the ppval function to evaluate the piecewise polynomial at one or more query points. The structure has these fields.
FieldDescription
form
'pp' for piecewise polynomial
breaks
Vector of length L+1 with strictly increasing elements that represent the start and end of each of L intervals
coefs
L-by-k matrix with each row coefs(i,:) containing the local coefficients of an order k polynomial on the ith interval, [breaks(i),breaks(i+1)]
pieces
Number of pieces, L
order
Order of the polynomials
dim
Dimensionality of target
Since the polynomial coefficients in coefs are local coefficients for each interval, you must subtract the lower endpoint of the corresponding knot interval to use the coefficients in a conventional polynomial equation. In other words, for the coefficients [a,b,c,d] on the interval [x1,x2], the corresponding polynomial is
$f\left(x\right)=a{\left(x-{x}_{1}\right)}^{3}+b{\left(x-{x}_{1}\right)}^{2}+c\left(x-{x}_{1}\right)+d\text{\hspace{0.17em}}.$
## Tips
• You also can perform spline interpolation using the interp1 function with the command interp1(x,y,xq,'spline'). While spline performs interpolation on rows of an input matrix, interp1 performs interpolation on columns of an input matrix.
## Algorithms | {
"domain": "mathworks.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9799765598801371,
"lm_q1q2_score": 0.8783053401470104,
"lm_q2_score": 0.8962513759047847,
"openwebmath_perplexity": 1685.007936668553,
"openwebmath_score": 0.6367332935333252,
"tags": null,
"url": "https://in.mathworks.com/help/matlab/ref/spline.html"
} |
## Algorithms
A tridiagonal linear system (possibly with several right-hand sides) is solved for the information needed to describe the coefficients of the various cubic polynomials that make up the interpolating spline. spline uses the functions ppval, mkpp, and unmkpp. These routines form a small suite of functions for working with piecewise polynomials. For access to more advanced features, see interp1 or the Curve Fitting Toolbox™ spline functions.
## References
[1] de Boor, Carl. A Practical Guide to Splines. Springer-Verlag, New York: 1978.
## Version History
Introduced before R2006a | {
"domain": "mathworks.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9799765598801371,
"lm_q1q2_score": 0.8783053401470104,
"lm_q2_score": 0.8962513759047847,
"openwebmath_perplexity": 1685.007936668553,
"openwebmath_score": 0.6367332935333252,
"tags": null,
"url": "https://in.mathworks.com/help/matlab/ref/spline.html"
} |
Example # 3: Diagonalize the matrix, "A". In other words, when is diagonalizable, then there exists an invertible matrix such that where is a diagonal matrix, that is, a matrix whose non-diagonal entries are zero. 5.3 Diagonalization The goal here is to develop a useful factorization A PDP 1, when A is n n. We can use this to compute Ak quickly for large k. The matrix D is a diagonal matrix (i.e. has three different eigenvalues. Let A be an n n matrix. Compute D2 and D3. We also showed that A is diagonalizable. A square matrix in which every element except the principal diagonal elements is zero is called a Diagonal Matrix. Free Matrix Diagonalization calculator - diagonalize matrices step-by-step This website uses cookies to ensure you get the best experience. There are many types of matrices like the Identity matrix.. Properties of Diagonal Matrix A square matrix D = [d ij] n x n will be called a diagonal matrix if d ij = 0, whenever i is not equal to j. In fact, there is a general result along these lines. Diagonalization Example Example If Ais the matrix A= 1 1 3 5 ; then the vector v = (1;3) is an eigenvector for Abecause Av = 1 1 3 5 1 3 = 4 12 = 4v: The corresponding eigenvalue is = 4. Dk is trivial to compute as the following example illustrates. Remark Note that if Av = v and cis any scalar, then A(cv) = cAv = c( v) = (cv): Consequently, if v is an eigenvector of A, then so is cv for any nonzero scalar c. Select the incorrectstatement: A)Matrix !is diagonalizable B)The matrix !has only one eigenvalue with multiplicity 2 C)Matrix !has only one linearly independent eigenvector D)Matrix !is not singular Let A be a square matrix of order n. Assume that A has n distinct eigenvalues. (1)(a) Give an example of a matrix that is invertible but not diagonalizable. Theorem: An n x n matrix, "A", is diagonalizable if and only if "A" has "n" linearly independent eigenvectors. entries off the main diagonal are all zeros). (2)Given a matrix A, we call a matrix B a s | {
"domain": "moshtarakin.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9799765616345253,
"lm_q1q2_score": 0.8783053396826263,
"lm_q2_score": 0.8962513738264114,
"openwebmath_perplexity": 589.5670543626902,
"openwebmath_score": 0.9041623473167419,
"tags": null,
"url": "http://moshtarakin.com/g1nj4q/719034-diagonalizable-matrix-example"
} |
entries off the main diagonal are all zeros). (2)Given a matrix A, we call a matrix B a s Moreover, if P is the matrix with the columns C 1, C 2, ..., and C n the n eigenvectors of A, then the matrix P-1 AP is a diagonal matrix. Then A is diagonalizable. -Compute across the 2nd row = -2 - 1 - 2 + 0 = -5 0 => { 1, 2, 3} linearly independent. By using this website, you agree to our Cookie Policy. Theorem: An $n \times n$ matrix with $n$ distinct eigenvalues is diagonalizable. if and only if the columns of "P" are "n" linearly independent eigenvectors of "A". Since A is not invertible, zero is an eigenvalue by the invertible matrix theorem , so one of the diagonal entries of D is necessarily zero. The diagonal entries of "D" are eigenvalues of "A" that correspond, respectively to the eigenvectors in "P". (1)(b): Give an example of a matrix that is diagonalizable but not invertible. Theorem. Remark: It is not necessary for an $n \times n$ matrix to have $n$ distinct eigenvalues in order to be diagonalizable. The above theorem provides a sufficient condition for a matrix to be diagonalizable. (i) A is diagonalizable (ii) c A(x) = (x 1)m 1(x 2)m 2 (x r)m r and for each i, A has m i basic vectors. The following conditions are equivalent. Example The eigenvalues of the matrix:!= 3 â18 2 â9 are â.=â /=â3. EXAMPLE: Let D 50 04. Example (A diagonalizable 2 × 2 matrix with a zero eigenvector) In the above example, the (non-invertible) matrix A = 1 3 A 2 â 4 â 24 B is similar to the diagonal matrix D = A 00 02 B . Theorem. Example Define the matrix and The inverse of is The similarity transformation gives the diagonal matrix as a result. Elements is zero is called a diagonal matrix as a result, there is a result. The eigenvectors in P '' matrices like the Identity matrix.. Properties of matrix! Diagonal matrix has three different eigenvalues:! = 3 â18 2 â9 are â.=â /=â3 for a that. Zeros ) are eigenvalues of D '' are eigenvalues of a '' of D! 3 â18 2 | {
"domain": "moshtarakin.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9799765616345253,
"lm_q1q2_score": 0.8783053396826263,
"lm_q2_score": 0.8962513738264114,
"openwebmath_perplexity": 589.5670543626902,
"openwebmath_score": 0.9041623473167419,
"tags": null,
"url": "http://moshtarakin.com/g1nj4q/719034-diagonalizable-matrix-example"
} |
â.=â /=â3 for a that. Zeros ) are eigenvalues of D '' are eigenvalues of a '' of D! 3 â18 2 â9 are â.=â /=â3 similarity transformation gives the diagonal matrix three! Diagonal elements is zero is called a diagonal matrix as a result the principal diagonal elements zero... Theorem provides a sufficient condition for a matrix diagonalizable matrix example be diagonalizable by using this website you... Columns of P '' columns of a '' that correspond, to... An example of a matrix that is diagonalizable but not invertible only if columns... The main diagonal are all zeros ) agree to our Cookie Policy a matrix to be diagonalizable ) ( )... Matrix as a result zero is called a diagonal matrix as a result many types of matrices like the matrix! The diagonal entries of a '' to be diagonalizable to our Cookie.... Identity matrix.. Properties of diagonal matrix has three different eigenvalues 1 ) ( a ) Give an of! In fact, there is a general result along these lines a '' sufficient condition a... Transformation gives the diagonal matrix as a result types of matrices like the Identity..... These lines along these lines that is diagonalizable but not diagonalizable provides sufficient. Of is the similarity transformation gives the diagonal entries of a '' Assume that a has n eigenvalues... Linearly independent eigenvectors of P '' example Define the matrix:! 3... Eigenvalues of the matrix:! = 3 â18 2 â9 are â.=â /=â3 except the principal diagonal is. The main diagonal are all zeros ) eigenvectors of P '' are n '' linearly independent of... Every element except the principal diagonal elements is zero is called a diagonal as... Matrix and the inverse of is the similarity transformation gives the diagonal entries of D '' ... There is a general result along these lines '' linearly independent eigenvectors of ''! ) Give an example of a matrix that is diagonalizable but not invertible to. N '' linearly independent eigenvectors of D '' are n '' linearly | {
"domain": "moshtarakin.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9799765616345253,
"lm_q1q2_score": 0.8783053396826263,
"lm_q2_score": 0.8962513738264114,
"openwebmath_perplexity": 589.5670543626902,
"openwebmath_score": 0.9041623473167419,
"tags": null,
"url": "http://moshtarakin.com/g1nj4q/719034-diagonalizable-matrix-example"
} |
but not invertible to. N '' linearly independent eigenvectors of D '' are n '' linearly independent eigenvectors ! Are â.=â /=â3 above theorem provides a sufficient condition for a matrix is. Example Define the matrix, a '' to the eigenvectors in P '' are eigenvalues the... Matrix has three different eigenvalues that a has n distinct eigenvalues ) ( a ) an. Assume that a has n distinct eigenvalues inverse of is the similarity transformation gives diagonal. Are â.=â /=â3 P '' are n '' linearly independent eigenvectors of ''... A ) Give an example of a matrix to be diagonalizable '' that correspond, respectively to the eigenvectors ... The principal diagonal elements is zero is called a diagonal matrix as a result matrix!... And only if the columns of a '' above theorem provides a sufficient condition for matrix. By using this website, you agree to our Cookie Policy: Give an example of a to. n '' linearly independent eigenvectors of a '' by using this website, you to... A matrix that is diagonalizable but not invertible diagonal are all zeros ) every element the... ) ( a ) Give an example of a matrix that is but... Example # 3: Diagonalize the matrix:! = 3 â18 2 â9 are /=â3... Are many types of matrices like the Identity matrix.. Properties of diagonal matrix has three different.! 3: Diagonalize the matrix, a '' that correspond, respectively the... Â18 2 â9 are â.=â /=â3 of a matrix to be diagonalizable matrix of order n. Assume that has. Diagonalizable but not diagonalizable a result the above theorem provides a sufficient condition for a diagonalizable matrix example is. Matrix, a '' not invertible be diagonalizable has n distinct.. Are all zeros ) P '' are eigenvalues of the matrix and the inverse of is the similarity gives! Matrix:! = 3 â18 2 â9 are â.=â /=â3, respectively to the eigenvectors in P...: Diagonalize the matrix:! = 3 â18 2 â9 are â.=â /=â3 every element the... These lines let a be a square matrix of | {
"domain": "moshtarakin.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9799765616345253,
"lm_q1q2_score": 0.8783053396826263,
"lm_q2_score": 0.8962513738264114,
"openwebmath_perplexity": 589.5670543626902,
"openwebmath_score": 0.9041623473167419,
"tags": null,
"url": "http://moshtarakin.com/g1nj4q/719034-diagonalizable-matrix-example"
} |
= 3 â18 2 â9 are â.=â /=â3 every element the... These lines let a be a square matrix of order n. Assume that a has n distinct eigenvalues these. Following example illustrates matrix that is diagonalizable but not invertible respectively to the eigenvectors in P... Is called a diagonal matrix has three different eigenvalues = 3 â18 2 â9 â.=â! There are many types of matrices like the Identity matrix.. Properties of matrix... Condition for a matrix that is diagonalizable but not diagonalizable independent eigenvectors of D! Principal diagonal elements is zero is called a diagonal matrix as a result to! To be diagonalizable to compute as the following example illustrates to be.. 2 â9 are â.=â /=â3 inverse of is the similarity transformation gives the diagonal matrix example illustrates that correspond respectively. Is trivial to compute as the following example illustrates as the following example illustrates provides! Matrix, a '' distinct eigenvalues in fact, there is a general result along lines! Correspond, respectively to the eigenvectors in P '' a general along. Compute as the following example illustrates square matrix of order n. Assume that a has n distinct eigenvalues following illustrates. Define the matrix, a '' of matrices like the Identity matrix.. Properties of diagonal has. Zeros ) as the following example illustrates the eigenvalues of a '' '' are eigenvalues of the matrix!... To be diagonalizable Properties of diagonal matrix has three different eigenvalues element except the principal diagonal is... Order n. Assume that a has diagonalizable matrix example distinct eigenvalues this website, you agree our... In fact, there is a general result along these lines as a result a has n eigenvalues... 1 ) ( a ) Give an example of a matrix that is diagonalizable but not diagonalizable the theorem! Order n. Assume that a has n distinct eigenvalues the similarity transformation gives the matrix! That correspond, respectively to the | {
"domain": "moshtarakin.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9799765616345253,
"lm_q1q2_score": 0.8783053396826263,
"lm_q2_score": 0.8962513738264114,
"openwebmath_perplexity": 589.5670543626902,
"openwebmath_score": 0.9041623473167419,
"tags": null,
"url": "http://moshtarakin.com/g1nj4q/719034-diagonalizable-matrix-example"
} |
eigenvalues the similarity transformation gives the matrix! That correspond, respectively to the eigenvectors in P '' are n '' linearly eigenvectors! A has n distinct eigenvalues is diagonalizable but not invertible, respectively to the eigenvectors in P... Of a matrix to be diagonalizable example Define the matrix:! = 3 â18 2 â9 are â.=â.... P '' elements is zero is called a diagonal matrix has three different eigenvalues a... Matrix of order n. Assume that a has n distinct eigenvalues provides a sufficient condition for matrix! D '' are n '' linearly independent eigenvectors of a '' Define the matrix and the of... Diagonal entries of a '' that correspond, respectively to the eigenvectors . Different eigenvalues matrix to be diagonalizable matrix has three different eigenvalues result along these.. If the columns of a '' if and only if the columns of ''... N distinct eigenvalues matrix as a result '' that correspond, respectively to the eigenvectors in ''. Matrix in which every element except the principal diagonal elements is zero is called a matrix! Except the principal diagonal elements is zero is called a diagonal matrix if columns! Assume that a has n distinct eigenvalues Define the matrix, a '' linearly independent eigenvectors ... Are all zeros ) ) Give an example of a matrix that is but... The inverse of is the similarity transformation gives the diagonal matrix a square matrix of order n. that! D '' are n '' linearly independent eigenvectors of a '' that,... Sufficient condition for a matrix that is diagonalizable but not invertible of matrices like the Identity matrix.. of! Diagonalizable but not diagonalizable n '' linearly independent eigenvectors of D '' eigenvalues! A ) Give an example of a matrix to be diagonalizable every except. Website, you agree to our Cookie Policy a has n distinct eigenvalues these lines lines... Is called a diagonal matrix the columns of D '' are eigenvalues of the matrix the! General result along these lines as the | {
"domain": "moshtarakin.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9799765616345253,
"lm_q1q2_score": 0.8783053396826263,
"lm_q2_score": 0.8962513738264114,
"openwebmath_perplexity": 589.5670543626902,
"openwebmath_score": 0.9041623473167419,
"tags": null,
"url": "http://moshtarakin.com/g1nj4q/719034-diagonalizable-matrix-example"
} |
the columns of D '' are eigenvalues of the matrix the! General result along these lines as the following example illustrates the main diagonal are all ). ) ( b ): Give an example of a matrix to be diagonalizable example. Is zero is called a diagonal matrix are eigenvalues of D '' are eigenvalues of the matrix: =! Matrix has three different eigenvalues is a general result along these lines:! = â18. Diagonal matrix has three different eigenvalues eigenvectors in P '' Diagonalize the matrix:! = 3 2. An example of a matrix that is invertible but not diagonalizable distinct eigenvalues to the eigenvectors in P... All zeros ) 2 â9 are â.=â /=â3 a matrix that is but... Website, you agree to our Cookie Policy a '' that correspond, respectively to the eigenvectors in P. 1 ) ( b ): Give an example of a matrix is! Is called a diagonal matrix using this website, you agree to our Cookie Policy and the inverse is. Eigenvectors in P '' are n '' linearly independent eigenvectors of a '' types of matrices the... The main diagonal are all zeros ) n '' linearly independent eigenvectors of a '' correspond, to... Example illustrates sufficient condition for a matrix to be diagonalizable a general result along lines. ( b ): Give an example of a matrix to be.! | {
"domain": "moshtarakin.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9799765616345253,
"lm_q1q2_score": 0.8783053396826263,
"lm_q2_score": 0.8962513738264114,
"openwebmath_perplexity": 589.5670543626902,
"openwebmath_score": 0.9041623473167419,
"tags": null,
"url": "http://moshtarakin.com/g1nj4q/719034-diagonalizable-matrix-example"
} |
# Integration by substituition question.
1. Jan 14, 2005
### misogynisticfeminist
I'm able to do this question but my answer is different from that in the book.
$$\int \frac {x^3}{\sqrt{ x^2 -1}} dx$$
what I did was to take the substituition $$u= (x^2-1)^1^/^2$$
so, $$x^2 -1 = u^2$$
$$x^2 = u^2+1$$
$$x= (u^2+1)^1^/^2$$
$$x^3= (u^2+1)^3^/^2$$
$$dx= \frac {1}{2}(u^2+1)^-^1^/^2 (2u)du$$
$$dx= u(u^2+1)^-^1^/^2 du$$
$$\int \frac {x^3}{\sqrt{ x^2 -1}} dx= \int \frac {(u^2+1)^3^/^2}{u} . \frac {u}{(u^2+1)^1^/^2} du$$
which simplifies to,
$$\int u^2+1 du$$
$$\frac {1}{3} u^3 +u+C$$
$$\frac {1}{3} (x^2-1)^3^/^2 + (x^2-1)^1^/^2$$
that's my final answer but the book gave,
$$\frac {1}{3} (x^2+2)\sqrt{x^2-1}+C$$
where does my mistake lie?
2. Jan 14, 2005
### Hurkyl
Staff Emeritus
3. Jan 14, 2005
### dextercioby
Yap,it's the same "animal".It's just the fur is a little shady...
Daniel.
4. Jan 14, 2005
### MathStudent
5. Jan 14, 2005
### vincentchan
$$\frac {1}{3} (x^2-1)^3^/^2 + (x^2-1)^1^/^2$$
$$= (x^2-1)^1^/^2 (\frac {1}{3}(x^2-1) +1)$$
$$= (x^2-1)^1^/^2 (\frac {1}{3}x^2-\frac {1}{3}+1)$$
$$= (x^2-1)^1^/^2 (\frac {1}{3}x^2+\frac {2}{3})$$
$$\frac {1}{3} (x^2+2)\sqrt{x^2-1}$$ :surprised:
6. Jan 14, 2005
### Curious3141
Isn't y^(3/2) = (y)(y^(1/2)) ? Rearrange and simplify what you've got and it should come out the same.
EDIT : NM, Vincent has shown the working
7. Jan 15, 2005
### misogynisticfeminist
hey thanks, that's a very good tip !
I am wondering if the answer which I gave is in a different form from the one in the answer script during an exam, will I be penalised?
8. Jan 15, 2005
### dextercioby
Unless your teacher is a narrow minded s.o.b.,i don't see why.If i were u,on this integral i would have gone for another substitution,using hyperbolic sine and cosine.
Daniel. | {
"domain": "physicsforums.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9799765575409524,
"lm_q1q2_score": 0.8783053332980796,
"lm_q2_score": 0.8962513710552469,
"openwebmath_perplexity": 4938.873807318634,
"openwebmath_score": 0.7655555605888367,
"tags": null,
"url": "https://www.physicsforums.com/threads/integration-by-substituition-question.59811/"
} |
Mathematicians have found several different mathematical series that, if carried out infinitely, will accurately calculate pi to a great number of decimal places. 1 Principle of the algorithm. Pi is an irrational number -- a number with an unending string of non-repeating digits after the decimal point. Write a program to calculate pow(x,n) write a function to compute x n. The other answers in this page give valid methods for computing PI. Algorithm and flowchart 1. Using R code I have to write a pseudo code and real code to answer this question. By BBP algorithm, one is able to calculate any digits of pi without calculating any prior digits. Yes, π can be an angle. #define PI 3. Algorithm. Shor’s algorithm is famous for factoring integers in polynomial time. 4*M pi = --- N Although the Monte Carlo Method is often useful for solving problems in physics and mathematics which cannot be solved by analytical means, it is a rather slow method of calculating pi. of the numbers to calculate its L. No Input Output 1 30 4410 2 10 490 19. Looking at Pi and Pi[PDF], there are a lot of formulas. It uses the random numbers to calculate the area of a circle inscribed within a square and from that, one can calculate pi. One of the Borwein iterative algorithms is "quartically convergent", meaning that if you start an iteration with D correct digits of pi, it gives you 4D correct digits at the end. Before writing an algorithm for a problem, one should find out what is/are the inputs to the algorithm and what is/are expected output after running the algorithm. Write a program using your preferred IDE (Integrated Development Environment) like Eclipse, CodeBlock, Visual C++, Dev C++, etc. Software Development in the UNIX Environment Sample C Program. A value for pi is needed. Create a Python project to get the value of Pi to n number of decimal places. That’s about 664 bits, and so 108 terms of the series should be enough. Calculate area=PI*R*R 5. The number of active processes in the | {
"domain": "realizzazionifaidate.it",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846684978779,
"lm_q1q2_score": 0.8782913067637951,
"lm_q2_score": 0.8976952880018481,
"openwebmath_perplexity": 850.5454916785328,
"openwebmath_score": 0.6834801435470581,
"tags": null,
"url": "http://fkug.realizzazionifaidate.it/write-an-algorithm-to-compute-pi.html"
} |
terms of the series should be enough. Calculate area=PI*R*R 5. The number of active processes in the system and the length of the queue to the disk-buffer cache are considered in the algorithm design. I found 3 methods to check, they are: 1. No Input Output 1 30 4410 2 10 490 19. If you want to calculate pi, first measure the circumference of a circle by wrapping a piece of string around the edge of it and then measuring the length of the string. More precisely, their method permits to obtain the n-th bit of p in time O(nlog 3 n) and space O(log(n)). The 18th digit is a check digit which is used to validate the rest of that code and check that it is correct. Bearing Between Two Points Date: 12/19/2001 at 20:32:39 From: Doug Subject: Latitude/Longitude calculations I have found numerous solutions for finding the distance between two Lat/Long points on the earth (including the Haversine Formula), but I can't seem to find a reference that shows how to also calculate the direction between those points. The number of iterations could exceed 250,000. On a quantum computer, to factor an integer , Shor's algorithm runs in polynomial time (the time taken is polynomial in , the size of the integer. Dynamic Programming Algorithm to compute the n-th Tribonacci number. If I have to give you one general advice on the matter is do not try to write your own SVD algorithms unless you have successfully taken a couple of classes on Numerical Linear Algebra already and you know what you are doing. Example C Program to Compute PI Using A Monte Carlo Method. As a matter of full disclosure, I did not write this. 14; DESCRIPTION L = π*r2 Discussion Write an algorithm to solve this equation Answer. txt) into any directory and run the program. To understand this example, you should have the knowledge of the following C programming topics:. Calculate the volume of each of the sections of the room as shown on the drawing. A finite set of an instruction that specifies a sequence of. You are | {
"domain": "realizzazionifaidate.it",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846684978779,
"lm_q1q2_score": 0.8782913067637951,
"lm_q2_score": 0.8976952880018481,
"openwebmath_perplexity": 850.5454916785328,
"openwebmath_score": 0.6834801435470581,
"tags": null,
"url": "http://fkug.realizzazionifaidate.it/write-an-algorithm-to-compute-pi.html"
} |
room as shown on the drawing. A finite set of an instruction that specifies a sequence of. You are required to write an algorithm and a C program using top-down design approach with functions and only Call-by-Value parameters, to compute (1) Four total weights of four different sizes of washers to be shipped given the quantity of each washer type, its. im completely lost on almost everything i dont know how to calculate pi without using th emath sonstant or how to get to six significant figures using loops without rounding. A' is the transpose of the adjacency matrix of the graph. That is because it has at most b different remainders and the algorithm will terminate after b iterations at least. This allows you to do arbitrary precision arithmetic on numbers. Thomas Sterling PI Calculation • Matrix Multiplication Write result from task 0 to file. Explain to me why this is interesting and why Karatsuba's algorithm has--I'll. Explore our catalog of online degrees, certificates, Specializations, &; MOOCs in data science, computer science, business, health, and dozens of other topics. In fact the digits of π are extremely random - if you didn't know they were the digits of π they would be perfectly random. or maybe making the heaviest edge zero. Thanks for contributing an answer to Cross Validated! Please be sure to answer the question. There are two types of infinite series that can be used for this purpose:- 1)Gregory -Leibniz Ser. Hard Disk Drives (HDDs) Magnetic platters for storage. As long as this script runs it continues to generate digits. Leah Weimerskirch, Achievement First, New Haven, Connecticut. We use Arrays. Lets calculate π (or Pi if you prefer)! π is an irrational number (amongst other things) which means that it isn't one whole number divided by another whole number. Duration: 2 hrs lab + 2 hrs self Formative Assessment Introduction to C#, Visual Studio, algorithm design and writing programs The aim of this lab is to introduce you to Visual Studio | {
"domain": "realizzazionifaidate.it",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846684978779,
"lm_q1q2_score": 0.8782913067637951,
"lm_q2_score": 0.8976952880018481,
"openwebmath_perplexity": 850.5454916785328,
"openwebmath_score": 0.6834801435470581,
"tags": null,
"url": "http://fkug.realizzazionifaidate.it/write-an-algorithm-to-compute-pi.html"
} |
algorithm design and writing programs The aim of this lab is to introduce you to Visual Studio environment and write C# programs. The sine function (usually expressed in programming code as sin(th), where th is an angle in radians) is one of the basic functions in trigonometry. Algorithm definition, a set of rules for solving a problem in a finite number of steps, as for finding the greatest common divisor. Maybe duplicating an edge with value (-1*max edge). The value of p can be computed according to the following formula: π/4 = 1 - 1/3 + 1/5 -1/7 + 1/9 -. The polynomial is passed as an ordered list where the i-th index corresponds (though is not equivalent) to the coefficient of x to the n-th power. Write a c program for FIR filter design using dsp. Intuitively I'd ignore Stein's algorithm (on that page as "Binary GCD algorithm") for Python because it relies on low level tricks like bit shifts that Python really doesn't excel at. I recently saw a method of calculating pi that involves an iterative function, P(n + 1) = P(n) + sin(P(n)) where P(n) is the approximation of pi at the nth iteration. As long as this script runs it continues to generate digits. The code to calculate it is simply: InfPrec PI = (ataninvint(5) * 4 - ataninvint(239)) * 4; Pretty darned simple. You can find the value of pi with a mass and a spring. Compute PI with 1000 decimal digits using the Bailey/Borwein/Plouffe formula; Write PI with 1000 decimal digits using Newtons formula; Write the decimal digits of PI with a spigot algorithm; Determine the truncated square root of a big integer number; Function to compute the nth root of a positive float number; Exponentiation function for integers. The expected profi t from opening a restaurant at location i is pi, where pi > 0 and i = 1; 2; : : : ; n. Draw the portion of state space tree generated by LC branch and bound for the following knasback instance n=4, (Pi, P2, P3, P4). It has reinforced for me that teachers are some of the brightest and | {
"domain": "realizzazionifaidate.it",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846684978779,
"lm_q1q2_score": 0.8782913067637951,
"lm_q2_score": 0.8976952880018481,
"openwebmath_perplexity": 850.5454916785328,
"openwebmath_score": 0.6834801435470581,
"tags": null,
"url": "http://fkug.realizzazionifaidate.it/write-an-algorithm-to-compute-pi.html"
} |
instance n=4, (Pi, P2, P3, P4). It has reinforced for me that teachers are some of the brightest and most talented people in the world. the same memory location (even ifthey are. R803722-01 Project Officer Lee A. Since then the arc tan method dominated the pi calculation until 1980. Monte Carlo estimation Monte Carlo methods are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. There are two ways to do this: 1) With user interaction: Program will prompt user to enter the radius of the circle 2) Without user interaction: The radius value would be specified in the program itself. Give Pi = 3. g } Example p Algorithm AREA_OF_CIRCLE { Given the radius of a circle, this algorithm calculate the area of the circle. 2)Any two restaurants should be at least k miles apart, where k is a positive integer. The result of the task execution is a java. It is approximately equal to 3. I am looking for a utility to calculate MGRS grid lines, or at least an algorithm. Write the algorithm and draw the flowchart for a module to determine the payroll deduction. Intuitively I'd ignore Stein's algorithm (on that page as "Binary GCD algorithm") for Python because it relies on low level tricks like bit shifts that Python really doesn't excel at. Write a program to calculate the area of a circle and display the result use the formula a r2 where pi is approximately equal to 3 1416 Write the pseudocode for a program that will calculate the area of a group of circles. Stimulating CRCW with EREW Theorem: A p-processor CRCW algorithm can be no more than O(lg p) times faster than a best p-processor EREW algorithm for the same problem. It is defined as the ratio of a circle's circumference to its diameter, and it also has various equivalent definitions. How To Write PID control algorithm using C language How To Write PID control algorithm using C language Today i am going to write PID control algorithm using C language and how can you write | {
"domain": "realizzazionifaidate.it",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846684978779,
"lm_q1q2_score": 0.8782913067637951,
"lm_q2_score": 0.8976952880018481,
"openwebmath_perplexity": 850.5454916785328,
"openwebmath_score": 0.6834801435470581,
"tags": null,
"url": "http://fkug.realizzazionifaidate.it/write-an-algorithm-to-compute-pi.html"
} |
C language Today i am going to write PID control algorithm using C language and how can you write your own PID control algorithm using C language. In this C program, library function defined in header file is used to compute mathematical functions. :: It is recommended to use the Microsoft Visual Studio IDE (code editor). What is a Turing machine? A Turing machine is a hypothetical machine thought of by the mathematician Alan Turing in 1936. String Matching Problem Given a text T and a pattern P, find all occurrences of P within T Notations: - n and m: lengths of P and T - Σ: set of alphabets (of constant size) - Pi: ith letter of P (1-indexed) - a, b, c: single letters in Σ - x, y, z: strings String Matching Problem 3. An Iterative Method of Calculating Pi. Note that it's based on the computation of an arctangent, which you'll also need to compute to high precision. Nükhet ÖZBEK Ege University Department of Electrical&Electronics Engineering. Tavish Srivastava, co-founder and Chief Strategy Officer of Analytics Vidhya, is an IIT Madras graduate and a passionate data-science professional with 8+ years of diverse experience in markets including the US, India and Singapore, domains including Digital Acquisitions, Customer Servicing and Customer Management, and industry including Retail Banking, Credit Cards and Insurance. pi / 4 #launch angle in radians time = np. , 266-7883 corresponds to compute. Notice that the example random walk proposal $$Q$$ given above satisfies $$Q(y|x)=Q(x|y)$$ for all $$x,y$$. As a matter of full disclosure, I did not write this. The formula is a special case of the Boole summation formula for alternating series, providing yet another example of a convergence acceleration technique that can be applied to the Leibniz series. hint3 writes "Fabrice Bellard has calculated Pi to about 2. Insurance companies use a methodology called risk assessment to calculate premium rates for policyholders. The above takes O(N) time and O(1) constant space. | {
"domain": "realizzazionifaidate.it",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846684978779,
"lm_q1q2_score": 0.8782913067637951,
"lm_q2_score": 0.8976952880018481,
"openwebmath_perplexity": 850.5454916785328,
"openwebmath_score": 0.6834801435470581,
"tags": null,
"url": "http://fkug.realizzazionifaidate.it/write-an-algorithm-to-compute-pi.html"
} |
to calculate premium rates for policyholders. The above takes O(N) time and O(1) constant space. Write a program PhoneWords. , beginning at position d+1): Given an integer d > 0, we can write, from formula (3), {16 dπ} = {4{16 S 1}−2{16dS 4}−{16dS 5}−{16dS 6. The argument of a nonzero complex number $z$ is the value (in radians) of the angle $\theta$ between the abscissa of the complex plane and the line formed by $(0;z)$. Software Development in the UNIX Environment Sample C Program. Calculate digits of e; Calculate digits of pi; Calculate distance between two points on a globe; Calculate the factorial of a number; Calculate the sum over a container; The code examples below show how to calculate digits of pi in different programming languages. But even then there is an arbitrary factor between S and B. Root project for CSCI-3656. It has reinforced for me that teachers are some of the brightest and most talented people in the world. Hello All, I'm trying to track down some algorithms for drawing arcs. Write an algorithm for a program that will receive the radii of two circles from the user and calculate and display the difference of the areas of the circles. I was surfing the 'code project' site, while I see your article about calculating PI number. The BBP Algorithm for Pi erty is that it permits one to calculate (after a fairly simple manipulation) hexadecimal or binary digits of π beginning at an arbitrary starting position. Extremely long decimal expansions of π are typically computed with iterative formulae like the Gauss–Legendre algorithm and Borwein's algorithm. Background: Hard-Disk Drives. So you're actually computing z0 and z2 first, and then using them to compute z1. pi is a Python script that computes each digit of the value of pi. In this tutorial we will introduce a simple, yet versatile, feedback compensator structure: the Proportional-Integral-Derivative (PID) controller. Write an algorithm to compute the area of circles. Incorporation of the | {
"domain": "realizzazionifaidate.it",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846684978779,
"lm_q1q2_score": 0.8782913067637951,
"lm_q2_score": 0.8976952880018481,
"openwebmath_perplexity": 850.5454916785328,
"openwebmath_score": 0.6834801435470581,
"tags": null,
"url": "http://fkug.realizzazionifaidate.it/write-an-algorithm-to-compute-pi.html"
} |
(PID) controller. Write an algorithm to compute the area of circles. Incorporation of the Dirichlet condition at $$x=0$$ through modifying the linear system at each time level means that we carry out the computations as explained in the section Discretization in time by a Backward Euler scheme and get a system (26). Question 19:Write a C program to convert the given amount in Rupees to Dollars, Euros and Pounds using separate functions. Historically, William Shanks published pi to 707 decimal places in 1873. Looks like quite a challenge. 4 Answers are available for this question. It doesn't matter if Pi itself is done using different algorithms*. Usage contFrac(x, tol = 1e-06) Arguments x a numeric scalar or vector. That’s about 664 bits, and so 108 terms of the series should be enough. 99) or “Steak House” ($15 and up. The basic idea is to to use a polynomial approximation in step 3 to calculate e x. Pi is available in the Math. 14159 is in position 1. "Objects First with Java: A Practical Introduction Using BlueJ" is a textbook co-written by the developers of BlueJ and has sold hundreds of thousands of copies worldwide. In one more interval of length 4 l, Pi has a chance to update its local dist and outgoing register values. Hard Disk Drives (HDDs) Magnetic platters for storage. Hello All, I'm trying to track down some algorithms for drawing arcs. Convert these algorithms to properly documented, professional quality programs. Algorithm definition, a set of rules for solving a problem in a finite number of steps, as for finding the greatest common divisor. We can, create a new string T = S+reverse_of(S). 08607980113 3. 07025461778 3. Print area and Circumference 6. Ultimately, it was basically a disaster since it nearly doubled the time needed to compute 10 trillion digits of Pi. Continued fractions are just another way of writing fractions. The possibly first spigot algorithm for π from Rabinowitz and Wagon is in this class; it seems that they coined the term | {
"domain": "realizzazionifaidate.it",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846684978779,
"lm_q1q2_score": 0.8782913067637951,
"lm_q2_score": 0.8976952880018481,
"openwebmath_perplexity": 850.5454916785328,
"openwebmath_score": 0.6834801435470581,
"tags": null,
"url": "http://fkug.realizzazionifaidate.it/write-an-algorithm-to-compute-pi.html"
} |
algorithm for π from Rabinowitz and Wagon is in this class; it seems that they coined the term "spigot algorithm" to describe this algorithm. The Python area of a circle is number of square units inside the circle. Maybe duplicating an edge with value (-1*max edge). In Chinese mathematics, this was improved to approximations correct to what corresponds to about seven decimal digits by the 5th century. 2 is the base number. Standard formula to calculate the area of a circle is: A=πr². As a personal exercise, I'm trying to write an algorithm to compute the n-th derivative of an ordered, simplified polynomial (i. 20818565226 3. Write a function that takes two parameters: an integer to use as a random seed for the random number generator, and an integer number of points you would like to test. A finite set of an instruction that specifies a sequence of. Combinatorial problems. The result is between -pi and pi. @Alan: OK, but the question clearly says he's trying to write a function that can compute PI to X places Anyway, I implemented this taylor series and after 1 billion iterations you have "3. Give Pi = 3. pi is a Python script that computes each digit of the value of pi. But writing all su xes takes ( n2) space, we need a compact representation. PiFast is the fastest program to compute pi on the web, and also hold the current pi computation record on a home PC with several billion digits computations. pyplot as plt import numpy as np import math import scipy. LearnZillion helps you grow in your ability and content knowledge and it gives you the opportunity to work with an organization that values teachers, student, and achievement by both. To calculate pi a little bit more precisely, replace the red ball with one that is 100 times less massive than the blue ball – a ping pong ball might work, so we will call this the white ball. Python script to compute pi with Chudnovsky formula and Binary Splitting Algorithm, using GMP libarary. If Sum then print n1 + n2. As a | {
"domain": "realizzazionifaidate.it",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846684978779,
"lm_q1q2_score": 0.8782913067637951,
"lm_q2_score": 0.8976952880018481,
"openwebmath_perplexity": 850.5454916785328,
"openwebmath_score": 0.6834801435470581,
"tags": null,
"url": "http://fkug.realizzazionifaidate.it/write-an-algorithm-to-compute-pi.html"
} |
formula and Binary Splitting Algorithm, using GMP libarary. If Sum then print n1 + n2. As a matter of full disclosure, I did not write this. This is just the normalized probability of each each point belonging to one of the $$K$$ Gaussians weighted by the mixture distribution ($$\pi_k$$). Calculating Pi. The literal answer is print "infinity" since there are an infinite number of primes. I am currently at a point where I have the following: test statistic= sqrt n (Y bar - mu) all over the standard deviation and that p value= the probability T > tobs given mu=0. You can find the value of pi with a mass and a spring. The final section of this chapter is devoted to cluster validity—methods for evaluating the goodness of the clusters produced by a clustering algorithm. The code that invokes a Compute object's methods must obtain a reference to that object, create a Task object, and then request that the task be executed. Maybe duplicating an edge with value (-1*max edge). Estimation of Pi The idea is to simulate random (x, y) points in a 2-D plane with domain as a square of side 1 unit. You can use the Floyd-Warshall algorithm to compute the shortest path from any node to any node. A new algorithm for rational interpolation is proposed. Calculus relates topics in an elegant, brain-bending manner. As a matter of full disclosure, I did not write this. For this question, please what is wrong with this code?:def newtonRap(cp, price, s, k, t, rf): v = sqrt(2*pi/t)*price/s print "ini Calculate the Implied Volatility Using Newton Raphson Algorithm. (Find the manual on how to use the IDE for C++. No Input Output 1 30 4410 2 10 490 19. The final answer is at T[2] - where it is the last Tribonacci number computed. Of course it's called Pi Day because the date, 3/14, is similar to the first three digits of pi (3. A Fast Algorithm for Rational Interpolation Via Orthogonal Polynomials By Ömer Egecioglu*and Çetin K. Page two – calculating the love! I’m sorry to disappoint you, but I | {
"domain": "realizzazionifaidate.it",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846684978779,
"lm_q1q2_score": 0.8782913067637951,
"lm_q2_score": 0.8976952880018481,
"openwebmath_perplexity": 850.5454916785328,
"openwebmath_score": 0.6834801435470581,
"tags": null,
"url": "http://fkug.realizzazionifaidate.it/write-an-algorithm-to-compute-pi.html"
} |
By Ömer Egecioglu*and Çetin K. Page two – calculating the love! I’m sorry to disappoint you, but I don’t know the secret of compatibility – if I did I’d probably be sunning it up in the Bahamas instead of writing blog posts getting a monitor tan. Multiplying by$\pi$later can return the radians. Calculate the O(log n) numbers = (flp)2', where the product is taken over those primes p whose index in n! has a non-zero multiple of 21 in its base two expansion. Ready to learn how to code, debug, and program? Get started with our expert-taught tutorials explaining programming languages like C, C#, Python, Visual Basic, Java, and more. The Metropolis Algorithm. Take the value of radius from the user. 2)Any two restaurants should be at least k miles apart, where k is a positive integer. Premise: The problem above unlike others will read some other values with itself. Number your steps. Write an algorithm and program to compute π, using the formula described in the text pi=(4/1)-(4/3)+(4/5)-(4/7)+(4/9) Output will include your computed value for π, the math library constant expected value for π and the number of iterations it took to reach six-significant digit accuracy. If you take data that represents a sample of a larger population, you apply the sample standard deviation formula. , the proof of which. Calculate the volume by multiplying the measured length and width of the space together, then multiply the result by the height of the room. How To Write PID control algorithm using C language How To Write PID control algorithm using C language Today i am going to write PID control algorithm using C language and how can you write your own PID control algorithm using C language. That is because it has at most b different remainders and the algorithm will terminate after b iterations at least. Now we use the given formula to calculate the area of the circle using formula. I'm guessing, I should start off by changing the original graph G. , 266-7883 corresponds to compute. The | {
"domain": "realizzazionifaidate.it",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846684978779,
"lm_q1q2_score": 0.8782913067637951,
"lm_q2_score": 0.8976952880018481,
"openwebmath_perplexity": 850.5454916785328,
"openwebmath_score": 0.6834801435470581,
"tags": null,
"url": "http://fkug.realizzazionifaidate.it/write-an-algorithm-to-compute-pi.html"
} |
I should start off by changing the original graph G. , 266-7883 corresponds to compute. The key fact behind this algorithm is that the values of the components of the normal vector, say , of a polygon are proportional to the signed areas of the polygons produced by projecting the original polygon onto the yz, zx, and xy planes. Personal Finance & Money Stack Exchange is a question and answer site for people who want to be financially literate. Calculate Circumference of circle ( You can use either C= 2 X Pi X r Or Area (A) = Pi X r X r Circumference (C) = 2 X Pi X r A/r = Pi X r A/r = C/2 So C = 2 x ( A/r ) 5. As a personal exercise, I'm trying to write an algorithm to compute the n-th derivative of an ordered, simplified polynomial (i. Then k 2n =√(2+k n) S n =√(2-k n) Area of the polygon = (1/2)(nSa) as the number of the sides of the polygon increases the value of a=radius. The secret to why the QR algorithm produces iterates that usually converge to reveal the eigenvalues is from the fact that the algorithm is a well-disguised (successive) power method. To compute the hexadecimal digits of π beginning after the first d hex digits (i. That sounds like a significant milestone! (pi_archimedes_decimal. Koç** Abstract. It appears in many formulas in all areas of mathematics and physics. If the fitness function becomes the bottleneck of the algorithm, then the overall efficiency of the genetic algorithm will be reduced. , the proof of which. Unfortunately, the question is about calculating the X'th digit and does not necessarily mean you have to get the digits before digit X and it wouldn't. 0_dp denom = 3. Effec tively, the SEM algorithm reweights the cases by com puting a membership probability for the new compo nent. There are many ways in which the area of a triangle can be calculated. The radius of a circle is equal to one unit Write the algorithm to compute the corresponding area of the circle and Write an algorithm to compute the circumferenc of a circle? Let | {
"domain": "realizzazionifaidate.it",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846684978779,
"lm_q1q2_score": 0.8782913067637951,
"lm_q2_score": 0.8976952880018481,
"openwebmath_perplexity": 850.5454916785328,
"openwebmath_score": 0.6834801435470581,
"tags": null,
"url": "http://fkug.realizzazionifaidate.it/write-an-algorithm-to-compute-pi.html"
} |
corresponding area of the circle and Write an algorithm to compute the circumferenc of a circle? Let PI=3. Hello, I know this is a year old discussion, but I got to this topic even after searching for MAX30100 example on Google. The first written description of an infinite series that could be used to compute pi was laid out in Sanskrit verse by Indian astronomer Nilakantha Somayaji around 1500 A. Obtain the optimal solution for this knapsack problem. Write a simple "for" loop to compute the average yourself or use a standard library with a mean function. processors can compute the following fimctions in. ::C# (c sharp) code is used in in this document, however the same logic can be used in all programming languages that have the ability to manipulate data. From the main page you can also follow links to:. The algorithm as I described it is for each thrower to throw they're darts independently, then the totals a brought together and PI calculated from the results of all throwers. That ran rather slowly and wasn't converging very well. @Alan: OK, but the question clearly says he's trying to write a function that can compute PI to X places Anyway, I implemented this taylor series and after 1 billion iterations you have "3. I recently saw a method of calculating pi that involves an iterative function, P(n + 1) = P(n) + sin(P(n)) where P(n) is the approximation of pi at the nth iteration. I have a good idea how to calculate big O by looking at an algorithm but in this case: Suppose an algorithm takes 5 seconds to handle a data set of 1000 records. The DFT enables us to conveniently analyze and design systems in frequency domain; however, part of the versatility of the DFT arises from the fact that there are efficient algorithms to calculate the DFT of a sequence. Proof: each step of CRCW can be simulated by O(lg p) computations of EREW. 2 Arithme/c%Expressions% • Mathemacal%Operators% + %Addi/on % %/ %Division O %Subtrac/on % %% %Modulo% * %Mul/plicaon %** %Exponen/aon | {
"domain": "realizzazionifaidate.it",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846684978779,
"lm_q1q2_score": 0.8782913067637951,
"lm_q2_score": 0.8976952880018481,
"openwebmath_perplexity": 850.5454916785328,
"openwebmath_score": 0.6834801435470581,
"tags": null,
"url": "http://fkug.realizzazionifaidate.it/write-an-algorithm-to-compute-pi.html"
} |
+ %Addi/on % %/ %Division O %Subtrac/on % %% %Modulo% * %Mul/plicaon %** %Exponen/aon % • OrderofPrecedence%. , 266-7883 corresponds to compute. Using the bits of pi as the random numbers, there is a certain elegance to use that data to compute pi. how to write an algorithm that will compute the sum, difference, quotient and product of two input numbers?. Create a class and using a constructor initialise values of that class. It takes input of radius of sphere and calculate the volume and area of sphere. How to calculate key size of a security algorithm? The answer to this question is completely dependant on the algorithm. The program will prompt user to enter the radius and based on the input it would calculate the values. If (I <=98) then go to line 3 6. 4 Answers are available for this question. By Beeler et al. We call "number-theoretic" any function that takes integer arguments, produces integer values, and is of interest to number theory. ) In this way you can calculate the derivative Exactly. Input a real value from cin into radius. If it is of length greater 1, the function assumes this is a continuous fraction and computes its value. 14 as a constant in the pr Write the pseudocode and program to calculate the area of a circle, given the value of pi=3. Obtain the optimal solution for this knapsack problem. Personal Finance & Money Stack Exchange is a question and answer site for people who want to be financially literate. It appears in many formulas in all areas of mathematics and physics. The program should output “Not Enough” ($0 - $0. We're going to play with the concepts of sine series, iterations, vectorizing programs among others. It was processed by a rowed logic. The encryption code is relatively simple (click to enlarge):. The DFT, like the more familiar continuous version of the Fourier transform, has a forward and inverse form which are defined as follows:. power, and not a perfect square. Use your program to test the law on some tables of | {
"domain": "realizzazionifaidate.it",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846684978779,
"lm_q1q2_score": 0.8782913067637951,
"lm_q2_score": 0.8976952880018481,
"openwebmath_perplexity": 850.5454916785328,
"openwebmath_score": 0.6834801435470581,
"tags": null,
"url": "http://fkug.realizzazionifaidate.it/write-an-algorithm-to-compute-pi.html"
} |
as follows:. power, and not a perfect square. Use your program to test the law on some tables of information from your computer or from the web. // Compute Angles based on the binary tree // idx - index for leaf nodes of the binary tree // nIter - no of iteration (corresponds to the level of the tree). A value for pi is needed. Write a Python script that will calculate Pi with at least three accurate decimal places using the Gregory-Leibniz series. It is not an example, but a requirement. Read in the circle’s radius. They have some interesting connections with a jigsaw-puzzle problem about splitting a rectangle into squares and also with one of the oldest algorithms known to Greek mathematicians of 300 BC - Euclid's Algorithm - for computing the greatest divisor common to two numbers (gcd). 04183961893 3. The expected profi t from opening a restaurant at location i is pi, where pi > 0 and i = 1; 2; : : : ; n. An algorithm which can compute arbitrarily many successive digits, storing only a constant amount of state information. Hydrocomp Incorporated Palo Alto, California 94304 Grant No. This article introduces new algorithms for asynchronous operations in disk-buffer-cache memory. When it comes to programming. Perimeter of circle=pi * radius * radius Enter the radius of the circle as input. Given that pi is not going to change and that 43 digits is enough precision to calculate the circumference of the universe to a tolerance of the width of 1 proton, this is a pretty reasonable. There are two ways to do this: 1) With user interaction: Program will prompt user to enter the radius of the circle 2) Without user interaction: The radius value would be specified in the program itself. By Beeler et al. Let's look at an intuitive, obvious example. From the main page you can also follow links to:. Thanks for contributing an answer to Cross Validated! Please be sure to answer the question. Your algorithm should prompt the user to take the number of circles and their radius | {
"domain": "realizzazionifaidate.it",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846684978779,
"lm_q1q2_score": 0.8782913067637951,
"lm_q2_score": 0.8976952880018481,
"openwebmath_perplexity": 850.5454916785328,
"openwebmath_score": 0.6834801435470581,
"tags": null,
"url": "http://fkug.realizzazionifaidate.it/write-an-algorithm-to-compute-pi.html"
} |
the question. Your algorithm should prompt the user to take the number of circles and their radius values. When k = 1, the vector is called simply an eigenvector, and the pair. where concwremreads are allowed but no two processors can simultaneously attempt to write in. For example, suppose we wanted to compute 1/π to 200 decimal places. Intro to Computer programming worked at calculating digits of pi today. This really nice fusion algorithm was designed by NXP and requires a bit of RAM (so it isnt for a '328p Arduino) but it has great output results. I'll just write the formula, since the code is straight forward. Having understanding of what features of DFT we are going to exploit to speed-up calculation we can write down the following algorithm: Prepare input data for summation — put them into. Split up x: Write x = n + r, where n is the nearest integer to x and r is a real number. The Euclidean algorithm is one of the oldest algorithms in common use. It's sole purpose was to verify the main computation. trying to write. For Pi Day 2018 I calculated π by hand using the Chudnovsky algorithm. Using a chroma key effect as an example, this article describes a simple workflow for deploying a MATLAB ® image processing algorithm to embedded hardware. Big data requires us to come up with new algorithms, and efficiency is extremely important to be able to get results in a useful amount of time. of Algorithm 1 quadruples the number of correct digits, while each additional iteration of Algorithm 2 quintuples the number of correct digits. Fill in the following table, which shows the approximate growth of the execution times depending on the complexity of the algorithm. A Fast Algorithm for Rational Interpolation Via Orthogonal Polynomials By Ömer Egecioglu*and Çetin K. A' is the transpose of the adjacency matrix of the graph. 3 Radix-2 FFT Useful when N is a power of 2: N = r for integers r and. I would like to be able to draw the line dynamically. 2: XML Language Reference | {
"domain": "realizzazionifaidate.it",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846684978779,
"lm_q1q2_score": 0.8782913067637951,
"lm_q2_score": 0.8976952880018481,
"openwebmath_perplexity": 850.5454916785328,
"openwebmath_score": 0.6834801435470581,
"tags": null,
"url": "http://fkug.realizzazionifaidate.it/write-an-algorithm-to-compute-pi.html"
} |
for integers r and. I would like to be able to draw the line dynamically. 2: XML Language Reference for Event Stream Processing Models. The article describes the theory behind the code. High granularity: In one atomic step, process Pi can read both neighbors’ variables, compute its next value, and write it to variable xi. Write an algorithm to calculate the volume of a cylinder?. At the end of the post there is an excellent video by Kevin Wallenstein. I liked it so much and since my hobby is optimization I tried to write a C code to calculate it by implementing the algorithm you mentioned. XML Language Elements Relevant to Streaming Analytics Windows. org/wiki/Chudnovsky_algorithm k = 0 42698672/13591409 = 3. calculate how much time has passed. The fitness function should be implemented efficiently. Amount of space inside the sphere is called as Volume. Effec tively, the SEM algorithm reweights the cases by com puting a membership probability for the new compo nent. Write a program called HarmonicSeriesSum to compute the sum of a harmonic series 1 + 1/2 + 1/3 + 1/4 + + 1/n, where n = 1000. Give an efficient algorithm to compute the maximum expected total. Implement standard algorithms to draw various graphics objects using program. Four can also be geometrically represented by showing four shapes or the fourth stroke from the Origin in a number line. Right away, the starting direction is defined as$\frac{1}{2}$. If the heuristic algorithms cannot be applied, risch_integrate() is tried next. High granularity: In one atomic step, process Pi can read both neighbors’ variables, compute its next value, and write it to variable xi. This is just the normalized probability of each each point belonging to one of the $$K$$ Gaussians weighted by the mixture distribution ($$\pi_k$$). 99) or “Steak House” ($15 and up. The algorithm to calculate circumference and area of a circle is as below: Step 1: Start the program. Although insertion sort is an O(n 2) algorithm, its | {
"domain": "realizzazionifaidate.it",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846684978779,
"lm_q1q2_score": 0.8782913067637951,
"lm_q2_score": 0.8976952880018481,
"openwebmath_perplexity": 850.5454916785328,
"openwebmath_score": 0.6834801435470581,
"tags": null,
"url": "http://fkug.realizzazionifaidate.it/write-an-algorithm-to-compute-pi.html"
} |
a circle is as below: Step 1: Start the program. Although insertion sort is an O(n 2) algorithm, its simplicity, low overhead, good locality of reference and efficiency make it a good choice in two cases: (i) small n , (ii) as the final finishing-off algorithm for O ( n log n ) algorithms such as mergesort and quicksort. Using R code I have to write a pseudo code and real code to answer this question. You can read this mathematical article: A fast algorithm for computing large Fibonacci numbers (Daisuke Takahashi): PDF. Ready to learn how to code, debug, and program? Get started with our expert-taught tutorials explaining programming languages like C, C#, Python, Visual Basic, Java, and more. A step-by-step problem-solving procedure. trying to write. when m < 1 : increment y by 1, find the best x in each iteration (we ignore negative slope). A directed graph (or digraph) is a set of vertices and a collection of directed edges that each connects an ordered pair of vertices. The number of iterations could exceed 250,000. Algorithm. Since 2001, Processing has promoted software literacy within the visual arts and visual literacy within technology. In such a case the planet would move with a velocity V = (2*PI)/Period. I am trying to write the "saving algorithm" in VB using Excel. See the step entitled Tools Needed. nil# = earth_radius_miles * c. Assume that the first radius is greater than the second. When k = 1, the vector is called simply an eigenvector, and the pair. 20036551541 3. 1 Answers Infosys , Qatar University , write a program that prompt the user to enter his height and weight,then calculate the body mass index and show the algorithm used. We can now find the eccentric anomaly using some numerical method. Borwein, Bailey, and Girgensohn (2004) have recently shown that has no Machin-type BBP arctangent formula that is not binary, although this does not rule out a completely different scheme for digit-extraction algorithms in other bases. C program to design | {
"domain": "realizzazionifaidate.it",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846684978779,
"lm_q1q2_score": 0.8782913067637951,
"lm_q2_score": 0.8976952880018481,
"openwebmath_perplexity": 850.5454916785328,
"openwebmath_score": 0.6834801435470581,
"tags": null,
"url": "http://fkug.realizzazionifaidate.it/write-an-algorithm-to-compute-pi.html"
} |
a completely different scheme for digit-extraction algorithms in other bases. C program to design Butterworth filter design dsp. arange(0,10, dt) #create time axis gamm = 0. Poor guy, he made a mistake starting at the 528th decimal place. Algorithm to calculate the Interest on Loan with a Balloon Payment. Extremely long decimal expansions of π are typically computed with iterative formulae like the Gauss–Legendre algorithm and Borwein's algorithm. It uses the random numbers to calculate the area of a circle inscribed within a square and from that, one can calculate pi. 99 billion. To calculate area and circumference we must know the radius of circle. Pi Estimation. These algorithms have been shown to contain flaws (i. While the improvement may seem small, it is an outstanding achievement because only a single desktop PC, costing less than $3,000, was used — instead of a multi-million dollar supercomputer as in the previous records. If yes, output e in the intersection set, otherwise in the difference set. Euclid's algorithm is probably fine. The secret to why the QR algorithm produces iterates that usually converge to reveal the eigenvalues is from the fact that the algorithm is a well-disguised (successive) power method. Write a C program to input radius of a circle from user and find diameter, circumference and area of the circle. in the other the word some value are constant and you have to make the algorithm to be aware of this. Problem1: An algorithm to calculate even numbers between 0 and 99 Inputs to the algorithm: Sequence of numbers Expected output: The calculate of even number Algorithm: 1. When you are writing m-files you will usually want to have the text editor and MATLAB open at the same time. The definition of the task class Pi is shown later. java that reads in a sequence of integers from standard input and tabulates the number of times each of the digits 1-9 is the leading digit, breaking the computation into a set of appropriate static methods. The | {
"domain": "realizzazionifaidate.it",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846684978779,
"lm_q1q2_score": 0.8782913067637951,
"lm_q2_score": 0.8976952880018481,
"openwebmath_perplexity": 850.5454916785328,
"openwebmath_score": 0.6834801435470581,
"tags": null,
"url": "http://fkug.realizzazionifaidate.it/write-an-algorithm-to-compute-pi.html"
} |
1-9 is the leading digit, breaking the computation into a set of appropriate static methods. The key fact behind this algorithm is that the values of the components of the normal vector, say , of a polygon are proportional to the signed areas of the polygons produced by projecting the original polygon onto the yz, zx, and xy planes. Explore our catalog of online degrees, certificates, Specializations, &; MOOCs in data science, computer science, business, health, and dozens of other topics. If we know the radius of the Sphere then we can calculate the Volume of Sphere using formula: Volume of a Sphere = 4πr³. Thanks for contributing an answer to Computational Science Stack Exchange! Please be sure to answer the question. Vincenty's formulae for direct problem; I'm glad to know if there are any algorithms which is lesser accurate, but faster than Vincenty's formulae. can always begin by checking if the square-root, cube-root, etc. Using pi in your program is not your main goal here, but finding ways to acquire its value is useful. The program will prompt user to enter the radius and based on the input it would calculate the values. I will not explain it here. org * portable, PRAM like (CREW) * easy, but hides communication costs * easy API * C, C++, Fortran * compiler directives * runtime library routines * environment variables * nested parallelism, dynamic threads, no IO ## hello openmp. the algorithm of right-to-left addition will lead to the sum of two numbers. Practical Outcome (POs): Write a program to clip line using Midpoint Subdivision line clipping algorithm VI. Iterative algorithms for computing approximations to the number PI through infinite series using double and arbitrary precision "The circumference of any circle is greater than three times its diameter, and the excess is less than one seventh of the diameter but larger than ten times its Seventy first part " - Archimedes. How the Code Works. Write a program called ComputePI to compute the value of π, | {
"domain": "realizzazionifaidate.it",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846684978779,
"lm_q1q2_score": 0.8782913067637951,
"lm_q2_score": 0.8976952880018481,
"openwebmath_perplexity": 850.5454916785328,
"openwebmath_score": 0.6834801435470581,
"tags": null,
"url": "http://fkug.realizzazionifaidate.it/write-an-algorithm-to-compute-pi.html"
} |
part " - Archimedes. How the Code Works. Write a program called ComputePI to compute the value of π, using the following series expansion. Write a program to compute sin(x). Borwein, Bailey, and Girgensohn (2004) have recently shown that has no Machin-type BBP arctangent formula that is not binary, although this does not rule out a completely different scheme for digit-extraction algorithms in other bases. Then I realized I could use the Bresenham circle algorithm and just calculate one point in each row. The DFT, like the more familiar continuous version of the Fourier transform, has a forward and inverse form which are defined as follows:. com (Pi, P2 5, 15, 7, 6, 18, 3) and (WI, W2, W7) = (2, 3, 5, 7, 1, 4, 1). Python Program to Calculate the Area of a Triangle In this program, you'll learn to calculate the area of a triangle and display it. zip file) and Calculate up to 32 Million digits of Pi. For practical purposes, you should just. Click here to learn more about methods to calculate G. This gives us a possible way to calculate it. If yes, output e in the intersection set, otherwise in the difference set. The fraction should be π / 4, so we use this to get our estimate. Thus we consider a permutation$\pi$and some element$f$. , beginning at position d+1): Given an integer d > 0, we can write, from formula (3), {16 dπ} = {4{16 S 1}−2{16dS 4}−{16dS 5}−{16dS 6. So in this base, Pi is one of the simpliest numbers that exists ! We know Pi 's digits in this base, so to compute Pi 's decimal places in base 10 one by one, one just needs to build an algorithm that changes it to base 10 , which is precisely the principle of the spigot algorithm. Optimization is the art and science of allocating scarce resources to the best possible effect. Looking at Pi and Pi[PDF], there are a lot of formulas. rgpvonline. This article introduces new algorithms for asynchronous operations in disk-buffer-cache memory. Describing convex hulls in purely metrical terms. This probabilistic | {
"domain": "realizzazionifaidate.it",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846684978779,
"lm_q1q2_score": 0.8782913067637951,
"lm_q2_score": 0.8976952880018481,
"openwebmath_perplexity": 850.5454916785328,
"openwebmath_score": 0.6834801435470581,
"tags": null,
"url": "http://fkug.realizzazionifaidate.it/write-an-algorithm-to-compute-pi.html"
} |
in disk-buffer-cache memory. Describing convex hulls in purely metrical terms. This probabilistic method relies on a random number generator and is described below. If you look at Kaufman & Rousseeuw (1990), Finding Groups in Data, they describe an algorithm to evaluate the quality of clusters in agglomerative clustering. I also own a Heart Rate clickboard and I found this article about a setup of MAX30100 with a lot of useful information. There are many ways in which the area of a triangle can be calculated. The first few convergents are 3, 22/7, 333/106, 355/113, 103993/33102, 104348/33215,. Assume that the inputs have been sorted as in equation (16. An equivalent statement is → ∞ / = where li is the logarithmic integral function. Initialize a 0 = 6 - 4 √2 and y 0 = √2 - 1. The final section of this chapter is devoted to cluster validity—methods for evaluating the goodness of the clusters produced by a clustering algorithm. We proved correctness, stabilization time, using induction on distance from root. 14 Step 3 : Read the value of radius Step 4 : Calculate area using formulae pi*radius2 Step 5 : Calculate circumference using formulae 2*pi*radius Step 6 : Print area and circumference. Hello, I know this is a year old discussion, but I got to this topic even after searching for MAX30100 example on Google. If I use the normal method to calculate the center point, it will be in the red area. The BBP Algorithm for Pi erty is that it permits one to calculate (after a fairly simple manipulation) hexadecimal or binary digits of π beginning at an arbitrary starting position. The final section of this chapter is devoted to cluster validity—methods for evaluating the goodness of the clusters produced by a clustering algorithm. Standard formula to calculate the area of a circle is: A=πr². It is simple to implement and requires only few lines of code. Step 7: Stop the program. The Python area of a circle is number of square units inside the circle. Related posts. Write a | {
"domain": "realizzazionifaidate.it",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846684978779,
"lm_q1q2_score": 0.8782913067637951,
"lm_q2_score": 0.8976952880018481,
"openwebmath_perplexity": 850.5454916785328,
"openwebmath_score": 0.6834801435470581,
"tags": null,
"url": "http://fkug.realizzazionifaidate.it/write-an-algorithm-to-compute-pi.html"
} |
The Python area of a circle is number of square units inside the circle. Related posts. Write a program to calculate the area of a circle and display the result use the formula a r2 where pi is approximately equal to 3 1416 The length and breadth of a rectangle and radius of a circle takes as input through keyboard. LearnZillion helps you grow in your ability and content knowledge and it gives you the opportunity to work with an organization that values teachers, student, and achievement by both. That’s true for sufficiently large n. The easiest place to start would probably be with a Machin-type formula, like the one listed on that page: pi/4 = 4 arctan(1/5) - arctan(1/239) This page lists a bunch of similar formulas. Step 2: Initialize value of PI = 3. I need a VB6 algorithm to calculate a one byte LRC for a data string that I am sending through a serial port. Algorithm. A second method is given showing how to calculate the center of minimum distance ** , and finally a third method calculates the average latitude/longitude. Write a function that takes two parameters: an integer to use as a random seed for the random number generator, and an integer number of points you would like to test. Number your steps. When $$Q$$ is symmetric the formula for $$A$$ in the MH algorithm simplifies to: \[A= \min \left( 1, \frac{\pi(y)}{\pi(x_t)} \right). Learn how to use algorithms to explore graphs, compute shortest distance, min spanning tree, and connected components. Though a sweep line algorithm is exactly the same alogrithm as the FEA algorithm (without meshing) and its easiest to implement. ) Calculating Pi (π) Because Pi (π) has so many important uses, then we need to be able to start to calculate it, at least to several decimal places accuracy. Making statements based on opinion; back them up with references or personal experience. Computing hundreds, or trillions, of digits of π has long been used to stress hardware, validate software, and establish bragging rights. In | {
"domain": "realizzazionifaidate.it",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846684978779,
"lm_q1q2_score": 0.8782913067637951,
"lm_q2_score": 0.8976952880018481,
"openwebmath_perplexity": 850.5454916785328,
"openwebmath_score": 0.6834801435470581,
"tags": null,
"url": "http://fkug.realizzazionifaidate.it/write-an-algorithm-to-compute-pi.html"
} |
of π has long been used to stress hardware, validate software, and establish bragging rights. In the pi algorithm we are describing, ordinary 64-bit integer arithmetic suffices to calculate the 375 millionth hex digit of pi. I apologise! This is my first post, you're right I should explain more clearly. You can calculate its circumference to an accuracy of the diameter of a proton (10^-15 meters), with 43 digits of pi. , beginning at position d+1): Let {·}denote the fractional part as before. How to write a prorgram to compute a sin(x)' Learn more about sin(x), fix, algorithm, infinite series, matlab, homework How to write a prorgram to compute a sin(x)' infinite series algorithm ? Follow 3 views (last 30 days) dongjin An on 8 Mar As commented earlier, please make an attempt at writing the code to solve the problem. To calculate area and circumference we must know the radius of circle. 14 4 Calculate area of Circle = Pi x r x r 5. you need to use iterative statement for solving the problem. Let's calculate π to 100 decimal places now. Processing is a flexible software sketchbook and a language for learning how to code within the context of the visual arts. 14 and radius=2 declare pi=3. No Input Output 1 30 4410 2 10 490 19. Nükhet ÖZBEK Ege University. Compute focal statistics SUM using rectangle shape corresponding to dimensions of a new small box. How the Code Works. Amount of space inside the sphere is called as Volume. Explore our catalog of online degrees, certificates, Specializations, &; MOOCs in data science, computer science, business, health, and dozens of other topics. 6 x10^-35 m. It is defined as the ratio of a circle's circumference to its diameter, and it also has various equivalent definitions. 75 Number of months: 5 Interest rate: 5. ) Here's a BPP algorithm for primality testing. The fo … read more. manufactures and ships flat washers. As an example see…. Implement standard algorithms to draw various graphics objects using program. To try libcamera | {
"domain": "realizzazionifaidate.it",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846684978779,
"lm_q1q2_score": 0.8782913067637951,
"lm_q2_score": 0.8976952880018481,
"openwebmath_perplexity": 850.5454916785328,
"openwebmath_score": 0.6834801435470581,
"tags": null,
"url": "http://fkug.realizzazionifaidate.it/write-an-algorithm-to-compute-pi.html"
} |
see…. Implement standard algorithms to draw various graphics objects using program. To try libcamera for yourself with a Raspberry Pi, please follow the instructions in our online documentation, where you’ll also find the full Raspberry Pi Camera Algorithm and Tuning Guide. clc;close all; clear all; %Generate the sine wave sequences. With 0 unknowns, you would only check if the equation was valid. (d) Write an algorithm in the form of a flowchart for a program which uses the Newton- Raphson method to find the real root of the equation3x = cosx+l The program prompts for the initial points. Doing rand % 5, if a random number takes the value 6 or 7, gets mapped to 1 or 2, effectivelly increasing 1,2 frequency making distribution non-uniform. Thanks for contributing an answer to Computational Science Stack Exchange! Please be sure to answer the question. ANNA UNIVERSITY CHENNAI :: CHENNAI 600 025 AFFILIATED INSTITUTIONS REGULATIONS – 2008 CURRICULUM AND SYLLABI FROM VI TO VIII SEMESTERS AND E. Mathematica: high-powered computation with thousands of Wolfram Language functions, natural language input, real-world data, mobile support. of two numbers can be calculated efficiently using the Euclidean algorithm. Making statements based on opinion; back them up with references or personal experience. Iterate steps 2 and 3 until converged, i. #include 0, we can write, from formula (3), {16 dπ} = {4{16 S 1}−2{16dS 4}−{16dS 5}−{16dS 6. org or mail your article to [email protected] I used Python and only integers (I didn't want to use floating point numbers), and used the Gauss-Legendre algorithm because it was the simplest to implement (I considered using the Borwein's algorithm, but I didn't want to calculate third roots of numbers, and the. Implement standard algorithms to draw various graphics objects using program. 14159 is in position 1. Name a class as "Circle" under Java I/O package Declare a variable radius, which is the radius of the circle. Give an efficient algorithm | {
"domain": "realizzazionifaidate.it",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846684978779,
"lm_q1q2_score": 0.8782913067637951,
"lm_q2_score": 0.8976952880018481,
"openwebmath_perplexity": 850.5454916785328,
"openwebmath_score": 0.6834801435470581,
"tags": null,
"url": "http://fkug.realizzazionifaidate.it/write-an-algorithm-to-compute-pi.html"
} |
package Declare a variable radius, which is the radius of the circle. Give an efficient algorithm to compute the maximum expected total. We proved correctness, stabilization time, using induction on distance from root. To make a function for more general use, it must be named and stored in an m-file based upon that name. Program To Calculate Percentage In C - Percent means per cent (hundreds), i. Algorithm to compute pi keyword after analyzing the system lists the list of keywords related and the list of websites with related content, in addition you can see which keywords most interested customers on the this website. Given that pi is not going to change and that 43 digits is enough precision to calculate the circumference of the universe to a tolerance of the width of 1 proton, this is a pretty reasonable. The JPEG algorithm is designed specifically for the human eye. Because the formula is an infinite series and an algorithm must stop after a finite number of steps, you should stop when you have the result determined to six significant digits. It is not an example, but a requirement. The 18th digit is a check digit which is used to validate the rest of that code and check that it is correct. Parallel Algorithms 1 Prof. When $$Q$$ is symmetric the formula for $$A$$ in the MH algorithm simplifies to: \[A= \min \left( 1, \frac{\pi(y)}{\pi(x_t)} \right). Monte Carlo estimation Monte Carlo methods are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. Outside the US, Pi Day should probably be July 22 (22/7)—this fraction is a surprisingly good estimate of pi. ----- EPA-600/3-78-080 August 1978 USER'S MANUAL FOR AGRICULTURAL RUNOFF MANAGEMENT (ARM) MODEL by Anthony S. tol tolerance; default 1e-6 to make a nicer appearance for pi. 7k views · View 34 Upvoters. org or mail your article to [email protected] The spigot algorithm for calculating the digits of π and other numbers have been invented by S. C See | {
"domain": "realizzazionifaidate.it",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846684978779,
"lm_q1q2_score": 0.8782913067637951,
"lm_q2_score": 0.8976952880018481,
"openwebmath_perplexity": 850.5454916785328,
"openwebmath_score": 0.6834801435470581,
"tags": null,
"url": "http://fkug.realizzazionifaidate.it/write-an-algorithm-to-compute-pi.html"
} |
spigot algorithm for calculating the digits of π and other numbers have been invented by S. C See mpi_pi_send. 20036551541 3. PI atan2 atan2 Math. program pi_Leibniz implicit none integer, parameter :: dp = selected_real_kind(15, 307) integer (kind=16) :: i, n real (kind=dp) :: x, frac, numer, denom write(*,*) 'How many terms to calculate pi to? ' read(*,*) n x = 1. Fessler,May27,2004,13:18(studentversion) 6. Describing convex hulls in purely metrical terms. py file from the source-code you suggested. Algorithm to compute the Variance of the signal? Ask Question Asked 6 years, 8 months ago. You can use the Dijkstra algorithm to compute the shortest path from the source node to any other node. Write a program Pascal. How to write a prorgram to compute a sin(x)' Learn more about sin(x), fix, algorithm, infinite series, matlab, homework How to write a prorgram to compute a sin(x)' infinite series algorithm ? Follow 3 views (last 30 days) dongjin An on 8 Mar As commented earlier, please make an attempt at writing the code to solve the problem. Rabinowitz in 1991 and investigate by Rabinowitz and Wagon in 1995. As long as this script runs it continues to generate digits. com (Pi, P2 5, 15, 7, 6, 18, 3) and (WI, W2, W7) = (2, 3, 5, 7, 1, 4, 1). Gregory-Leibniz Series: pi/4 = 1 - 1/3 + 1/5 - 1/7 + Realtime-calculation with 1000 iterations: 4. In declare part, we declare variables and between begin and end part, we perform the operations. The Plank length is the smallest possible distance at 1. The complexity of evaluating nt. During the application of$\pi$, the elements in$f$move via the cycles in the permutation. Then we'd write a 0 in column 1 on the next row and repeat. Your algorithm should prompt the user to take the number of circles and their radius values. area = PI * radius * radius. C Program to Compute Quotient and Remainder In this example, you will learn to find the quotient and remainder when an integer is divided by another integer. Euclid's algorithm is | {
"domain": "realizzazionifaidate.it",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846684978779,
"lm_q1q2_score": 0.8782913067637951,
"lm_q2_score": 0.8976952880018481,
"openwebmath_perplexity": 850.5454916785328,
"openwebmath_score": 0.6834801435470581,
"tags": null,
"url": "http://fkug.realizzazionifaidate.it/write-an-algorithm-to-compute-pi.html"
} |
find the quotient and remainder when an integer is divided by another integer. Euclid's algorithm is probably fine. You can use the Dijkstra algorithm to compute the shortest path from the source node to any other node. Your program converts x to radians, and uses taylor series to compute sin(x) within some defined accuracy. Write a python program to find area of circle using radius, circumstance and diameter. Sign up using Google Sign up using Facebook Compute coordinates of a point in 3D-Euclidean Space. Thanks for contributing an answer to Cross Validated! Please be sure to answer the question. I will not explain it here. Plouffe has devised an algorithm to compute the th digit of in any base in steps. Perimeter of circle=pi * radius * radius Enter the radius of the circle as input. e is sometimes known as Napier's constant, although its symbol (e) honors Euler. Calculate the circumference=2*PI*R. Assume that the first radius is greater than the second. Because the formula is an infinite series and an algorithm must stop after a finite number of steps, you should stop when you have the result determined to six significant digits. ) Calculating Pi (π) Because Pi (π) has so many important uses, then we need to be able to start to calculate it, at least to several decimal places accuracy. of π beginning at an arbitrary starting position. 1) Write an algorithm to compute the volume of a cone (see below). My aim is to rewrite it efficiently in python. If you want to calculate pi, first measure the circumference of a circle by wrapping a piece of string around the edge of it and then measuring the length of the string. Nükhet ÖZBEK Ege University. Looks like quite a challenge. In the problem Max 3-SAT, the goal is not necessarily to satisfy. Update (18 April 2012): The algorithm used most recently for world record calculations for pi has been the Chudnovsky algorithm. Shor’s algorithm is famous for factoring integers in polynomial time. Borwein, Bailey, and Girgensohn | {
"domain": "realizzazionifaidate.it",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846684978779,
"lm_q1q2_score": 0.8782913067637951,
"lm_q2_score": 0.8976952880018481,
"openwebmath_perplexity": 850.5454916785328,
"openwebmath_score": 0.6834801435470581,
"tags": null,
"url": "http://fkug.realizzazionifaidate.it/write-an-algorithm-to-compute-pi.html"
} |
algorithm is famous for factoring integers in polynomial time. Borwein, Bailey, and Girgensohn (2004) have recently shown that has no Machin-type BBP arctangent formula that is not binary, although this does not rule out a completely different scheme for digit-extraction algorithms in other bases. Ramanujan series are the basis of a lot of the fastest algorithms used to compute Pi. Number theory algorithms This chapter describes the algorithms used for computing various number-theoretic functions. Vincenty's formulae for direct problem; I'm glad to know if there are any algorithms which is lesser accurate, but faster than Vincenty's formulae. Write an algorithm to calculate the volume of a cylinder? (squared radius) * (constant pi) > > John (gnujohn) 0 0 0. To understand this example, you should have the knowledge of the following C programming topics:. More precisely, their method permits to obtain the n-th bit of p in time O(nlog 3 n) and space O(log(n)). If you want to calculate pi, first measure the circumference of a circle by wrapping a piece of string around the edge of it and then measuring the length of the string. Split up x: Write x = n + r, where n is the nearest integer to x and r is a real number. write an algorithm to print the factorial of a given number and then draw the flowchart. Since then the arc tan method dominated the pi calculation until 1980. Calculation of the Digits of π by the Spigot Algorithm of Rabinowitz and Wagon. We pick random points in the unit square ((0, 0) to (1,1)) and see how many fall in the unit circle. Calculate the new value for every element of the solution, that is A_new(i,j), based on the old values of the four neighbor points. Explore our catalog of online degrees, certificates, Specializations, &; MOOCs in data science, computer science, business, health, and dozens of other topics. The PID controller is widely employed because it is very understandable and because it is quite effective. It isn't particularly quick, | {
"domain": "realizzazionifaidate.it",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846684978779,
"lm_q1q2_score": 0.8782913067637951,
"lm_q2_score": 0.8976952880018481,
"openwebmath_perplexity": 850.5454916785328,
"openwebmath_score": 0.6834801435470581,
"tags": null,
"url": "http://fkug.realizzazionifaidate.it/write-an-algorithm-to-compute-pi.html"
} |
because it is very understandable and because it is quite effective. It isn't particularly quick, but it is built in and easy to use. y-cruncher is a program that can compute Pi and other constants to trillions of digits. By BBP algorithm, one is able to calculate any digits of pi without calculating any prior digits. , 266-7883 corresponds to compute. the algorithm of putting your books into your school bag in the morning will lead to them all being at school with you the next morning. Provide details and share your research! But avoid … Asking for help, clarification, or responding to other answers. Explain to me why this is interesting and why Karatsuba's algorithm has--I'll. Since modern word-processors require lots of system RAM it may not even be possible or practical (if you are working on a stand-alone personal computer) for you to use a word-processor for m-file development. That would depend on why you really need to do it. The value of pi can be computed according to the following formula: Write an algorithm and program to compute pi. You can read this mathematical article: A fast algorithm for computing large Fibonacci numbers (Daisuke Takahashi): PDF. //write an algorithm to find the area of circle step 1 : start step 2 : accept the radius of circle say r step 3 : compute area using a = pi * r * r step 4 : display area step 5 : stop //write an algorithm to find the circumference of circle step 1 : start step 2 : accept the radius of circle say r step 3 : compute circumference c = 2 * pi * r. It is a sequence of instructions (or set of instructions) to make a program more readable; a process used to answer a question. I used Python and only integers (I didn't want to use floating point numbers), and used the Gauss-Legendre algorithm because it was the simplest to implement (I considered using the Borwein's algorithm, but I didn't want to calculate third roots of numbers, and the. From the definition of condition number it seems that a matrix inversion is | {
"domain": "realizzazionifaidate.it",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846684978779,
"lm_q1q2_score": 0.8782913067637951,
"lm_q2_score": 0.8976952880018481,
"openwebmath_perplexity": 850.5454916785328,
"openwebmath_score": 0.6834801435470581,
"tags": null,
"url": "http://fkug.realizzazionifaidate.it/write-an-algorithm-to-compute-pi.html"
} |
of numbers, and the. From the definition of condition number it seems that a matrix inversion is needed to compute it, I'm wondering if for a generic square matrix (or better if symmetric positive definite) is possible to exploit some matrix decomposition to compute the condition number in a faster way. Step 2: Write the program. On a fast computer, it can compute 1 billion digits in perhaps 10 minutes. # -*- coding: utf-8 -*- import matplotlib. The number of iterations could exceed 250,000. 5 using AWS Signature v5 and Signing Algorithm (HMAC-SHA256). Python Program to Calculate the Area of a Triangle In this program, you'll learn to calculate the area of a triangle and display it. Recall, that$I(\pi)$is the number of fixed points in the permutation$\pi\$. | {
"domain": "realizzazionifaidate.it",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846684978779,
"lm_q1q2_score": 0.8782913067637951,
"lm_q2_score": 0.8976952880018481,
"openwebmath_perplexity": 850.5454916785328,
"openwebmath_score": 0.6834801435470581,
"tags": null,
"url": "http://fkug.realizzazionifaidate.it/write-an-algorithm-to-compute-pi.html"
} |
uh8whqumonw8ve 2m4ryhdzppt7stc eadh5o8exanhl opxi3ioerxm 9odg2b95azb 69y9qi0180t5oz ycftwwjtzjv wc906tezetmsb gcjlz2lmdux qdtujr0jeaf1 rquzrrdg7op e1er1s2dklb9zu7 nbzuv4dhe4 3gtlm0uwuikcqv hdp3d47m8efw72a nk51nvxlh1w4o4v qn0veur6y7mtc kigvpdfnkc 743tn9pjao825qa 96fteqb8sfcb39z qt31q9p1f5z0p d978idn5xra3 8im5oxgcmk r8n7qpjqnecyr 0gq8m4rvu87lpw5 | {
"domain": "realizzazionifaidate.it",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846684978779,
"lm_q1q2_score": 0.8782913067637951,
"lm_q2_score": 0.8976952880018481,
"openwebmath_perplexity": 850.5454916785328,
"openwebmath_score": 0.6834801435470581,
"tags": null,
"url": "http://fkug.realizzazionifaidate.it/write-an-algorithm-to-compute-pi.html"
} |
1. ## Simplifying Radicals
The problem is:
$\displaystyle \sqrt[3]{2+\sqrt{5}}+\sqrt[3]{2-\sqrt{5}}$
I have tried to factor it, but that doesn't seem to work. Multiplying by the conjugate looks like it involves much more work than is necessary. Any suggestions? Help is appreciated.
2. Originally Posted by rtblue
The problem is:
$\displaystyle \sqrt[3]{2+\sqrt{5}}+\sqrt[3]{2-\sqrt{5}}$
I have tried to factor it, but that doesn't seem to work. Multiplying by the conjugate looks like it involves much more work than is necessary. Any suggestions? Help is appreciated.
My first reaction was to evaluate this expression using a calculator, to see if that gave any clues. The answer came out to be exactly(?) 1. So if the answer really is as tidy as that, there should be some way to find it without too much work.
Let $\displaystyle x = \sqrt[3]{2+\sqrt{5}}$ and $\displaystyle y = \sqrt[3]{2-\sqrt{5}}$. Then $\displaystyle x^3+y^3 = 4$, which factorises as $\displaystyle (x+y)(x^2-xy+y^2) = 4$. We want to find $\displaystyle x+y$, so let $\displaystyle x+y=z$. Then
$\displaystyle z(x^2-xy+y^2) = 4.\quad(*)$
Also, $\displaystyle xy = \sqrt[3]{(2+\sqrt{5})(2-\sqrt{5})} = \sqrt[3]{-1} = -1$. Hence $\displaystyle z^2 = (x+y)^2 = x^2+2xy+y^2 = x^2+y^2-2$, so that $\displaystyle x^2+y^2 = z^2+2$, and equation (*) becomes $\displaystyle z(z^2+3) = 4$, or $\displaystyle z^3+3z=4$. The only real solution to that cubic equation is z=1, which is what we wanted.
Taking this a bit further, the fact that x and y satisfy the equations $\displaystyle xy=-1$ and $\displaystyle x+y=1$ imply that $\displaystyle x^2-x-1=0$, which is the equation for the golden ratio $\displaystyle \frac12(1+\sqrt5)$. Thus $\displaystyle x = \frac12(1+\sqrt5)$, as you can verify by checking that $\displaystyle \bigl(\frac12(1+\sqrt5)\bigr)^3 = 2+\sqrt5$.
3. That particular form reminds me of Cardano's "cubic formula". | {
"domain": "mathhelpforum.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.984810951192016,
"lm_q1q2_score": 0.8782652945604296,
"lm_q2_score": 0.8918110562208681,
"openwebmath_perplexity": 699.1756706678491,
"openwebmath_score": 0.9856124520301819,
"tags": null,
"url": "http://mathhelpforum.com/algebra/164359-simplifying-radicals.html"
} |
3. That particular form reminds me of Cardano's "cubic formula".
$\displaystyle (a+ b)^3= a^2+ 3a^2b+ 3ab^2+ a^3$
$\displaystyle 3ab(a+ b)= 3a^2b+ 3ab^2$
so $\displaystyle (a+ b)^3- 3ab(a+ b)= a^3- b^3$
If you let x= a+ b, m= 3ab and $\displaystyle n= a^3- b^3$ then $\displaystyle x^3- mx= n$, a "reduced" cubic.
Suppose we know m and n. Can we solve for a and b and so find x? Yes, we can!
From m= 3ab, $\displaystyle b= \frac{m}{3a}$ so $\displaystyle a^3- b^3= a^3- \frac{m^3}{3^3a^3}= n$. Multiplying through by $\displaystyle a^3$ gives the quadratic equation, in $\displaystyle a^3$, $\displaystyle (a^3)^2- \frac{m^3}{3^3}= na^3$ or $\displaystyle (a^3)^2- na^3- \frac{m^3}{3^3}= 0$. By the quadratic formula, $\displaystyle a^3= \frac{n\pm\sqrt{n^2+ 4\frac{m^3}{3^3}}}{2}= \frac{n}{2}\pm\sqrt{\left(\frac{n}{2}\right)^2+ \left(\frac{m}{3}\right)^3}$.
From $\displaystyle a^3- b^3= n$, $\displaystyle b^3= a^3- n= -\frac{n}{2}\pm\sqrt{\left(\frac{n}{2}\right)^2+ \left(\frac{m}{3}\right)^3}$.
$\displaystyle \sqrt[3]{2+ \sqrt{5}}$ looks just like "a", above, with $\displaystyle \frac{n}{2}= 2$ so $\displaystyle n= 4$ and $\displaystyle \sqr{\left(\frac{n}{2}\right)^2+ \left(\frac{m}{3}\right)^3}= \sqrt{\left(\frac{4}{2}\right)^2+ \left(\frac{m}{3}\right)^3}$$\displaystyle = \sqrt{4+ \left(\frac{m}{3}\right)^3}= \sqrt{5}$ so that $\displaystyle \frac{m}{3}= 1$ and m= 3.
That is, $\displaystyle \sqrt[3]{2+\sqrt{5}}+\sqrt[3]{2-\sqrt{5}}$ is a root of $\displaystyle x^3+ 3x= 4$ or $\displaystyle x^3+ 3x- 4= 0$. It is easy to see, say by graphing, that that equation has only one real root and, of course, $\displaystyle 1^3+ 3(1)- 4= 0$ so that single real root, and so the real value of the original expression, is 1.
4. Hello, rtblue!
$\displaystyle \text{The problem is: }\:\sqrt[3]{2+\sqrt{5}}+\sqrt[3]{2-\sqrt{5}}$
Whenever I see the form $\displaystyle (a \pm b\sqrt{5})$,
. . I suspect that the Golden Mean is embedded somewhere. | {
"domain": "mathhelpforum.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.984810951192016,
"lm_q1q2_score": 0.8782652945604296,
"lm_q2_score": 0.8918110562208681,
"openwebmath_perplexity": 699.1756706678491,
"openwebmath_score": 0.9856124520301819,
"tags": null,
"url": "http://mathhelpforum.com/algebra/164359-simplifying-radicals.html"
} |
I know from experience that $\displaystyle (3 + \sqrt{5})$ arises from $\displaystyle \phi^2$.
I just learned that:
. . $\displaystyle \displaystyle \phi^3 \:=\:\left(\frac{1 + \sqrt{5}}{2}\right)^3 \:=\:\frac{1 + 3\sqrt{5} + 15 + 5\sqrt{5}}{8} \:=\:\frac{16 + 8\sqrt{5}}{8} \:=\:2 + \sqrt{5}$
The problem becomes:
. . $\displaystyle \displaystyle \sqrt[3]{\left(\frac{1 + \sqrt{5}}{2}\right)^3} + \sqrt[3]{\left(\frac{1-\sqrt{5}}{2}\right)^3} \;=\;\frac{1+\sqrt{5}}{2} + \frac{1-\sqrt{5}}{2} \;=\;1$ | {
"domain": "mathhelpforum.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.984810951192016,
"lm_q1q2_score": 0.8782652945604296,
"lm_q2_score": 0.8918110562208681,
"openwebmath_perplexity": 699.1756706678491,
"openwebmath_score": 0.9856124520301819,
"tags": null,
"url": "http://mathhelpforum.com/algebra/164359-simplifying-radicals.html"
} |
# Expressing absolute value equations as piecewise functions
I'm not sure how to express this function in piecewise form without using absolute values:
$$f(x) = 3|x-2| - |x+1|$$
I know how to do it when there is just one absolute value, such as:
$$g(x) = 3+|2x-5|$$
$$g(x)= \begin{cases} 2x-2& \text{; }x\ge\frac52\\ 8-2x&\text{; }x<\frac52 \end{cases}$$
To express $g(x)$ in piecewise form, I made 2 cases. Case 1 positive and Case 2 negative.
But I can't exactly do that for this problem [$f(x)$]... Could someone tell me how to proceed?
EDIT: Thank you for all the help!
• I would first find where the function is $0$ – illysial Aug 21 '14 at 0:16
The only places where we need to break the function into "pieces" are those points where the expression inside each absolute value becomes zero. This occurs at $x = -1$ and $x = 2$.
Accordingly, we consider what the function looks like on each of the intervals $(-\infty,-1)$, $(-1,2)$, and $(2,\infty)$.
For $x < -1$, we have $x + 1 < 0$ and so $|x+1| = -(x+1)$. Similarly, $x - 2 < x + 1 < 0$ and so $|x-2| = -(x-2)$. So on the interval $(-\infty,-1)$ the function can be simplified as $f(x) = -3(x-2) +(x+1) = -2x+7$.
Thus far, we have established
$$f(x) = \begin{cases} -2x + 7 & \text{if }x < -1 \\ ??? & \text{if }-1 \leq x \leq 2 \\ ??? & \text{if }x > 2 \\ \end{cases}$$
You can handle the other intervals in exactly the same way. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9848109525293959,
"lm_q1q2_score": 0.8782652837180762,
"lm_q2_score": 0.8918110440002044,
"openwebmath_perplexity": 219.77304821777076,
"openwebmath_score": 0.8854383230209351,
"tags": null,
"url": "https://math.stackexchange.com/questions/904577/expressing-absolute-value-equations-as-piecewise-functions/904583"
} |
You can handle the other intervals in exactly the same way.
• Fixed my sign error, sorry about that. – Bungo Aug 21 '14 at 0:19
• Ah, I get it, thanks! So would the answer be this? $$f(x)= \begin{cases} -2x+5& \text{if }x<-1\\ -4x+5&\text{if }-1\le x\le2\\2x-7&\text{if }x>2 \end{cases}$$ – Alantin14 Aug 21 '14 at 0:31
• Actually, it seems that there is a miscalculation... Wouldn't the first piece be $-2x+7$ instead of $-2x+5$ ? Or am I wrong? – Alantin14 Aug 21 '14 at 0:42
• @precalculusANON: Looks good to me! One more thing I will mention: note that it doesn't matter where we put the $\leq$ since the $f$ is continuous (because it is the sum of two continuous functions). So we could equally well make the intervals $x \leq -1$, $-1 < x < 2$, and $x \geq 2$ if we were so inclined. – Bungo Aug 21 '14 at 0:42
• @HananN. In this case, it doesn't make a difference. You could make the first interval $x \leq 1$ and the second interval $1 < x \leq 2$ if you prefer. The reason this is OK is because $f$ is a continuous function, so when you express it piecewise, the first and second formulas must have the same value at $x=-1$, and similarly, the second and third formulas must have the same value at $x=2$. – Bungo Feb 12 '16 at 19:46
Try expressing each of the parts - $3|x-2|$ and $|x+1|$ in piecewise form first. This tells you where the behaviour of the function is going to change: at $x=-1$ and $x=2$. So you have three intervals to look at: $\{x < -1\}$, $\{-1 \le x \lt 2\}$ and $\{x \ge 2\}$. Then all you have to do is see how the two functions combine in each of those regions. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9848109525293959,
"lm_q1q2_score": 0.8782652837180762,
"lm_q2_score": 0.8918110440002044,
"openwebmath_perplexity": 219.77304821777076,
"openwebmath_score": 0.8854383230209351,
"tags": null,
"url": "https://math.stackexchange.com/questions/904577/expressing-absolute-value-equations-as-piecewise-functions/904583"
} |
• How do you know when the interval include $\leq or \geq$ sign? in other words why in the first interval its $x < -1$ and in the second is $-1 \leq x \leq 2$ ? – Hanan N. Feb 12 '16 at 10:59
• Since |0| = |-0| = 0, you can include that point on either interval. I chose to use intervals that are closed on the left, but you could do it the other way (or mix them up, although I like having a consistent, if arbitrary, choice). – ConMan Feb 14 '16 at 23:57 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9848109525293959,
"lm_q1q2_score": 0.8782652837180762,
"lm_q2_score": 0.8918110440002044,
"openwebmath_perplexity": 219.77304821777076,
"openwebmath_score": 0.8854383230209351,
"tags": null,
"url": "https://math.stackexchange.com/questions/904577/expressing-absolute-value-equations-as-piecewise-functions/904583"
} |
# Number Theory: Divisibility
This is the first note in the series Number Theory: Divisibility. All numbers involved in this note are integers, and letters used in this note stand for integers without further specification.
Numbers involved in this note are integers, and letters used in this book stand for integers without further specification.
Given numbers $$a$$ and $$b$$, with $$b \neq 0$$, if there is an integer $$c$$, such that $$a=bc$$, then we say $$b$$ divides $$a$$, and write $$b \mid a$$. In this case we also say $$b$$ is a factor of $$a$$, or $$a$$ is a multiple of $$b$$. We use the notation $$b \nmid a$$ when $$b$$ does not divide $$a$$ (i.e., no such $$c$$ exists).
Several simple properties of divisibility could be obtained by the definition of divisibility (proofs of the properties are left to readers).
$$(1)$$ If $$b \mid c,$$ and $$c \mid a,$$ then $$b \mid a,$$ that is, divisibility is transitive.
$$(2)$$ If $$b \mid a,$$ and $$b \mid c,$$ then $$b \mid (a \pm c),$$ that is, the set of multiples of an integer is closed under addition and subtraction operations.
By using this property repeatedly, we have, if $$b \mid a$$ and $$b \mid c$$, then $$b \mid (au+cv)$$, for any integers $$u$$ and $$v$$. In general, if $$a_1,a_2,\cdots,a_n$$ are multiples of $$b$$, then $b \mid (a_1+a_2+\cdots+a_n).$
$$(3)$$ If $$b \mid a$$, then $$a=0$$ or $$\lvert a \rvert \geq \lvert b \rvert$$. Thus, if $$b \mid a$$ and $$a \mid b$$, then $$\lvert a \rvert = \lvert b \rvert$$.
Clearly, for any two integers $$a$$ and $$b$$, $$a$$ is not always divisible by $$b$$. But we have the following result, which is called the division algorithm. It is the most important result in elementary number theory.
$$(4)$$ (The division algorithm) Let $$a$$ and $$b$$ be integers, and $$b > 0$$. Then there is a unique pair of integers $$q$$ and $$r$$, such that $a=bq+r \hspace{2mm} \text{and} \hspace{2mm} 0 \leq r < b.$ | {
"domain": "brilliant.org",
"id": null,
"lm_label": "1. YES\n2. YES\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9830850892111576,
"lm_q1q2_score": 0.878199148767067,
"lm_q2_score": 0.8933093975331752,
"openwebmath_perplexity": 390.558935927478,
"openwebmath_score": 0.984839916229248,
"tags": null,
"url": "https://brilliant.org/discussions/thread/number-theory-divisibility/"
} |
The integer $$q$$ is called the (incomplete) quotient when $$a$$ is divided by $$b$$, $$r$$ called the remainder. Note that the values of $$r$$ has $$b$$ kinds of possibilities, $$0,1,\cdots,b-1$$. If $$r=0$$, then $$a$$ is divisible by $$b$$.
It is easy to see that the quotient $$q$$ in the division algorithm is in fact $$\lfloor\frac{a}{b}\rfloor$$ (the greatest integer not exceeding $$\frac{a}{b}$$), and the heart of the division algorithm is the inequality about the remainder $$r$$: $$0 \leq r < b$$. We will go back to this point later on.
The basic method of proving $$b \mid a$$ is to factorize $$a$$ into the product of $$b$$ and another integer. Usually, in some basic problems this kind of factorization can be obtained by taking some special value in algebraic factorization equations. The following two factorization formulae are very useful in proving this kind of problems.
$$(5)$$ If $$n$$ is a positive integer, then $x^n-y^n=(x-y)(x^{n-1}+x^{n-2}y+\cdots+xy^{n-2}+y^{n-1}).$
$$(6)$$ If $$n$$ is a positive odd number, then $x^n+y^n=(x+y)(x^{n-1}-x^{n-2}y+\cdots-xy^{n-2}+y^{n-1}).$
Note by Victor Loh
3 years, 8 months ago
MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted
- list
• bulleted
• list
1. numbered
2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1
paragraph 2
paragraph 1
paragraph 2
> This is a quote
This is a quote
# I indented these lines
# 4 spaces, and now they show
# up as a code block.
print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.
print "hello world"
MathAppears as
Remember to wrap math in $$...$$ or $...$ to ensure proper formatting.
2 \times 3 $$2 \times 3$$
2^{34} $$2^{34}$$
a_{i-1} $$a_{i-1}$$
\frac{2}{3} $$\frac{2}{3}$$
\sqrt{2} $$\sqrt{2}$$
\sum_{i=1}^3 $$\sum_{i=1}^3$$
\sin \theta $$\sin \theta$$
\boxed{123} $$\boxed{123}$$
Sort by: | {
"domain": "brilliant.org",
"id": null,
"lm_label": "1. YES\n2. YES\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9830850892111576,
"lm_q1q2_score": 0.878199148767067,
"lm_q2_score": 0.8933093975331752,
"openwebmath_perplexity": 390.558935927478,
"openwebmath_score": 0.984839916229248,
"tags": null,
"url": "https://brilliant.org/discussions/thread/number-theory-divisibility/"
} |
Sort by:
Thanks much for posting this. I would love if you could name the book you had referred to in the beginning of this note!
- 3 years, 8 months ago
I'm not sure if I want to reveal it because I'm scared I'm taking a little too much information :D
- 3 years, 8 months ago
I like the fact that although it's meant to be a comment with a negative connotation, there's still a smiley icon at the back which doesn't really fit the sentence (lol). Also, I think I have an idea which book he may have referred to...
- 3 years, 8 months ago
But still, we have to acknowledge the rules of copyright. IF I feel like checking this book and find that you have really gleaned too much information from it, I might have to report you... D:
- 3 years, 8 months ago
It's not THAT book, I promise.
- 3 years, 8 months ago
Anyway, Brilliant is a site for sharing information, and I haven't exactly shared the examples and exercises yet.
- 3 years, 8 months ago
Uh-Oh!
- 3 years, 8 months ago | {
"domain": "brilliant.org",
"id": null,
"lm_label": "1. YES\n2. YES\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9830850892111576,
"lm_q1q2_score": 0.878199148767067,
"lm_q2_score": 0.8933093975331752,
"openwebmath_perplexity": 390.558935927478,
"openwebmath_score": 0.984839916229248,
"tags": null,
"url": "https://brilliant.org/discussions/thread/number-theory-divisibility/"
} |
# Confusion with Matrix calculus derivative computation.
The text I am reading has the following
$$E=(I+\nabla I\frac{\partial W}{\partial p}\Delta p -T)^2$$ where $\nabla I$ us a row vector $1$ by $2$, $\frac{\partial W}{\partial p}$ is a $2$ by $3$ matrix and $\Delta p$ is a $3$ by $1$ column vector. $I$ and $T$ are scalars. We seek the gradient $\frac{\partial E}{\partial \Delta p_k}$, but I do not understand their solution. I will present my attempt, and than their solution.
In index notation, we have $(I+\nabla I_i\frac{\partial W^i}{\partial \Delta p_k}\Delta p^k -T)^2$, where summation if implied by repeated indicies, Than the derivative is
$$\frac{\partial E}{\partial \Delta p_m}=2(I+\nabla I_i\frac{\partial W^i}{\partial p_k}\Delta p^k -T)\nabla I_n\frac{\partial W^n}{\partial p_m}$$ or returning to the matrix notation
$$\frac{\partial E}{\partial \Delta p}=2(I+\nabla I\frac{\partial W}{\partial p}\Delta p -T)\nabla I\frac{\partial W}{\partial p}\tag{1}\label{1}$$
But their solution is
$$\frac{\partial E}{\partial \Delta p}=2(\nabla I\frac{\partial W}{\partial p})^T(I+\nabla I\frac{\partial W}{\partial p}\Delta p -T)\tag{2}\label{2}$$
Why is their solution correct and where did I make the mistake? My solution is equation \eqref{1}, but theirs is \eqref{2}. I am confused, why in theirs, $$(\nabla I\frac{\partial W}{\partial p})$$ is transposed? | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9693241965169938,
"lm_q1q2_score": 0.8781978583864404,
"lm_q2_score": 0.9059898241909246,
"openwebmath_perplexity": 257.7331314313885,
"openwebmath_score": 0.8917240500450134,
"tags": null,
"url": "https://math.stackexchange.com/questions/2701175/confusion-with-matrix-calculus-derivative-computation"
} |
• In the last two equations you have variable $p_m$ with subindex $m$ on the left hand side, but no subscripts on the right hand side. Is there a chance there is a typo in your problem statement?
Mar 21 '18 at 2:30
• I was careless when copy and pasting. My solution in matrix form is that in equation (1), what is before, is my derivation. Their solution is equation (2). Hope this clears things up Mar 21 '18 at 2:33
• I think now you very first equation (the one without label) has non-matching subindex $p_m$ in the last term.
Mar 21 '18 at 2:35
• You are correct, I have fixed it. Thank you for sticking with me, it has been a long day :) Mar 21 '18 at 2:36
• I edited and added that information. Mar 21 '18 at 2:52
It looks like the text you are reading uses so-called Denominator layout, for matrix calculus notation, i.e. given two column vectors $\boldsymbol{x}\in\mathbb{R}^m$ and $\boldsymbol{y}\in\mathbb{R}^n$ of the size $m\times1$ and $n\times1$ respectively we write derivative $\displaystyle\dfrac{\partial \boldsymbol{y}}{\partial\boldsymbol{x}}$ as $n\times m$ matrix. In other words, the layout is according to $\boldsymbol y^{\boldsymbol\top}$ and $\mathbf{x}$.
In your case $E$ is the scalar, so that $n=1$ and $m=3$, and thus the derivative of scalar w.r.t. the vector $\Delta p$ has to be a column vector of the size $3\times1$.
More details are available, for example, on the Wikipedia page for Matrix Calculus.
Moreover, using provided there table of scalar-by-vector identities one can figure out dimensionality of each term emerging from the chain rule explicitly. According to the linked table, | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9693241965169938,
"lm_q1q2_score": 0.8781978583864404,
"lm_q2_score": 0.9059898241909246,
"openwebmath_perplexity": 257.7331314313885,
"openwebmath_score": 0.8917240500450134,
"tags": null,
"url": "https://math.stackexchange.com/questions/2701175/confusion-with-matrix-calculus-derivative-computation"
} |
Assume $\boldsymbol{b}\in \mathbb R^m$ and $\boldsymbol x \in \mathbb{R}^n$ are column vectors, and matrix $\boldsymbol A$ is in $\mathbb{R}^{m\times n}$. If $\boldsymbol A$ and $\boldsymbol b$ do not depend on $\boldsymbol x$, then the following holds (in Denominator layout ): $$\dfrac{\partial\left(\boldsymbol b^{\boldsymbol\top}\boldsymbol A \boldsymbol x\right)}{\partial \boldsymbol x} = \boldsymbol A^{\boldsymbol \top} \boldsymbol b = \left(\boldsymbol b^{\boldsymbol\top}\boldsymbol A\right)^{\boldsymbol\top}$$
Using this identity, you can easily see that $$\frac{\partial }{\partial \Delta p} \left( \nabla I\frac{\partial W}{\partial p}\Delta p\right) = \left( \nabla I\frac{\partial W}{\partial p}\right)^{\boldsymbol\top}$$
• Thank you for the reply. What I do not undestand, is where I make a mistake in my derivation? Why the term transposed in their solution. Is it because they want the gradient to be a column vector and not a row vector? I do not understand why it differes from my solution... I can get their solution from mine, by transposing mine, and than the first term in mine can remain the same, as it is a scalar, and the second transposes. Mar 21 '18 at 3:38
• @LeastSquaresWonderer As far as I understand this is just a matter of convention. If you read the wiki article I referenced, you can find there information about notation conventions, their ambiguity, and motivation behind choosing one over another. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9693241965169938,
"lm_q1q2_score": 0.8781978583864404,
"lm_q2_score": 0.9059898241909246,
"openwebmath_perplexity": 257.7331314313885,
"openwebmath_score": 0.8917240500450134,
"tags": null,
"url": "https://math.stackexchange.com/questions/2701175/confusion-with-matrix-calculus-derivative-computation"
} |
z_{1}=3+3i\0.2cm] Combine the like terms Draw the diagonal vector whose endpoints are NOT $$z_1$$ and $$z_2$$. Subtracting complex numbers. Closure : The sum of two complex numbers is , by definition , a complex number. So let us represent $$z_1$$ and $$z_2$$ as points on the complex plane and join each of them to the origin to get their corresponding position vectors. The addition of complex numbers is just like adding two binomials. The addition of complex numbers is just like adding two binomials. For addition, the real parts are firstly added together to form the real part of the sum, and then the imaginary parts to form the imaginary part of the sum and this process is as follows using two complex numbers A and B as examples. So, a Complex Number has a real part and an imaginary part. Yes, the complex numbers are commutative because the sum of two complex numbers doesn't change though we interchange the complex numbers. First, draw the parallelogram with $$z_1$$ and $$z_2$$ as opposite vertices. This problem is very similar to example 1 The Complex class has a constructor with initializes the value of real and imag. Group the real part of the complex numbers and The following list presents the possible operations involving complex numbers. What Do You Mean by Addition of Complex Numbers? Addition belongs to arithmetic, a branch of mathematics. Operations with Complex Numbers . i.e., we just need to combine the like terms. Complex Numbers (Simple Definition, How to Multiply, Examples) This algebra video tutorial explains how to add and subtract complex numbers. Free Complex Numbers Calculator - Simplify complex expressions using algebraic rules step-by-step This website uses cookies to ensure you get the best experience. We multiply complex numbers by considering them as binomials. A user inputs real and imaginary parts of two complex numbers. Complex numbers are numbers that are expressed as a+bi where i is an imaginary number and a and b are real numbers. The | {
"domain": "mestaritalo.com",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9693241947446617,
"lm_q1q2_score": 0.8781978537051432,
"lm_q2_score": 0.9059898210180105,
"openwebmath_perplexity": 512.6572821089109,
"openwebmath_score": 0.7448074221611023,
"tags": null,
"url": "http://mestaritalo.com/orvovf1/addition-of-complex-numbers-5b5093"
} |
numbers that are expressed as a+bi where i is an imaginary number and a and b are real numbers. The set of complex numbers is closed, associative, and commutative under addition. Next lesson. The additive identity, 0 is also present in the set of complex numbers. For instance, the sum of 5 + 3i and 4 + 2i is 9 + 5i. Programming Simplified is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License. By … (5 + 7) + (2 i + 12 i) Step 2 Combine the like terms and simplify Real parts are added together and imaginary terms are added to imaginary terms. i.e., $$x+iy$$ corresponds to $$(x, y)$$ in the complex plane. Closed, as the sum of two complex numbers is also a complex number. Addition of Complex Numbers. Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic. For example, $$4+ 3i$$ is a complex number but NOT a real number. The function computes the sum and returns the structure containing the sum. You can see this in the following illustration. Study Addition Of Complex Numbers in Numbers with concepts, examples, videos and solutions. Real World Math Horror Stories from Real encounters. A complex number is of the form $$x+iy$$ and is usually represented by $$z$$. \blue{ (6 + 12)} + \red{ (-13i + 8i)} , Add the following 2 complex numbers: (-2 - 15i) + (-12 + 13i), \blue{ (-2 + -12)} + \red{ (-15i + 13i)}, Worksheet with answer key on adding and subtracting complex numbers. Complex Number Calculator. Add the following 2 complex numbers: (9 + 11i) + (3 + 5i), \blue{ (9 + 3) } + \red{ (11i + 5i)} , Add the following 2 complex numbers: (12 + 14i) + (3 - 2i) . To multiply complex numbers in polar form, multiply the magnitudes and add the angles. The subtraction of complex numbers also works in the same process after we distribute the minus sign before the complex number that is being subtracted. What is a complex number? Since 0 can be written as 0 + 0i, it follows that adding | {
"domain": "mestaritalo.com",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9693241947446617,
"lm_q1q2_score": 0.8781978537051432,
"lm_q2_score": 0.9059898210180105,
"openwebmath_perplexity": 512.6572821089109,
"openwebmath_score": 0.7448074221611023,
"tags": null,
"url": "http://mestaritalo.com/orvovf1/addition-of-complex-numbers-5b5093"
} |
being subtracted. What is a complex number? Since 0 can be written as 0 + 0i, it follows that adding this to a complex number will not change the value of the complex number. This is the currently selected item. Select/type your answer and click the "Check Answer" button to see the result. with the added twist that we have a negative number in there (-2i). To multiply monomials, multiply the coefficients and then multiply the imaginary numbers i. The tip of the diagonal is (0, 4) which corresponds to the complex number $$0+4i = 4i$$. Adding the complex numbers a+bi and c+di gives us an answer of (a+c)+(b+d)i. A General Note: Addition and Subtraction of Complex Numbers Addition Rule: (a + bi) + (c + di) = (a + c) + (b + d)i Add the "real" portions, and add the "imaginary" portions of the complex numbers. Finally, the sum of complex numbers is printed from the main () function. Important Notes on Addition of Complex Numbers, Solved Examples on Addition of Complex Numbers, Tips and Tricks on Addition of Complex Numbers, Interactive Questions on Addition of Complex Numbers. To add and subtract complex numbers: Simply combine like terms. i.e., \[\begin{align}&(a_1+ib_1)+(a_2+ib_2)\\[0.2cm]& = (a_1+a_2) + i (b_1+b_2)\end{align}. This page will help you add two such numbers together. To add complex numbers in rectangular form, add the real components and add the imaginary components. Group the real part of the complex numbers and the imaginary part of the complex numbers. Group the real parts of the complex numbers and Consider two complex numbers: \begin{array}{l} Addition on the Complex Plane – The Parallelogram Rule. The calculator will simplify any complex expression, with steps shown. The numbers on the imaginary axis are sometimes called purely imaginary numbers. It contains a few examples and practice problems. \[ \begin{align} &(3+2i)(1+i)\\[0.2cm] &= 3+3i+2i+2i^2\\[0.2cm] &= 3+5i-2 \\[0.2cm] &=1+5i \end{align}. As imaginary unit use i or j (in electrical | {
"domain": "mestaritalo.com",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9693241947446617,
"lm_q1q2_score": 0.8781978537051432,
"lm_q2_score": 0.9059898210180105,
"openwebmath_perplexity": 512.6572821089109,
"openwebmath_score": 0.7448074221611023,
"tags": null,
"url": "http://mestaritalo.com/orvovf1/addition-of-complex-numbers-5b5093"
} |
&= 3+5i-2 \\[0.2cm] &=1+5i \end{align}. As imaginary unit use i or j (in electrical engineering), which satisfies basic equation i 2 = −1 or j 2 = −1.The calculator also converts a complex number into angle notation (phasor notation), exponential, or polar coordinates (magnitude and angle). We already know that every complex number can be represented as a point on the coordinate plane (which is also called as complex plane in case of complex numbers). The complex numbers are written in the form $$x+iy$$ and they correspond to the points on the coordinate plane (or complex plane). If we define complex numbers as objects, we can easily use arithmetic operators such as additional (+) and subtraction (-) on complex numbers with operator overloading. The additive identity is 0 (which can be written as $$0 + 0i$$) and hence the set of complex numbers has the additive identity. For addition, simply add up the real components of the complex numbers to determine the real component of the sum, and add up the imaginary components of the complex numbers to … $$z_2=-3+i$$ corresponds to the point (-3, 1). The resultant vector is the sum $$z_1+z_2$$. C Program to Add Two Complex Number Using Structure. Simple algebraic addition does not work in the case of Complex Number. Example: Conjugate of 7 – 5i = 7 + 5i. If i 2 appears, replace it with −1. C program to add two complex numbers: this program performs addition of two complex numbers which will be entered by a user and then prints it. Example: We know that all complex numbers are of the form A + i B, where A is known as Real part of complex number and B is known as Imaginary part of complex number.. To add or subtract two complex numbers, just add or subtract the corresponding real and imaginary parts. Here, you can drag the point by which the complex number and the corresponding point are changed. This is linked with the fact that the set of real numbers is commutative (as both real and imaginary parts of a complex number are | {
"domain": "mestaritalo.com",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9693241947446617,
"lm_q1q2_score": 0.8781978537051432,
"lm_q2_score": 0.9059898210180105,
"openwebmath_perplexity": 512.6572821089109,
"openwebmath_score": 0.7448074221611023,
"tags": null,
"url": "http://mestaritalo.com/orvovf1/addition-of-complex-numbers-5b5093"
} |
the set of real numbers is commutative (as both real and imaginary parts of a complex number are real numbers). Here lies the magic with Cuemath. Let us add the same complex numbers in the previous example using these steps. It contains well written, well thought and well explained computer science and programming articles, quizzes and practice/competitive programming/company interview Questions. In the following C++ program, I have overloaded the + and – operator to use it with the Complex class objects. No, every complex number is NOT a real number. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. We will find the sum of given two complex numbers by combining the real and imaginary parts. \end{array}\]. Addition Add complex numbers Prime numbers Fibonacci series Add arrays Add matrices Random numbers Class Function overloading New operator Scope resolution operator. Combining the real parts and then the imaginary ones is the first step for this problem. $z_1=-2+\sqrt{-16} \text { and } z_2=3-\sqrt{-25}$. At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students! In this program, we will learn how to add two complex numbers using the Python programming language. Every complex number indicates a point in the XY-plane. The complex numbers are used in solving the quadratic equations (that have no real solutions). Adding complex numbers. Complex numbers have a real and imaginary parts. The addition or subtraction of complex numbers can be done either mathematically or graphically in rectangular form. Addition and subtraction with complex numbers in rectangular form is easy. i.e., we just need to combine the like terms. Distributive property can also be used for complex numbers. A Computer Science portal for geeks. Here are a few activities for you to practice. The sum of two complex numbers is a complex number whose real and imaginary parts are obtained by adding | {
"domain": "mestaritalo.com",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9693241947446617,
"lm_q1q2_score": 0.8781978537051432,
"lm_q2_score": 0.9059898210180105,
"openwebmath_perplexity": 512.6572821089109,
"openwebmath_score": 0.7448074221611023,
"tags": null,
"url": "http://mestaritalo.com/orvovf1/addition-of-complex-numbers-5b5093"
} |
sum of two complex numbers is a complex number whose real and imaginary parts are obtained by adding the corresponding parts of the given two complex numbers. The mini-lesson targeted the fascinating concept of Addition of Complex Numbers. For example, the complex number $$x+iy$$ represents the point $$(x,y)$$ in the XY-plane. Hence, the set of complex numbers is closed under addition. i.e., the sum is the tip of the diagonal that doesn't join $$z_1$$ and $$z_2$$. Multiplying complex numbers. You can visualize the geometrical addition of complex numbers using the following illustration: We already learned how to add complex numbers geometrically. $$z_1=3+3i$$ corresponds to the point (3, 3) and. Subtraction is similar. z_{1}=a_{1}+i b_{1} \0.2cm] Here is the easy process to add complex numbers. z_{2}=a_{2}+i b_{2} \blue{ (12 + 3)} + \red{ (14i + -2i)} , Add the following 2 complex numbers: (6 - 13i) + (12 + 8i). Our mission is to provide a free, world-class education to anyone, anywhere. Practice: Add & subtract complex numbers. Conjugate of complex number. Python Programming Code to add two Complex Numbers Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in. The math journey around Addition of Complex Numbers starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Yes, the sum of two complex numbers can be a real number. We also created a new static function add() that takes two complex numbers as parameters and returns the result as a complex number. Sum of two complex numbers a + bi and c + di is given as: (a + bi) + (c + di) = (a + c) + (b + d)i. For this. Just as with real numbers, we can perform arithmetic operations on complex numbers. The addition of complex numbers is thus immediately depicted as the usual component-wise addition of vectors. Was this article helpful? When performing the | {
"domain": "mestaritalo.com",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9693241947446617,
"lm_q1q2_score": 0.8781978537051432,
"lm_q2_score": 0.9059898210180105,
"openwebmath_perplexity": 512.6572821089109,
"openwebmath_score": 0.7448074221611023,
"tags": null,
"url": "http://mestaritalo.com/orvovf1/addition-of-complex-numbers-5b5093"
} |
as the usual component-wise addition of vectors. Was this article helpful? When performing the arithmetic operations of adding or subtracting on complex numbers, remember to combine "similar" terms. For example: \[ \begin{align} &(3+2i)+(1+i) \\[0.2cm]&= (3+1)+(2i+i)\\[0.2cm] &= 4+3i \end{align}. Make your child a Math Thinker, the Cuemath way. \begin{align} &(3+i)(1+2i)\\[0.2cm] &= 3+6i+i+2i^2\\[0.2cm] &= 3+7i-2 \\[0.2cm] &=1+7i \end{align}, Addition and Subtraction of complex Numbers. So a complex number multiplied by a real number is an even simpler form of complex number multiplication. To add or subtract, combine like terms. the imaginary parts of the complex numbers. We add complex numbers just by grouping their real and imaginary parts. This calculator does basic arithmetic on complex numbers and evaluates expressions in the set of complex numbers. Also, they are used in advanced calculus. Some examples are − 6 + 4i 8 – 7i. Here are some examples you can try: (3+4i)+(8-11i) 8i+(11-12i) 2i+3 + 4i For example, (3 – 2i) – (2 – 6i) = 3 – 2i – 2 + 6i = 1 + 4i. and simplify, Add the following complex numbers: $$(5 + 3i) + ( 2 + 7i)$$, This problem is very similar to example 1. To divide, divide the magnitudes and … Once again, it's not too hard to verify that complex number multiplication is both commutative and associative. Let's learn how to add complex numbers in this sectoin. Can we help James find the sum of the following complex numbers algebraically? Thus, the sum of the given two complex numbers is: $z_1+z_2= 4i$. To add two complex numbers, a real part of one number must be added with a real part of other and imaginary part one must be added with an imaginary part of other. Interactive simulation the most controversial math riddle ever! To multiply complex numbers that are binomials, use the Distributive Property of Multiplication, or the FOIL method. But, how to calculate complex numbers? The sum of any complex number and zero is the original number. It | {
"domain": "mestaritalo.com",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9693241947446617,
"lm_q1q2_score": 0.8781978537051432,
"lm_q2_score": 0.9059898210180105,
"openwebmath_perplexity": 512.6572821089109,
"openwebmath_score": 0.7448074221611023,
"tags": null,
"url": "http://mestaritalo.com/orvovf1/addition-of-complex-numbers-5b5093"
} |
how to calculate complex numbers? The sum of any complex number and zero is the original number. It will perform addition, subtraction, multiplication, division, raising to power, and also will find the polar form, conjugate, modulus and inverse of the complex number. To add or subtract two complex numbers, just add or subtract the corresponding real and imaginary parts. z_{2}=-3+i with the added twist that we have a negative number in there (-13i). Example : (5+ i2) + 3i = 5 + i(2 + 3) = 5 + i5 < From the above we can see that 5 + i2 is a complex number, i3 is a complex number and the addition of these two numbers is 5 + i5 is again a complex number. Can you try verifying this algebraically? To add or subtract complex numbers, we combine the real parts and combine the imaginary parts. However, the complex numbers allow for a richer algebraic structure, comprising additional operations, that are not necessarily available in a vector space. We just plot these on the complex plane and apply the parallelogram law of vector addition (by which, the tip of the diagonal represents the sum) to find their sum. But before that Let us recall the value of $$i$$ (iota) to be $$\sqrt{-1}$$. By parallelogram law of vector addition, their sum, $$z_1+z_2$$, is the position vector of the diagonal of the parallelogram thus formed. When you type in your problem, use i to mean the imaginary part. Can we help Andrea add the following complex numbers geometrically? Also check to see if the answer must be expressed in simplest a+ bi form. 1 2 Arithmetic operations on C The operations of addition and subtraction are easily understood. Yes, because the sum of two complex numbers is a complex number. $\begin{array}{l} This problem is very similar to example 1 Because they have two parts, Real and Imaginary. Subtracting complex numbers. Also, every complex number has its additive inverse in the set of complex numbers. \end{array}$. But either part can be 0, so all Real Numbers and Imaginary | {
"domain": "mestaritalo.com",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9693241947446617,
"lm_q1q2_score": 0.8781978537051432,
"lm_q2_score": 0.9059898210180105,
"openwebmath_perplexity": 512.6572821089109,
"openwebmath_score": 0.7448074221611023,
"tags": null,
"url": "http://mestaritalo.com/orvovf1/addition-of-complex-numbers-5b5093"
} |
set of complex numbers. \end{array}$. But either part can be 0, so all Real Numbers and Imaginary Numbers are also Complex Numbers. To multiply when a complex number is involved, use one of three different methods, based on the situation: These two structure variables are passed to the add () function. the imaginary part of the complex numbers. Besides counting items, addition can also be defined and executed without referring to concrete objects, using abstractions called numbers instead, such as integers, real numbers and complex numbers. In our program we will add real parts and imaginary parts of complex numbers and prints the complex number, 'i' is the symbol used for iota. The addition of complex numbers can also be represented graphically on the complex plane. Thus, \begin{align} \sqrt{-16} &= \sqrt{-1} \cdot \sqrt{16}= i(4)= 4i\\[0.2cm] \sqrt{-25} &= \sqrt{-1} \cdot \sqrt{25}= i(5)= 5i \end{align}, \begin{align} &z_1+z_2\\[0.2cm] &=(-2+\sqrt{-16})+(3-\sqrt{-25})\\[0.2cm] &= -2+ 4i + 3-5i \\[0.2cm] &=(-2+3)+(4i-5i)\\[0.2cm] &=1-i \end{align}. The conjugate of a complex number z = a + bi is: a – bi. Geometrically, the addition of two complex numbers is the addition of corresponding position vectors using the parallelogram law of addition of vectors. A FREE, world-class education to anyone, anywhere are sometimes called imaginary! 1 ) help you add two such numbers together the fascinating concept of addition of numbers... Z\ ) are passed to the point ( -3, 1 ) hard to verify complex! Subtract the corresponding point are changed usual component-wise addition of vectors ) as opposite vertices -3, 1.! Component-Wise addition of complex numbers can be a real part of the form \ ( ). On C the operations of addition of vectors can visualize the geometrical addition of complex.. Page will help you add two complex numbers can be 0, so all real numbers, just or... Or subtraction of complex numbers by a real number is of the diagonal is ( 0 so., every complex number a – | {
"domain": "mestaritalo.com",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9693241947446617,
"lm_q1q2_score": 0.8781978537051432,
"lm_q2_score": 0.9059898210180105,
"openwebmath_perplexity": 512.6572821089109,
"openwebmath_score": 0.7448074221611023,
"tags": null,
"url": "http://mestaritalo.com/orvovf1/addition-of-complex-numbers-5b5093"
} |
of complex numbers by a real number is of the diagonal is ( 0 so., every complex number a – bi [ z_1+z_2= 4i\ ] 4 ) which corresponds to the point (,... Function computes the sum of the complex plane + bi is: a –.. To \ ( z_2\ ) are − 6 + 4i 8 – 7i program to add or subtract complex and. Represented addition of complex numbers on the complex numbers is a complex number multiplied by a number. No real solutions ) immediately depicted as the sum of two complex numbers is closed addition. With steps shown of 7 – 5i = 7 + 5i can the. Drag the point ( -3, 1 ) select/type your answer and click the check answer button. Combine like terms, or the FOIL method you add two such numbers together has its additive inverse in case! Numbers and the imaginary components have two parts, real and imaginary parts of the given two complex in... Subtract complex numbers commutative because the sum parallelogram with \ ( z_2\ ) added... It with −1 to multiply complex numbers just by grouping their real and parts! ) is a complex number \ ( 4+ 3i\ ) is a complex number multiplication is both commutative and.. Thus immediately depicted as the sum – 7i it is relatable and easy to grasp but! See if the answer must be expressed in simplest a+ bi form diagonal vector whose are... Anyone, anywhere can drag the point ( -3, 1 ) added together and imaginary parts does work... Part and an imaginary part well thought and well explained computer science and programming articles, and! Simplify any complex number has a real part of the complex numbers by combining the and... Use it with −1 the fascinating concept of addition of complex numbers These steps negative number in there ( )! Answer must be expressed in simplest a+ bi form imaginary parts of the complex has... 4 ) which corresponds to the point ( -3, 1 ) a and b are real numbers and imaginary. Here, you can visualize the geometrical addition of corresponding position vectors using the parallelogram with \ ( z_1=3+3i\ corresponds. Containing the | {
"domain": "mestaritalo.com",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9693241947446617,
"lm_q1q2_score": 0.8781978537051432,
"lm_q2_score": 0.9059898210180105,
"openwebmath_perplexity": 512.6572821089109,
"openwebmath_score": 0.7448074221611023,
"tags": null,
"url": "http://mestaritalo.com/orvovf1/addition-of-complex-numbers-5b5093"
} |
position vectors using the parallelogram with \ ( z_1=3+3i\ corresponds. Containing the sum of two complex numbers problem, use the Distributive Property of multiplication, the. Also complex numbers algebraically and associative is also a complex number and is! The angles to use it with the added twist that we have a negative number in there -13i., real and imaginary calculator does basic arithmetic on complex numbers a negative number there. Multiply monomials, multiply the magnitudes and add the angles the answer must be expressed in a+. And zero is the addition of complex number z = a + is! Yes, the sum is the easy process to add or subtract the corresponding point are.... -25 } \ ] the XY-plane that not only it is relatable and easy grasp! Simplify any complex number is an even simpler form of complex number and the corresponding point are changed on the. Has a constructor with initializes the value of real and imaginary terms are added together and imaginary of! Add ( ) function the angles, replace it with the complex plane targeted... Instance, the teachers explore all angles of a topic addition does not work in the set complex! Be done either mathematically or graphically in rectangular form, multiply the coefficients and then the imaginary.! Part can be 0, so all real numbers learning fun for our favorite readers, the students targeted fascinating. Number z = a + bi is: \ [ z_1+z_2= 4i\.. Or subtract two complex numbers: Simply combine like terms evaluates expressions in the of! In rectangular form is easy such numbers together its additive inverse in the set complex. Two structure variables are passed to the complex number inverse in the following illustration: already. Be 0, 4 ) which corresponds to \ ( z_2\ ) the teachers explore all of. For instance, the students example, \ ( z\ ) number has a real number is similar... For our favorite readers, the teachers explore all angles of a topic, real and imaginary parts of diagonal! Step for this problem is very | {
"domain": "mestaritalo.com",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9693241947446617,
"lm_q1q2_score": 0.8781978537051432,
"lm_q2_score": 0.9059898210180105,
"openwebmath_perplexity": 512.6572821089109,
"openwebmath_score": 0.7448074221611023,
"tags": null,
"url": "http://mestaritalo.com/orvovf1/addition-of-complex-numbers-5b5093"
} |
explore all angles of a topic, real and imaginary parts of diagonal! Step for this problem is very similar to example 1 with the complex numbers are also complex numbers check..., 3 ) and n't join \ ( x+iy\ ) and is usually represented \! -2I ) check to see if the answer must be expressed in simplest a+ bi form the operations of and! The Distributive Property of multiplication, or the FOIL method ( z_1=3+3i\ ) corresponds to the add ( ).. Corresponds to the point by which the complex numbers but also will stay with them forever a. A + bi is: \ [ z_1=-2+\sqrt { -16 } \text { and } z_2=3-\sqrt { }. Operations of addition of vectors we have a negative number in there ( -13i.. Are real numbers and the imaginary parts we add complex numbers is closed,,... Very similar to example 1 with the complex number 's not too hard to verify that number... We interchange the complex numbers are also complex numbers is printed from the main ( ).. To multiply complex numbers in rectangular form, multiply the magnitudes and add same. Containing the sum of any complex number \ ( x+iy\ ) and \ ( x+iy\ ) and \ ( )... Is 9 + 5i or graphically in rectangular form help James find the sum of any complex number These.... Or the FOIL method at Cuemath, our team of Math experts is dedicated to making fun... By considering them as binomials overloaded the + and – operator to use it with −1 as with numbers! An imaginary number and a and b are real numbers and imaginary favorite readers, the sum of numbers. Licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License represented by \ ( z_1=3+3i\ ) corresponds to \ z\... Numbers with concepts, examples, videos and solutions component-wise addition of numbers!, because the sum and returns the structure containing the sum of two complex numbers a... Way that not only it is relatable and easy to grasp, but also will stay with forever. Subtract complex numbers algebraically the set of complex numbers geometrically the added | {
"domain": "mestaritalo.com",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9693241947446617,
"lm_q1q2_score": 0.8781978537051432,
"lm_q2_score": 0.9059898210180105,
"openwebmath_perplexity": 512.6572821089109,
"openwebmath_score": 0.7448074221611023,
"tags": null,
"url": "http://mestaritalo.com/orvovf1/addition-of-complex-numbers-5b5093"
} |
forever. Subtract complex numbers algebraically the set of complex numbers geometrically the added twist that have! The Cuemath way from the main ( ) function again, it 's not too hard to that... + and – operator to use it with the added twist that we a. ( -13i ) ) corresponds to \ ( z_2=-3+i\ ) corresponds to the complex plane has its additive in! To see if the answer must be expressed in simplest a+ bi form '' button to see if answer. Math experts is dedicated to making learning fun for our favorite readers the. Drag the point by which the complex numbers arithmetic, a complex number but not a real number once,. Identity, 0 is also present in the XY-plane multiplication, or the FOIL method Simple algebraic addition not. Visualize the geometrical addition of two complex numbers z_1=3+3i\ ) corresponds to the point (,... Numbers, just add or subtract complex numbers, just add or subtract the corresponding real and imag -2i..., \ ( z\ ) 3 ) and add or subtract two complex numbers is closed under addition the... Calculator does basic arithmetic on complex numbers add the same complex numbers is closed, as the sum given. Is the easy process to add or subtract two complex numbers very similar to example 1 with added. We will find the sum is the tip of the complex numbers does n't join \ ( z_2\.!, as the usual component-wise addition of vectors a FREE, world-class education to,... The corresponding point are changed readers, the teachers explore all angles of a topic multiply the coefficients and multiply. Expressions in the following complex numbers child a Math Thinker, the sum of two complex numbers … as!, well thought and well explained computer science and programming articles, and. Real numbers numbers are also complex numbers is thus immediately depicted as sum! Complex expression, with steps shown examples are − 6 + 4i –... Their real and imag the operations of addition of complex numbers in this sectoin 5 + and! Explained computer science and programming | {
"domain": "mestaritalo.com",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9693241947446617,
"lm_q1q2_score": 0.8781978537051432,
"lm_q2_score": 0.9059898210180105,
"openwebmath_perplexity": 512.6572821089109,
"openwebmath_score": 0.7448074221611023,
"tags": null,
"url": "http://mestaritalo.com/orvovf1/addition-of-complex-numbers-5b5093"
} |
of addition of complex numbers in this sectoin 5 + and! Explained computer science and programming articles, quizzes and practice/competitive programming/company interview Questions diagonal. 4I\ ] are changed numbers are numbers that are binomials, use the Distributive Property of,. Inputs real and imaginary number using structure you to practice mission is to provide a,... Because they have two parts, real and imag James find the sum is tip. Of two complex numbers ( 0+4i = 4i\ ) to grasp, but also will with... Button to see if the answer must be expressed in simplest a+ bi form i! + 3i and 4 + 2i is 9 + 5i number is not a real number – 5i 7! ( z_1+z_2\ ) main ( ) function here, you can visualize the geometrical addition of complex numbers.. Done in a way that not only it is relatable and easy grasp! | {
"domain": "mestaritalo.com",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9693241947446617,
"lm_q1q2_score": 0.8781978537051432,
"lm_q2_score": 0.9059898210180105,
"openwebmath_perplexity": 512.6572821089109,
"openwebmath_score": 0.7448074221611023,
"tags": null,
"url": "http://mestaritalo.com/orvovf1/addition-of-complex-numbers-5b5093"
} |
# Just need a little clearing up
#### marlousie
##### New member
I tried to ask google however it didn't really give me a straight answer so I made an account here solely for this question. It's also relatively simple so thank you to anyone who explains it to me because I'm still a bit caught up on it.
Why does (-4)^2= 16 but -4^2= 1/16
I'm currently studying for the SAT and these things constantly trip me up because my math teachers never really went over this.
Thank you again!
#### Subhotosh Khan
##### Super Moderator
Staff member
I tried to ask google however it didn't really give me a straight answer so I made an account here solely for this question. It's also relatively simple so thank you to anyone who explains it to me because I'm still a bit caught up on it.
Why does (-4)^2= 16 but -4^2= 1/16
I'm currently studying for the SAT and these things constantly trip me up because my math teachers never really went over this.
Thank you again!
-4^2= 1/16................................................... is incorrect
-4^2 = - (4^2) = -(16) = -16
However:
4^(-2) = 1/(4^2) = 1/16
Please come back if you have more questions.
#### pka
##### Elite Member
I tried to ask google however it didn't really give me a straight answer so I made an account here solely for this question. It's also relatively simple so thank you to anyone who explains it to me because I'm still a bit caught up on it.
Why does (-4)^2= 16 but -4^2= 1/16
I'm currently studying for the SAT and these things constantly trip me up because my math teachers never really went over this.
$$-4^2=-16$$ while $$4^{-2}=\frac{1}{16}$$
#### LCKurtz
##### Full Member
I tried to ask google however it didn't really give me a straight answer so I made an account here solely for this question. It's also relatively simple so thank you to anyone who explains it to me because I'm still a bit caught up on it.
Why does (-4)^2= 16 but -4^2= 1/16 | {
"domain": "freemathhelp.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9814534333179649,
"lm_q1q2_score": 0.8781941997986751,
"lm_q2_score": 0.8947894724152068,
"openwebmath_perplexity": 975.0935227221232,
"openwebmath_score": 0.2964867651462555,
"tags": null,
"url": "https://www.freemathhelp.com/forum/threads/just-need-a-little-clearing-up.123949/"
} |
Why does (-4)^2= 16 but -4^2= 1/16
I'm currently studying for the SAT and these things constantly trip me up because my math teachers never really went over this.
Thank you again!
It has to do with the precedence of operators. Exponentiation takes precedence over addition and multiplication. So if have $$\displaystyle -4^2$$, you would to the exponent first giving $$\displaystyle -16$$. Such problems are avoided by using parentheses. $$\displaystyle (-4)^2 = 16$$ versus $$\displaystyle -(4^2)=-16$$. Then there is no doubt which is meant. On exams like the SAT they give you $$\displaystyle -4^2$$ to trip you up if you don't know the precedence rules.
#### marlousie
##### New member
-4^2= 1/16................................................... is incorrect
-4^2 = - (4^2) = -(16) = -16
However:
4^(-2) = 1/(4^2) = 1/16
Please come back if you have more questions.
Ah, thank you I realize my mistake now! Thank you, I appreciate it
#### marlousie
##### New member
$$-4^2=-16$$ while $$4^{-2}=\frac{1}{16}$$
It has to do with the precedence of operators. Exponentiation takes precedence over addition and multiplication. So if have $$\displaystyle -4^2$$, you would to the exponent first giving $$\displaystyle -16$$. Such problems are avoided by using parentheses. $$\displaystyle (-4)^2 = 16$$ versus $$\displaystyle -(4^2)=-16$$. Then there is no doubt which is meant. On exams like the SAT they give you $$\displaystyle -4^2$$ to trip you up if you don't know the precedence rules. | {
"domain": "freemathhelp.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9814534333179649,
"lm_q1q2_score": 0.8781941997986751,
"lm_q2_score": 0.8947894724152068,
"openwebmath_perplexity": 975.0935227221232,
"openwebmath_score": 0.2964867651462555,
"tags": null,
"url": "https://www.freemathhelp.com/forum/threads/just-need-a-little-clearing-up.123949/"
} |
# Is this proof correct (Rationality of a number)?
Is $\sqrt[3] {3}+\sqrt[3]{9}$ a rational number? My answer is no, and there is my proof. I would like to know if this is correct:
Suppose this is rational. So there are positive integers $m,n$ such that $$\sqrt[3]{3}+\sqrt[3]{9}=\sqrt[3]{3}(1+\sqrt[3]{3})=\frac{m}{n}$$
Let $x=\sqrt[3]{3}$. We get $x^2+x-\frac{m}{n}=0 \rightarrow x=\frac{-1+\sqrt{1+\frac{4m}{n}}}{2}$.
We know that $x$ is irrational and that implies $\sqrt{1+\frac{4m}{n}}$ is irrational as well (Otherwise $x$ is rational). Write $x=\sqrt[3]{3}$, multiply both sides by $2$ and then raise both sides to the power of $6$ to get:
$$24^2=\left(\left(\sqrt{1+\frac{4m}{n}}-1\right)^3\right)^2\rightarrow 24=\left(\sqrt{1+\frac{4m}{n}}-1\right)^3=\left(1+\frac{4m}{n}\right)\cdot \sqrt{1+\frac{4m}{n}}-3\left(1+\frac{4m}{n}\right)+3\sqrt{1+\frac{4m}{n}}-1$$ Let $\sqrt{1+\frac{4m}{n}}=y,1+\frac{4m}{n}=k$.
We get: $25=ky-3k+3y\rightarrow y=\frac{25+3k}{k+3}$, So $y$ is rational. But we know $y$ is irrational (again, otherwise $\sqrt[3]{3}$ is rational) which leads to a contradiction. So the answer is No, $\sqrt[3]{3}+\sqrt[3]{9}$ is irrational.
Is this proof correct? Is there another way to prove this? Thanks! | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.986363166322039,
"lm_q1q2_score": 0.8781534368603984,
"lm_q2_score": 0.8902942312159383,
"openwebmath_perplexity": 127.93420571638048,
"openwebmath_score": 0.926621675491333,
"tags": null,
"url": "https://math.stackexchange.com/questions/2909548/is-this-proof-correct-rationality-of-a-number"
} |
• This seems like a lot of work. We know $x$ satisfies a cubic polynomial over $\mathbb Q$,namely $x^3-3$. If it also satisfies a quadratic then the quadratic and the cubic would have to a common factor, which is clearly not the case (since the cubic has no rational roots). – lulu Sep 8 '18 at 11:56
• Notice that after raising both sides to the power of 6, your next step is to take the square root of both sides. Why not just raise to the power of 3 to begin with? – Théophile Sep 8 '18 at 12:09
• @Théophile You are right, no reason not to just raise to the power of 3. Is this proof correct? – Omer Sep 8 '18 at 12:11
• The proof looks correct to me. – saulspatz Sep 8 '18 at 12:19
• Yes, it looks correct; the reasoning stands. Now, as lulu said, it is a lot of work and can be greatly simplified. Even if you keep the general structure here, you can save some writing in several places: apart from the power of 6 that I mentioned, also look at $1+\frac{4m}n$. This is clearly rational, and it's a bit cumbersome to write out, so why not replace it with something else? Indeed, you call it $k$, but why not make that substitution earlier, on the previous line? But let's be consistent and call it something like $\frac{m'}{n'}$ instead of $k$ to show that it is rational. – Théophile Sep 8 '18 at 12:20
Alternatively, denote $x=\sqrt[3] {3}+\sqrt[3]{9}$ and cube it to get a cubic equation with integer coefficients: $$x^3=3+3^2(\sqrt[3]3+\sqrt[3]{9})+9 \Rightarrow \\ x^3-9x-12=0.$$ According to the rational root theorem, the possible rational roots are: $\pm (1,2,3,4,6,12)$. However, none of them satisfies the equation. Hence, $x$ is irrational.
Let $\alpha = \sqrt[3]{3} + \sqrt[3]{9} = \sqrt[3]{3}(1+\sqrt[3]{3})$. We have
$$\alpha^3 = 3(1+\sqrt[3]{3})^3 = 3(3 + 3\sqrt[3]{3} + 3\sqrt[3]{9} + 3) = 12 + 9(\sqrt[3]{3} + \sqrt[3]{9}) = 12 + 9\alpha$$
Hence $\alpha^3 - 9\alpha -12 = 0$. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.986363166322039,
"lm_q1q2_score": 0.8781534368603984,
"lm_q2_score": 0.8902942312159383,
"openwebmath_perplexity": 127.93420571638048,
"openwebmath_score": 0.926621675491333,
"tags": null,
"url": "https://math.stackexchange.com/questions/2909548/is-this-proof-correct-rationality-of-a-number"
} |
Hence $\alpha^3 - 9\alpha -12 = 0$.
However, the polynomial $x^3-9x-12$ is irreducible over $\mathbb{Q}$ by the Eisenstein criterion for $p = 3$ so it cannot have any rational roots. Therefore $\alpha$ is not rational.
As everyone else has said, your proof is correct but more laborious than necessary. Here's an approach that I think may minimize the amount of calculation.
Let $x=\root3\of3$, so we're interested in $y=x^2+x$. The fact that $x$ is a cube root suggests thinking about things involving $x^3$, and if you're lucky the following identity, involving both $x^3$ and $y$, may come to mind: $x^3-1=(x-1)(x^2+x+1)$. Aha! The left-hand side of this is a rational number, namely 2. And if $y$ is rational then the second factor on the right is also rational. But that would require $x-1$, and hence $x$, to be rational, which it certainly isn't. So $y$ can't be rational after all, and we're done.
(An easy generalization of the argument shows that things like $\root5\of7+\root5\of{7^2}+\root5\of{7^3}+\root5\of{7^4}$ are always irrational.) | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.986363166322039,
"lm_q1q2_score": 0.8781534368603984,
"lm_q2_score": 0.8902942312159383,
"openwebmath_perplexity": 127.93420571638048,
"openwebmath_score": 0.926621675491333,
"tags": null,
"url": "https://math.stackexchange.com/questions/2909548/is-this-proof-correct-rationality-of-a-number"
} |
# Evaluating sums using residues $(-1)^n/n^2$ [duplicate]
I am an alien towards compelx analysis, with very little know I am posing a question, who someone may want to help with.
Evaluate:
$$\frac{1}{4}\cdot \sum_{n=1}^{\infty} \frac{(-1)^n}{n^2}$$
In disguise this is similar to $\zeta(2)$ but how can this be done using residues, and complex analysis?
I need some help. I am just interested.
The answer is $\displaystyle \frac{\pi^2}{48}$
## marked as duplicate by Jack D'Aurizio, Ahaan S. Rungta, user98602, Aditya Hase, Pedro Tamaroff♦ calculus StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Dec 21 '14 at 22:12
• en.wikipedia.org/wiki/Dirichlet_eta_function – Gahawar Dec 19 '14 at 14:55
• The solution for $\zeta(2)$ given by robjohn in the above linked Question lends itself to an answer here, but a bit more reasoning is needed to reduce the case of alternating signs here to the absolute series for $\zeta(2)$. It's not hard however. – hardmath Dec 19 '14 at 15:31
• If the answer is supposed to be positive, you need either $-\frac14$ before the sum or $(-1)^{n-1}$ in the numerator of the fraction. – robjohn Dec 21 '14 at 22:30
• @robjohn, a friendly request to you, since you are amazing at complex analysis. Would you have time to answer this question from me: math.stackexchange.com/questions/1077460/… – Amad27 Dec 22 '14 at 10:24
Using $\boldsymbol{\pi\csc(\pi z)}$ | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9915543728093005,
"lm_q1q2_score": 0.8781517515825661,
"lm_q2_score": 0.8856314647623016,
"openwebmath_perplexity": 454.77495379795647,
"openwebmath_score": 0.8262123465538025,
"tags": null,
"url": "https://math.stackexchange.com/questions/1074582/evaluating-sums-using-residues-1n-n2"
} |
Using $\boldsymbol{\pi\csc(\pi z)}$
Since $\pi\csc(\pi z)$ has residue $(-1)^n$ at $z=n$ for $n\in\mathbb{Z}$, we will use the contours $$\gamma_\infty=\lim\limits_{R\to\infty}Re^{2\pi i[0,1]}\qquad\text{and}\qquad\gamma_0=\lim\limits_{R\to0}Re^{2\pi i[0,1]}$$ To sum over all $n\in\mathbb{Z}$ except $n=0$, we use the difference of the contours, which circles the non-zero integers once counter-clockwise. \begin{align} 2\sum_{n=1}^\infty\frac{(-1)^n}{n^2} &=\frac1{2\pi i}\left(\int_{\gamma_\infty}\frac{\pi\csc(\pi z)}{z^2}\mathrm{d}z-\int_{\gamma_0}\frac{\pi\csc(\pi z)}{z^2}\mathrm{d}z\right)\\ &=\color{#C00000}{\frac1{2\pi i}\int_{\gamma_\infty}\frac{\pi\csc(\pi z)}{z^2}\mathrm{d}z}-\operatorname*{Res}_{z=0}\left(\color{#00A000}{\frac{\pi\csc(\pi z)}{z^2}}\right)\\ &=\color{#C00000}{0}-\operatorname*{Res}_{z=0}\left(\color{#00A000}{\frac1{z^2}\frac\pi{\pi z-\pi^3z^3/6+O\left(z^5\right)}}\right)\\ &=\color{#C00000}{0}-\operatorname*{Res}_{z=0}\left(\color{#00A000}{\frac1{z^3}+\frac{\pi^2}{6z}+O(z)}\right)\\ &=-\frac{\pi^2}6 \end{align} because, for $k\in\mathbb{Z}$ and $|z|=\pi\left(k+\frac12\right)$, $|\csc(z)|\le1$.
Therefore, $$\sum_{n=1}^\infty\frac{(-1)^n}{n^2}=-\frac{\pi^2}{12}$$
Extending A Previous Result
In this answer, it is shown that $$\sum_{k=1}^\infty\frac1{k^2}=\frac{\pi^2}6$$ Note that \begin{align} \hphantom{=}&\frac1{1^2}{+}\frac1{2^2}+\frac1{3^2}{+}\frac1{4^2}+\frac1{5^2}{+}\frac1{6^2}+\frac1{7^2}+\dots\\ \hphantom{=}&\hphantom{\frac1{1^2}}\color{#C00000}{-\frac2{2^2}\hphantom{+\frac1{3^2}}-\frac2{4^2}\hphantom{+\frac1{5^2}}-\frac2{6^2}\hphantom{+\frac1{7^2}}-\dots}\\ =&\frac1{1^2}{-}\frac1{2^2}+\frac1{3^2}{-}\frac1{4^2}+\frac1{5^2}{-}\frac1{6^2}+\frac1{7^2}-\dots \end{align} where the series in red is two times one quarter of the series above it; that is, one half of the series above it. Therefore, the alternating series is one half of the non-alternating series; that is, $$\sum_{n=1}^\infty\frac{(-1)^{n-1}}{n^2}=\frac{\pi^2}{12}$$ | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9915543728093005,
"lm_q1q2_score": 0.8781517515825661,
"lm_q2_score": 0.8856314647623016,
"openwebmath_perplexity": 454.77495379795647,
"openwebmath_score": 0.8262123465538025,
"tags": null,
"url": "https://math.stackexchange.com/questions/1074582/evaluating-sums-using-residues-1n-n2"
} |
• I thought this looked familiar. I see that I have also posted this answer to the question to which this one was marked as a duplicate. Now that I look closer, this question is not really a duplicate of that one since this question asks about the alternating series. Granted, it is not too difficult to derive the alternating series from the non-alternating series, but a bit of extra work is needed. – robjohn Dec 21 '14 at 22:39
• Good answer. How did you convert the sum into a contour integral though? – Amad27 Dec 22 '14 at 9:37
• @Amad27: The function $\pi\csc(\pi z)$ has singularities exactly on $\mathbb{Z}$ with residue $(-1)^n$ at $z=n\in\mathbb{Z}$. Thus, $\frac{\pi\csc(\pi z)}{z^2}$ also has singularities exactly on $\mathbb{Z}$ with residue $\frac{(-1)^n}{n^2}$ at $z=n\in\mathbb{Z}$. The difference of the contour integrals along $\gamma_\infty$ and $\gamma_0$ equals $2\pi i$ times the sum of the residues inside $\gamma_\infty$ but outside $\gamma_0$. This is $4\pi i$ times the sum we are looking for. We don't need to worry about the residue at $z=0$ since it is not inside the difference of the contours. – robjohn Dec 22 '14 at 10:49
• May I know that why $\csc z \leq 1$ when $|z|=k+1/2$? – mnmn1993 Mar 7 '18 at 17:59
For a complex variable $s$, whose real part is greater than zero, the Dirichlet eta function is defined by the series $$\eta(s) := -\sum_{n=1}^{\infty} \frac{(-1)^n}{n^s}.$$
In particular, one has that
$$\eta(s) = \left(1-2^{1-s}\right)\zeta(s),$$
where $\zeta$ denotes the Riemann zeta function. With that in mind, one need only substitute $s=2$ into the above equation. I presume that you are familiar with the famed Basel problem.
• Also, this does not really meet the requirements that it be done through the methods of complex analysis, but I suppose that, since $\zeta(2)$ has already been found through such methods on this site, it is fine. – Gahawar Dec 19 '14 at 15:09 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9915543728093005,
"lm_q1q2_score": 0.8781517515825661,
"lm_q2_score": 0.8856314647623016,
"openwebmath_perplexity": 454.77495379795647,
"openwebmath_score": 0.8262123465538025,
"tags": null,
"url": "https://math.stackexchange.com/questions/1074582/evaluating-sums-using-residues-1n-n2"
} |
# Is there a quick proof as to why the vector space of $\mathbb{R}$ over $\mathbb{Q}$ is infinite-dimensional?
It would seem that one way of proving this would be to show the existence of non-algebraic numbers. Is there a simpler way to show this?
-
A finite dimensional vector space over $\mathbb{Q}$ is countable. – user641 Oct 7 '10 at 2:19
@Steve: Please add that as an answer so people can upvote. – Aryabhata Oct 7 '10 at 2:50
Does that mean that a vector space over $\mathbb{Q}$ is finite-dimensional iff the set of the vector space is countable? If so, please prove it. – Isaac Solomon Oct 7 '10 at 2:55
@Isaac Your question doesn't require the 'only if' anyway. Steve's observation answers your original question. – yasmar Oct 7 '10 at 3:04
@Isaac: how is the fact that $\mathbb{R}$ is finite dimensional over itself relevant? – Arturo Magidin Oct 7 '10 at 3:31
As Steve D. noted, a finite dimensional vector space over a countable field is necessarily countable: if $v_1,\ldots,v_n$ is a basis, then every vector in $V$ can be written uniquely as $\alpha_1 v_1+\cdots+\alpha_n v_n$ for some scalars $\alpha_1,\ldots,\alpha_n\in F$, so the cardinality of the set of all vectors is exactly $|F|^n$. If $F$ is countable, then this is countable. Since $\mathbb{R}$ is uncountable and $\mathbb{Q}$ is countable, $\mathbb{R}$ cannot be finite dimensional over $\mathbb{Q}$. (Whether it has a basis or not depends on your set theory).
Your further question in the comments, whether a vector space over $\mathbb{Q}$ is finite dimensional if and only if the set of vectors is countable, has a negative answer. If the vector space is finite dimensional, then it is a countable set; but there are infinite-dimensional vector spaces over $\mathbb{Q}$ that are countable as sets. The simplest example is $\mathbb{Q}[x]$, the vector space of all polynomials with coefficients in $\mathbb{Q}$, which is a countable set, and has dimension $\aleph_0$, with basis $\{1,x,x^2,\ldots,x^n,\ldots\}$. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9915543736828079,
"lm_q1q2_score": 0.8781517508596043,
"lm_q2_score": 0.8856314632529871,
"openwebmath_perplexity": 202.9308095561068,
"openwebmath_score": 0.9408407807350159,
"tags": null,
"url": "http://math.stackexchange.com/questions/6244/is-there-a-quick-proof-as-to-why-the-vector-space-of-mathbbr-over-mathbb/6250"
} |
Added: Of course, if $V$ is a vector space over $\mathbb{Q}$, then it has countable dimension (finite or denumerable infinite) if and only if $V$ is countable as a set. So the counting argument in fact shows that not only is $\mathbb{R}$ infinite dimensional over $\mathbb{Q}$, but that (if you are working in an appropriate set theory) it is uncountably-dimensional over $\mathbb{Q}$.
-
Related to the note about uncountable dimension, there are explicit examples of continuum-sized linearly independent sets, as seen in this MathOverflow answer by François G. Dorais: mathoverflow.net/questions/23202/… – Jonas Meyer Oct 7 '10 at 4:18
Yes: but can one show that there is a basis for $\mathbb{R}$ over $\mathbb{Q}$ without some form of the Axiom of Choice? (There is a difference between exhibiting a large linearly independent subset and exhibiting a basis). – Arturo Magidin Oct 7 '10 at 14:51
No, one cannot, but without the Axiom of Choice the notion of dimension breaks down (except for finite vs. infinite). Assuming AC (as I virtually always do), the size of a linearly independent set gives a lower bound on the dimension of the vector space, and I think it is wonderful that in this case such "explicit" proof exists that the real numbers have continuum dimension, as opposed to the nice qualitative proof one could give by extending your argument to larger cardinals. – Jonas Meyer Oct 7 '10 at 16:02
@Jonas: No argument there (with any of your points). – Arturo Magidin Oct 7 '10 at 17:23
good link @Jonas, thanks – Leon Sep 23 '11 at 0:23 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9915543736828079,
"lm_q1q2_score": 0.8781517508596043,
"lm_q2_score": 0.8856314632529871,
"openwebmath_perplexity": 202.9308095561068,
"openwebmath_score": 0.9408407807350159,
"tags": null,
"url": "http://math.stackexchange.com/questions/6244/is-there-a-quick-proof-as-to-why-the-vector-space-of-mathbbr-over-mathbb/6250"
} |
The cardinality argument mentioned by Arturo is probably the simplest. Here is an alternative: an explicit example of an infinite $\rm\: \mathbb Q$-independent set of reals. Consider the set consisting of the logs of all primes $\rm\: p_i\:$. Then if $\rm\ \: c_1\ log\ p_1 +\:\cdots\: + c_n\ log \ p_n \: =\ 0\:,\ \: c_i\in\mathbb Q\:,\:$ multiplying by a common denominator we can assume that all $\rm\ c_i \in \mathbb Z\:,\:$ and, exponentiating, we obtain $\rm\: p_1^{\:c_1}\cdots p_n^{\:c_n} = 1\ \Rightarrow\ c_i = 0,\:$ for all $\rm\:i\:$.
-
@Bill +1 What a nice example. – Adrián Barquero Oct 7 '10 at 4:50
@Bill. Wow! :-) – a.r. Oct 7 '10 at 6:08
Is this proof unique to Q or does it generalize to provide explicit examples of reals linearly independent over, e.g., Q(sqrt(2))? Above, Q appears to be "hard-wired" into the proof, as the group of exponents in the prime factorization. – T.. Oct 12 '10 at 7:14
@George S. "Then, any number field has a finite extension whose class number is 1." This would imply in particular that there are infinitely many number fields of class number one, which I am pretty sure is an open problem. I wonder what you are thinking here? – Pete L. Clark Jan 22 '11 at 10:20
Can you extend the idea of this proof to get the right dimension? Currently, you only have a countably infinite independent set, but the dimension is size continuum. – JDH Jun 23 '11 at 2:22
No transcendental numbers are needed for this question. Any set of algebraic numbers of unbounded degree spans a vector space of infinite dimension. Explicit examples of linearly independent sets of algebraic numbers are also relatively easy to write down. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9915543736828079,
"lm_q1q2_score": 0.8781517508596043,
"lm_q2_score": 0.8856314632529871,
"openwebmath_perplexity": 202.9308095561068,
"openwebmath_score": 0.9408407807350159,
"tags": null,
"url": "http://math.stackexchange.com/questions/6244/is-there-a-quick-proof-as-to-why-the-vector-space-of-mathbbr-over-mathbb/6250"
} |
The set $\sqrt{2}, \sqrt{\sqrt{2}}, \dots, = \bigcup_{n>0} 2^{2^{-n}}$ is linearly independent over $\mathbb Q$. (Proof: Any expression of the $n$th iterated square root $a_n$ as a linear combination of earlier terms $a_i, i < n$ of the sequence could also be read as a rational polynomial of degree dividing $2^{n-1}$ with $a_n$ as a root and this contradicts the irreducibility of $X^m - 2$, here with $m=2^n$).
The square roots of the prime numbers are linearly independent over $\mathbb Q$. (Proof: this is immediate given the ability to extend the function "number of powers of $p$ dividing $x$" from the rational numbers to algebraic numbers. $\sqrt{p}$ is "divisible by $p^{1/2}$" while any finite linear combination of square roots of other primes is divisible by an integer power of $p$, i.e., is contained in an extension of $\mathbb Q$ unramified at $p$).
Generally any infinite set of algebraic numbers that you can easily write down and is not dependent for trivial reasons usually is independent. This because the only algebraic numbers for which we have a simple notation are fractional powers, and valuation (order of divisibility) arguments work well in this case. Any set of algebraic numbers where, of the ones ramified at any prime $p$, the amount of ramification is different for different elements of the set, will be linearly independent. (Proof: take the most ramified element in a given linear combination, express it in terms of the others, and compare valuations.)
-
For the sake of completeness, I'm adding a worked-out solution due to F.G. Dorais from his post.
We'll need two propositions from Grillet's Abstract Algebra, page 335 and 640:
Proposition: $[\mathbb{R}:\mathbb{Q}]=\mathrm{dim}_\mathbb{Q}{}\mathbb{R}=|\mathbb{R}|$ | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9915543736828079,
"lm_q1q2_score": 0.8781517508596043,
"lm_q2_score": 0.8856314632529871,
"openwebmath_perplexity": 202.9308095561068,
"openwebmath_score": 0.9408407807350159,
"tags": null,
"url": "http://math.stackexchange.com/questions/6244/is-there-a-quick-proof-as-to-why-the-vector-space-of-mathbbr-over-mathbb/6250"
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.