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I am brain dead. I've never been this stuck... on this kind of problem. XD
4. Originally Posted by chocole
The problem states how many presents would be received over any other number (n) of days.
I am brain dead. I've never been this stuck... on this kind of problem. XD
The formula for the sum of an arithmetic series is:
$S_{n} = \frac{n}{2} \left( 2a + (n - 1)d \right)$
a = 1 and d = 1
So:
$S_{n} = \frac{n}{2} \left( 2 + (n - 1) \right)$
$S_{n} = \frac{n}{2} \left( n + 1 \right)$
$S_{n} = \frac{n^2 + n}{2}$
Where n = the number of days.
So on the 1st day you would have 1 present, on the 2nd day, you would receive 3 presents, etc...
5. Hello, chocole!
The following questions are related to the Christmas song "The Twelve Days of Christmas".
We all know how to figure out the number of presents given over the 12 days.
It is 364 presents.
How many ways did you find to solve for 364 presents?
$\begin{array}{cccc}\text{Day} & & \text{Presents} \\
1 &1 & 1 \\
2 & 1+2 & 3 \\
3 & 1+2+3 & 6 \\
4 & 1+2+3+4 & 10 \\
\vdots & \vdots & \vdots
\end{array}$
. These are "triangular numbers."
On the $n^{th}$ day, my true love gave to me: . $\frac{n(n+1)}{2}$ gifts.
By the $n^{th}$ day, my true love gave to me a total of:
. . $\sum^n_{k=1}\frac{k(k+1)}{2} \;=\;\frac{n(n+1)(n+2)}{6}$ gifts. | {
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# Conditional probability that standard gaussian random variable is larger than another
Say there are two standard Gaussian random variables $$X$$ and $$Y$$. I am trying to evaluate the probability that the larger of the two is selected, given that it is known whether $$X$$ is positive or negative (the strategy is selecting $$X$$ if $$X$$ is positive and selecting $$Y$$ if $$X$$ is negative). In equation form this is $$Pr(X-Y>0|X>0) + Pr(X-Y<0|X<0)$$
How can this expression be evaluated? Numerically it appears to be $$\frac{3}{4}$$, and intuitively this makes sense.
I am also interested in this probability in the more general case, where the strategy involves selecting $$X$$ if $$X-S>0$$ and $$Y$$ otherwise, where $$S$$ is another independent Gaussian random variable.
• I don't understand your question. Did you mean $\text{Pr}(X-Y>0|X>0)\text{Pr}(X>0)+\text{Pr}(X-Y<0|X<0)\text{Pr}(X<0)$? – Angela Pretorius Jun 5 '19 at 4:32
• Yes, that's right! – tankerjeel Jun 5 '19 at 20:07
The following assumes $$X$$ and $$Y$$ are independent.
As Angela Richardson pointed out, you probably actually want to compute $$P(X-Y > 0 \mid X > 0) P(X>0) + P(X-Y <0 \mid X< 0) P(X<0)$$ which is $$3/4$$. (The quantity in your post is not $$3/4$$.)
For the first term, it suffices to compute $$P(X-Y > 0, X>0)$$. (Why?)
Consider the region of the plane that contains $$(x,y)$$ pairs satisfying $$x-y>0$$ and $$x>0$$. Then use rotational symmetry of the vector $$(X,Y)$$ to compute the probability.
The other term can be handled similarly. | {
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The other term can be handled similarly.
• Thanks. Is there a way to evaluate the more general case as well? I'm guessing symmetry no longer holds once $S$ is introduced. – tankerjeel Jun 5 '19 at 20:09
• In case anyone is curious, the stackexchange post in this link gives the solution as to how to compute the more general case. When $S$ is the standard normal, the probability turns out to be exactly $\frac{2}{3}$, and approaches $\frac{3}{4}$ as the variance of $S$ approaches $0$. – tankerjeel Jun 5 '19 at 23:57 | {
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# Thread: Volume of sphere with hole
1. ## Volume of sphere with hole
A hole of diameter d is drilled through a sphere of radius r in such a way that the axis of the hole passes through the centre of the sphere. Find the volume of the solid that remains.
I drew a picture of a sphere centred on the cartesian plane, with a hole going through its centre (in the shape of a cylinder).
I already knew that the volume of the sphere was $\frac {4}{3}\pi r^{3}$. I then worked out the volume of the hole (cylinder):
$
\int_{-r}^{r} \frac{1}{4}\pi d^{2}dy$
$= \frac{1}{2}\pi d^{2} r
$
Volume of solid remaining = $\frac {4}{3}\pi r^{3} - \frac{1}{2}\pi d^{2} r$
but the correct answer was $\frac{1}{6}\pi (4r^{2} - d^{2})^{\frac{3}{2}}$
I cant see where I went wrong cause surely the objective of this problem is to minus the hole (cylinder) from a sphere to get the remaining volume?
2. The volume subtracted out by drilling the hole is not precisely a cylinder, because a cylinder usually has flat ends. The material removed has curved ends (matches up with the sphere). So what do you get when you change this?
3. Very intresting problem. It sounds like Ackbeet is right in the fact that the figure removed is not
Spoiler:
only
a cylinder.
Good Luck!
4. Rather than subtracting the round-ended cylinder from the volume of the sphere, I think it's probably easier to set up the integral for the volume. You can use cylindrical shell elements of width $dx$, circumference $2 \pi x$, and height $2 \sqrt {R^2 - x^2}$, and integrate from x = d/2 to x = R.
5. I got the integral $V=-2\pi\int_{r^{2}-\frac{d^{2}}{4}}^{0} \sqrt{u} du$ after making the substitution of $u=r^{2} - x^{2}$ then my final answer was:
$\frac{4}{3}\pi(r^{2}-\frac{d^{2}}{4})^{\frac{3}{2}}$. Did I do something wrong or is there any way to simplify it to $\frac{1}{6}\pi(4r^{2} -d^{2)^\frac{3}{2}$ ?
6. They are the same answer. Proof:
$\displaystyle{\frac{4}{3}\pi(r^{2}-\frac{d^{2}}{4})^{\frac{3}{2}}}$ | {
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$\displaystyle{\frac{4}{3}\pi(r^{2}-\frac{d^{2}}{4})^{\frac{3}{2}}}$
$\displaystyle{=\frac{4}{3}\pi(4r^{2}-d^{2})^{\frac{3}{2}}\left(\frac{1}{4}\right)^{\fra c{3}{2}}}$
$\displaystyle{=\frac{4}{3}\pi(4r^{2}-d^{2})^{\frac{3}{2}}\,\frac{1}{4^{3/2}}}$
$\displaystyle{=\frac{1}{8}\,\frac{4}{3}\pi(4r^{2}-d^{2})^{\frac{3}{2}}}$
$\displaystyle{=\frac{1}{6}\pi(4r^{2}-d^{2})^{\frac{3}{2}}.}$ | {
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## Section1.2Statements
Statements are declarative sentences that are either true or false. The statements are formulated in such a way that any reader, who knows what all the words mean, can understand them.
1. “Victoria likes cookies.” is a declarative sentence, and it is either true or false, so it is a statement.
2. “Broccoli is green.” is a declarative sentence and it is true, so it is a statement.
3. “Broccoli is pink.” is a declarative sentence and it is false, so it is a statement.
4. “Cookies!” is not a declarative sentence, so it is not a statement.
From now on we concentrate on statements about the integers.
Consider the following:
1. “2 is equal to 3.” is a statement. It is false.
2. “2 plus 3 is equal to 5.” is a statement. It is true.
3. “2 plus 3” is not a statement, as it is not a declarative sentence; it is not even a sentence, as it does not contain a verb.
When we write a statement using the symbols $$=\text{,}$$ $$\ne\text{,}$$ $$\lt\text{,}$$ $$\le\text{,}$$ $$>\text{,}$$ or $$\ge\text{,}$$ the comparison symbol takes the place of the verb. A mathematical statement always has a verb or a symbol that takes the place of the verb, just as a sentence does.
A mathematical expression consists of objects and operations. The objects can be numbers or variables (see the next section) and the operations can be, for example $$+\text{,}$$ $$\cdot\text{,}$$ or $$-\text{.}$$ Unlike a statement, an expression has no comparison symbol, that means it has no “verb.” So expressions by themselves are not true or false, but expressions can be used in statements, as in Example 1.1.6.
We formulate Example 1.2.2 using symbols.
1. “2 = 3” is a statement. It is false.
2. “2 + 3 = 5” is a statement. It is true.
3. “2 + 3” is an expression. As it does not have a verb it is not a statement.
We identify whether statements about integers are true or false. Notice that all these examples when read out have a verb in them, namely “is”. | {
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1. $$2=2$$ is read “2 is equal to 2”. This is a true statement.
2. $$2=3$$ is read “2 is equal to 3”. This is a false statement.
3. $$2\gt 3$$ is read “2 is greater than”. This is a false statement.
4. $$2\ne 3$$ is read “2 is not equal to 3”. This is a true statement.
5. $$2\le 2$$ is read “2 is less than or equal to 2”. This is a true statement.
We give some examples of expressions and statements and identify them.
1. $$2 + 3$$” is an expression.
2. $$2+3 = 5$$” is a statement.
3. $$2 + 1 + 5$$” is an expression.
4. $$2+ 1 + 5 \lt 10$$” is a statement.
Decide whether the following are statements or not. If they are statements decide whether they are true or false.
1. “Sunflower”
2. “Stop signs are red.”
3. $$2$$ is equal to $$3\text{.}$$
4. $$\displaystyle (1+2)-4687$$
5. $$\displaystyle 2+3=7$$
6. $$\displaystyle 3 > -100$$
Solution.
1. “Sunflower” is not a sentence, so it is not a statement.
2. “Stop signs are red.” is a declarative sentence, so it is a statement. It is true.
3. “2 is equal to 3” is a declarative sentence, so it is a statement. As $$2\ne 3$$ the statement is false.
4. $$(1+2)-4687$$ is not a statement as it has no verb.
5. $$2+3=7$$ is a statement, the verb is ‘=’ (is equal to). As $$2+3=5$$ it is a false statement.
6. $$3 > -100$$ is a statement, the verb is “$$>$$” (is greater than). It is a true statement.
When a statement is true, we usually do not write “is true.” When a statement is false, always write “is false”.
In Checkpoint 1.2.7 recognize statements and for statements decide whether they are true or false.
Determine which of the following are mathematical statements.
For the statements decide whether they are true or false.
1. $$\displaystyle 13\le 38$$
2. $$\displaystyle -11-38$$
3. $$\displaystyle 13\cdot 38$$
4. $$\displaystyle 13\ne 38$$
### Subsection1.2.2Compound Statements | {
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4. $$\displaystyle 13\ne 38$$
### Subsection1.2.2Compound Statements
In mathematics we often deal with multiple statements that overlap. In these cases instead of writing each statement separately, we often write them as one string of statements. This allows us to connect the statements directly.
Instead of writing “$$2 + 3 = 5$$” and “$$5 = 1+4\text{,}$$” we write “$$2 + 3 = 5 = 1+4\text{.}$$
We can also do this with inequalities.
Writing “$$2+5 = 7 \lt 10$$” means both “2+5 = 7” and “$$7 \lt 10\text{.}$$” In words, “$$2$$ plus $$5$$ is $$7$$ and $$7$$ is less than $$10\text{.}$$
Compound statements are often used to prove identities, that is, when proving that two expressions are equal. The proofs of Theorem 1.4.6 and Theorem 1.4.7 and Theorem 1.4.12 in the next chapter is written that way. | {
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Derivation of $y=a(x-h)^2+k$ from $y=ax^2+bx+c$ given a vertex and a point
Derive $y=a(x-h)^2+k$ from $y=ax^2+bx+c$ given a vertex and a point.
Recently I have been solving a problem to which I could not find a solution. Google search of "quadratic equation given vertex and a point," yielded what I have been looking for. However, although I have already solved the problem, I am still wondering how to derive the equation I was looking for before I googled it.
I was given a vertex $V(-3, -2)$ and a point $P(-4, 0)$ of a parabola. Using $y=ax^2+bx+c$, I have derived an equation for finding the vertex for each parabola with $V(\frac{-b}{2a}, -a(\frac{b}{2a})^2+c)$.
I knew that given the same vertex, the parabola $y=x^2+6x+7$ was close to what I've been looking for. However, this parabola didn't go through $P(-4, 0)$, because it was too wide.
At this point it seemed as if I had enough information to derive an equation, that given a vertex and a point I would obtain a parabola according to the restrictions. However, from this point on I didn't know how to proceed further.
• Note that by symmetry, $(-2,0)$ is also a point on the parabola. – user137731 Jul 31 '16 at 23:53
• Is your goal to determine an equation of the parabola that has vertex $V(-3, -2)$ and passes through the point $P(-4, 0)$? – N. F. Taussig Jul 31 '16 at 23:56
• @N.F.Taussig Yes. – user270346 Jul 31 '16 at 23:56
• I was looking for a general equation, that given the vertex and a point of a parabola I could determine it's coefficients $a, b, c$ as in $y=ax^2+bx+c$. – user270346 Jul 31 '16 at 23:57
• Use the two points you're given and the one you get from symmetry. Plug each in to get a system of linear equations that you can hopefully then solve. – user137731 Jul 31 '16 at 23:58
$$y = ax^2 + bx + c$$
$$y - c = a\left(x^2 + \frac{b}{a}x\right)$$
$$y - c + a\frac{b^2}{4a^2}= a\left(x^2 + \frac{b}{a}x + \frac{b^2}{4a^2}\right)$$
$$y - c + \frac{b^2}{4a}= a\left(x+\frac{b}{2a}\right)^2$$ | {
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$$y - c + \frac{b^2}{4a}= a\left(x+\frac{b}{2a}\right)^2$$
$$y = a\left(x-\left(-\frac{b}{2a}\right)\right)^2 + \left(c- \frac{b^2}{4a}\right)$$
$$y = a\left(x-h\right)^2 + k$$
where $h = -\frac{b}{2a}$ and $k = c- \frac{b^2}{4a}$
If you're going in reverse (going from vertex $(h,k)$ and point $(x,y)$ to quadratic parameters $a,b,c$), then you can take the last few equations, isolate $a,b,c$, and translate them into terms of $h,k,x,y$:
$$a = \frac{y-k}{(h-x)^2}$$
$$b = -2ha = \frac{-2h(y-k)}{(h-x)^2}$$
$$c = k + \frac{b^2}{4a} = k + \frac{h^2(y-k)}{(h-x)^2}$$
• This answer followed by the @N. F. Taussig answer is the order what makes most sense to me after picking up on this problem from where I left it. Thanks! – user270346 Aug 1 '16 at 0:35
• @Marcus Stuhr, Regarding your edit, note that the parabola $y=x^2+6x+7$ doesn't have roots $x=-4$ and $x=-2$ as required. – user270346 Aug 1 '16 at 0:47
• I've done as you've requested. Please, take your time. Thanks. – user270346 Aug 1 '16 at 0:55
• @Matt I believe you're looking for $2x^2 + 12x + 16 = 0$, which goes through $(-4,0)$ and $(-2,0)$ and has vertex $(-3,-2)$. – Marcus Andrews Aug 1 '16 at 1:15
Since you know that the parabola has vertex $V(-3, -2)$, the vertex form of its equation is $$y = a[x - (-3)]^2 + (-2) = a(x + 3)^2 - 2$$ Since the parabola passes through the point $P(-4, 0)$, we can substitute $-4$ for $x$ and $0$ for $y$ to determine $a$. \begin{align*} 0 & = a(-4 + 3)^2 - 2\\ 2 & = a(-1)^2\\ 2 & = a \end{align*} Hence, $y = 2(x + 3)^2 - 2$. You can expand this expression to obtain the standard form of the equation. | {
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Addendum: In general, if you know the vertex $V(h, k)$ and a point $P(u, v) \neq V(h, k)$ on the parabola, you can write $$y = a(x - h)^2 + k$$ then substitute $u$ for $x$ and $v$ for $y$ to determine $a$. \begin{align*} v & = a(u - h)^2 + k\\ v - k & = a(u - h)^2\\ \frac{v - k}{(u - h)^2} & = a \end{align*} Then $$y = a(x - h)^2 + k = \frac{v - k}{(u - h)^2}(x - h)^2 + k$$ Again, expanding the expression to obtain its standard form enables you to determine the coefficients $a$, $b$, and $c$.
• Thanks for your answer, however that's not my question. Please forgive me if I was not clear enough. I was looking for a general equation, that given the vertex and a point of a parabola I could determine it's coefficients $a, b, c$ as in $y=ax^2+bx+c$, based on what I knew. So, it's more about how do you derive the general equation for solving this type of problems, rather than solving the problem itself. – user270346 Aug 1 '16 at 0:04
• This is especially useful due to the other point suggested by the symmetry. Thanks for your answer. – user270346 Aug 1 '16 at 0:37
• On a side note, do you have any idea as why the vertex equation for this parabola $y=2(x+3)^2-2$ and $y=x^2+6x+7$ following from it, don't have same roots? – user270346 Aug 1 '16 at 0:51
• If you expand $y = 2(x + 3)^2 - 2$, you obtain \begin{align} y & = 2(x + 3)^2 - 2\\ & = 2(x^2 + 6x + 9) - 2\\ & = 2x^2 + 12x + 18 - 2\\ & = 2x^2 + 12x + 16 \end{align} You forgot to multiply the terms in the parentheses by $2$. As you can verify, both $y = 2(x + 3)^2 - 2$ and $y = 2x^2 + 12x + 16$ have the roots $-4$ and $-2$. – N. F. Taussig Aug 1 '16 at 1:00
Generalized Form:
Given $y=ax^2+bx+c$, we can move the loose number $c$ to the other side and try completing the square! So from that, we get $$y-c=ax^2+bx\tag{1}$$ | {
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Factoring out $a$ gives us $y-c=a\left(x^2+\frac bax\right)$. Completing the square by dividing the coefficient of the $x$ term by $2$ and squaring it gives us $$y-c+a\left(\frac {b^2}{4a^2}\right)=a\left(x+\frac {b}{2a}\right)^2\tag{2}$$
Simplifying the left hand side and move it to the right hand side to obtain $$y=a\left(x-\left(-\frac {b}{2a}\right)^2\right)+\left(\frac {4ac-b^2}{4a}\right)\tag{3}$$
And since the Vertex follows the formula $(h,k)=\left(-\frac {b}{2a},\frac {4ac-b^2}{4a}\right)$, you get the Vertex formula by plugging it in. | {
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Sum of $\sum_{n=1}^{\infty}(-1)^{n+1}\frac{x^{2n-1}}{2n-1}$
I've been working with the series:
$$\sum_{n=1}^{\infty}(-1)^{n+1}\frac{x^{2n-1}}{2n-1}$$
From the ratio test it is clear that the series converges for $|x| < 1$, but I'm unable to obtain the sum of the series.
I'm looking for any hint of how to obtain the sum.
• Differentiate by x, use the geometric sum, integrate, and use that f(0)=0. Result is arctan(x) – Ákos Somogyi Aug 25 '15 at 17:11
Let $f(x)=\sum_{n=1}^{\infty}\frac{(-1)^{n+1}x^{2n-1}}{2n-1}$. Then we have
\begin{align} f'(x)=\sum_{n=1}^{\infty} (-1)^{n+1}x^{2n-2}&=\frac{-1}{x^2}\sum_{n=1}^{\infty} (-x^2)^{n}\\\\ &=\frac{1}{1+x^2} \tag 1 \end{align}
Integrating $(1)$ and using $f(0)=0$ reveals that
$$\bbox[5px,border:2px solid #C0A000]{f(x)=\arctan(x)}$$
• (+1) Similar to the approach I was going to take. I was going to note that it was the same as $$\sum_{n=0}^\infty(-1)^n\frac{x^{2n+1}}{2n+1}$$ and then show that the derivative of that was $\frac1{1+x^2}$ – robjohn Aug 25 '15 at 17:46
• @robjohn Thanks! I skipped any discussion of interval of convergence and also omitted discussing the legitimacy of differentiating term by term since it appeared that the OP was already aware of at least the first of these issued. – Mark Viola Aug 25 '15 at 17:50
Let $y$ be our sum so:
$$y=\sum_{n=1}^{\infty}{{(-1)}^{n+1}\frac{{x}^{2n-1}}{2n-1}}$$
Let's differentiate it to get:
$$y'=\sum_{n=1}^{\infty}{{(-1)}^{n+1}{x}^{2n-2}}\\ y'=\sum_{n=1}^{\infty}{{(-1)}^{n-1}{x}^{2n-2}}\\ y'=\sum_{n=1}^{\infty}{{(-1)}^{n+1}{(-x^2)}^{n-1}}\\ y'=\frac{1}{1+x^2}$$
Now integrate to get the original sum to get:
$$y=\arctan{x} +C$$
It is easy to see that for $x=0$ we have $y=0$, so $C=0$, hence the sum equals:
$$\sum_{n=1}^{\infty}{{(-1)}^{n+1}\frac{{x}^{2n-1}}{2n-1}}=\arctan{x}$$ | {
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$$\sum_{n=1}^{\infty}{{(-1)}^{n+1}\frac{{x}^{2n-1}}{2n-1}}=\arctan{x}$$
• Isn't that just a repeat of what's already posted? – Macavity Aug 25 '15 at 17:40
• I was typing my answer at the same time that's why I didn't notice the other answer – Oussama Boussif Aug 25 '15 at 17:43
• (+1) good answer. I've had it happen many times that a similar answer has been posted while I am typing up mine. In fact, it happened here, but I happened to see Dr. MV's answer before I posted. – robjohn Aug 25 '15 at 17:49
$$(-1)^{n+1}\dfrac{x^{2n-1}}{2n-1}=i^{2(n+1)}\cdot\dfrac{x^{2n-1}}{2n-1}=-i\cdot\dfrac{(ix)^{2n-1}}{2n-1}$$
If $S=\sum_{n=1}^\infty(-1)^{n+1}\dfrac{x^{2n-1}}{2n-1},$
$$i S=\sum_{n=1}^\infty(-1)^{n+1}\dfrac{(ix)^{2n-1}}{2n-1}$$
Now for $-1<y\le1,\ln(1+y)=y-\dfrac{y^2}2+\dfrac{y^3}3-\dfrac{y^4}4+\cdots$
$\ln(1-y)=-y-\dfrac{y^2}2-\dfrac{y^3}3-\dfrac{y^4}4-\cdots$
$\ln(1+y)-\ln(1-y)=?$
$$\implies2i S=\ln(1+ix)-\ln(1-ix)=\ln\dfrac{1+ix}{1-ix}$$
Let $1=r\cos A,x=r\sin A, x=\tan A$
Now, $$\ln\dfrac{1+ix}{1-ix}=\ln(e^{2iA})=2iA=2i\arctan x$$
• I know what you're trying to show, but the appearance of $x$ and $y$ and the fact that I think the sum for $iS$ looks like it has too many $i^{2n-1}$ in it, makes things a bit hard to follow. – robjohn Aug 25 '15 at 18:02 | {
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# quantile
Quantiles of a data set
## Syntax
``Y = quantile(X,p)``
``Y = quantile(X,N)``
``Y = quantile(___,'all')``
``Y = quantile(___,dim)``
``Y = quantile(___,vecdim)``
``Y = quantile(___,'Method',method)``
## Description
example
````Y = quantile(X,p)` returns quantiles of the elements in data vector or array `X` for the cumulative probability or probabilities `p` in the interval [0,1]. If `X` is a vector, then `Y` is a scalar or a vector having the same length as `p`.If `X` is a matrix, then `Y` is a row vector or a matrix where the number of rows of `Y` is equal to the length of `p`. For multidimensional arrays, `quantile` operates along the first nonsingleton dimension of `X`. ```
example
````Y = quantile(X,N)` returns quantiles for `N` evenly spaced cumulative probabilities (1/(`N` + 1), 2/(`N` + 1), ..., `N`/(`N` + 1)) for integer `N`>1. If `X` is a vector, then `Y` is a scalar or a vector with length `N`. If `X` is a matrix, then `Y` is a matrix where the number of rows of `Y` is equal to `N`.For multidimensional arrays, `quantile` operates along the first nonsingleton dimension of `X`. ```
example
````Y = quantile(___,'all')` returns quantiles of all the elements of `X` for either of the first two syntaxes.```
example
````Y = quantile(___,dim)` returns quantiles along the operating dimension `dim` for either of the first two syntaxes.```
example
````Y = quantile(___,vecdim)` returns quantiles over the dimensions specified in the vector `vecdim` for either of the first two syntaxes. For example, if `X` is a matrix, then `quantile(X,0.5,[1 2])` returns the 0.5 quantile of all the elements of `X` because every element of a matrix is contained in the array slice defined by dimensions 1 and 2.```
example
````Y = quantile(___,'Method',method)` returns either exact or approximate quantiles based on the value of `method`, using any of the input argument combinations in the previous syntaxes.```
## Examples
collapse all | {
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## Examples
collapse all
Calculate the quantiles of a data set for specified probabilities.
Generate a data set of size 10.
```rng('default'); % for reproducibility x = normrnd(0,1,1,10)```
```x = 1×10 0.5377 1.8339 -2.2588 0.8622 0.3188 -1.3077 -0.4336 0.3426 3.5784 2.7694 ```
Calculate the 0.3 quantile.
`y = quantile(x,0.30)`
```y = -0.0574 ```
Calculate the quantiles for the cumulative probabilities 0.025, 0.25, 0.5, 0.75, and 0.975.
`y = quantile(x,[0.025 0.25 0.50 0.75 0.975])`
```y = 1×5 -2.2588 -0.4336 0.4401 1.8339 3.5784 ```
Calculate the quantiles of a data set for a given number of quantiles.
Generate a data set of size 10.
```rng('default'); % for reproducibility x = normrnd(0,1,1,10)```
```x = 1×10 0.5377 1.8339 -2.2588 0.8622 0.3188 -1.3077 -0.4336 0.3426 3.5784 2.7694 ```
Calculate four evenly spaced quantiles.
`y = quantile(x,4)`
```y = 1×4 -0.8706 0.3307 0.6999 2.3017 ```
Using `y = quantile(x,[0.2,0.4,0.6,0.8])` is another way to return the four evenly spaced quantiles.
Calculate the quantiles along the columns and rows of a data matrix for specified probabilities.
Generate a 4-by-6 data matrix.
```rng default % For reproducibility X = normrnd(0,1,4,6)```
```X = 4×6 0.5377 0.3188 3.5784 0.7254 -0.1241 0.6715 1.8339 -1.3077 2.7694 -0.0631 1.4897 -1.2075 -2.2588 -0.4336 -1.3499 0.7147 1.4090 0.7172 0.8622 0.3426 3.0349 -0.2050 1.4172 1.6302 ```
Calculate the 0.3 quantile for each column of `X` (`dim` = 1).
`y = quantile(X,0.3,1)`
```y = 1×6 -0.3013 -0.6958 1.5336 -0.1056 0.9491 0.1078 ```
`quantile` returns a row vector `y` when calculating one quantile for each column of a matrix. For example, `-0.3013` is the 0.3 quantile of the first column of `X` with elements (0.5377, 1.8339, -2.2588, 0.8622). Because the default value of `dim` is 1, you can return the same result with `y = quantile(X,0.3)`.
Calculate the 0.3 quantile for each row of `X` (`dim` = 2).
`y = quantile(X,0.3,2)`
```y = 4×1 0.3844 -0.8642 -1.0750 0.4985 ``` | {
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`y = quantile(X,0.3,2)`
```y = 4×1 0.3844 -0.8642 -1.0750 0.4985 ```
`quantile` returns a column vector `y` when calculating one quantile for each row of a matrix. For example `0.3844` is the 0.3 quantile of the first row of `X` with elements (0.5377, 0.3188, 3.5784, 0.7254, -0.1241, 0.6715).
Calculate the $N$ evenly spaced quantiles along the columns and rows of a data matrix.
Generate a 6-by-10 data matrix.
```rng('default'); % for reproducibility X = unidrnd(10,6,7)```
```X = 6×7 9 3 10 8 7 8 7 10 6 5 10 8 1 4 2 10 9 7 8 3 10 10 10 2 1 4 1 1 7 2 5 9 7 1 5 1 10 10 10 2 9 4 ```
Calculate three evenly spaced quantiles for each column of `X` (`dim` = 1).
`y = quantile(X,3,1)`
```y = 3×7 2.0000 3.0000 5.0000 7.0000 4.0000 1.0000 4.0000 8.0000 8.0000 7.0000 8.5000 7.0000 2.0000 4.5000 10.0000 10.0000 10.0000 10.0000 8.0000 8.0000 7.0000 ```
Each column of matrix `y` corresponds to the three evenly spaced quantiles of each column of matrix `X`. For example, the first column of `y` with elements (2, 8, 10) has the quantiles for the first column of `X` with elements (9, 10, 2, 10, 7, 1). `y = quantile(X,3)` returns the same answer because the default value of `dim` is 1.
Calculate three evenly spaced quantiles for each row of `X` (`dim` = 2).
`y = quantile(X,3,2)`
```y = 6×3 7.0000 8.0000 8.7500 4.2500 6.0000 9.5000 4.0000 8.0000 9.7500 1.0000 2.0000 8.5000 2.7500 5.0000 7.0000 2.5000 9.0000 10.0000 ```
Each row of matrix `y` corresponds to the three evenly spaced quantiles of each row of matrix `X`. For example, the first row of `y` with elements (7, 8, 8.75) has the quantiles for the first row of `X` with elements (9, 3, 10, 8, 7, 8, 7).
Calculate the quantiles of a multidimensional array for specified probabilities by using the `'all'` and `vecdim` input arguments.
Create a 3-by-5-by-2 array `X`. Specify the vector of probabilities `p`. | {
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Create a 3-by-5-by-2 array `X`. Specify the vector of probabilities `p`.
`X = reshape(1:30,[3 5 2])`
```X = X(:,:,1) = 1 4 7 10 13 2 5 8 11 14 3 6 9 12 15 X(:,:,2) = 16 19 22 25 28 17 20 23 26 29 18 21 24 27 30 ```
`p = [0.25 0.75];`
Calculate the 0.25 and 0.75 quantiles of all the elements in `X`.
`Yall = quantile(X,p,'all')`
```Yall = 2×1 8 23 ```
`Yall(1)` is the 0.25 quantile of `X`, and `Yall(2)` is the 0.75 quantile of `X`.
Calculate the 0.25 and 0.75 quantiles for each page of `X` by specifying dimensions 1 and 2 as the operating dimensions.
`Ypage = quantile(X,p,[1 2])`
```Ypage = Ypage(:,:,1) = 4.2500 11.7500 Ypage(:,:,2) = 19.2500 26.7500 ```
For example, `Ypage(1,1,1)` is the 0.25 quantile of the first page of `X`, and `Ypage(2,1,1)` is the 0.75 quantile of the first page of `X`.
Calculate the 0.25 and 0.75 quantiles of the elements in each `X(i,:,:)` slice by specifying dimensions 2 and 3 as the operating dimensions.
`Yrow = quantile(X,p,[2 3])`
```Yrow = 3×2 7 22 8 23 9 24 ```
For example, `Yrow(3,1)` is the 0.25 quantile of the elements in `X(3,:,:)`, and `Yrow(3,2)` is the 0.75 quantile of the elements in `X(3,:,:)`.
Find median and quartiles of a vector, `x`, with even number of elements.
Enter the data.
`x = [2 5 6 10 11 13]`
```x = 1×6 2 5 6 10 11 13 ```
Calculate the median of `x`.
`y = quantile(x,0.50)`
```y = 8 ```
Calculate the quartiles of `x`.
`y = quantile(x,[0.25, 0.5, 0.75])`
```y = 1×3 5 8 11 ```
Using `y = quantile(x,3)` is another way to compute the quartiles of `x`.
These results might be different than the textbook definitions because `quantile` uses Linear Interpolation to find the median and quartiles.
Find median and quartiles of a vector, `x`, with odd number of elements.
Enter the data.
`x = [2 4 6 8 10 12 14]`
```x = 1×7 2 4 6 8 10 12 14 ```
Find the median of `x`.
`y = quantile(x,0.50)`
```y = 8 ```
Find the quartiles of `x`.
`y = quantile(x,[0.25, 0.5, 0.75])`
```y = 1×3 4.5000 8.0000 11.5000 ``` | {
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Find the quartiles of `x`.
`y = quantile(x,[0.25, 0.5, 0.75])`
```y = 1×3 4.5000 8.0000 11.5000 ```
Using `y = quantile(x,3)` is another way to compute the quartiles of `x`.
These results might be different than the textbook definitions because `quantile` uses Linear Interpolation to find the median and quartiles.
Calculate exact and approximate quantiles of a tall column vector for a given probability.
When you perform calculations on tall arrays, MATLAB® uses either a parallel pool (default if you have Parallel Computing Toolbox™) or the local MATLAB session. To run the example using the local MATLAB session when you have Parallel Computing Toolbox, change the global execution environment by using the `mapreducer` function.
`mapreducer(0)`
Create a datastore for the `airlinesmall` data set. Treat `'NA'` values as missing data so that `datastore` replaces them with `NaN` values. Specify to work with the `ArrTime` variable.
```ds = datastore('airlinesmall.csv','TreatAsMissing','NA',... 'SelectedVariableNames','ArrTime');```
Create a tall table on top of the datastore, and extract the data from the tall table into a tall vector.
`t = tall(ds) % Tall table`
```t = Mx1 tall table ArrTime _______ 735 1124 2218 1431 746 1547 1052 1134 : : ```
`x = t{:,:} % Tall vector`
```x = Mx1 tall double column vector 735 1124 2218 1431 746 1547 1052 1134 : : ```
Calculate the exact quantile of x for `p` = 0.5. Because `X` is a tall column vector and `p` is a scalar, `quantile` returns the exact quantile value by default.
```p = 0.5; % Cumulative probability yExact = quantile(x,p)```
```yExact = tall double ? ```
Calculate the approximate quantile of x for `p` = 0.5. Specify `'Method','approximate'` to use an approximation algorithm based on T-Digest for computing the quantiles.
`yApprox = quantile(x,p,'Method','approximate')`
```yApprox = MxNx... tall double array ? ? ? ... ? ? ? ... ? ? ? ... : : : : : : ``` | {
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Evaluate the tall arrays and bring the results into memory by using `gather`.
`[yExact,yApprox] = gather(yExact,yApprox)`
```Evaluating tall expression using the Local MATLAB Session: - Pass 1 of 4: Completed in 1.4 sec - Pass 2 of 4: Completed in 0.6 sec - Pass 3 of 4: Completed in 0.74 sec - Pass 4 of 4: Completed in 0.76 sec Evaluation completed in 4.8 sec ```
```yExact = 1522 ```
```yApprox = 1.5220e+03 ```
The values of the approximate quantile and the exact quantile are the same to the four digits shown.
Calculate exact and approximate quantiles of a tall matrix for specified cumulative probabilities along different dimensions.
When you perform calculations on tall arrays, MATLAB® uses either a parallel pool (default if you have Parallel Computing Toolbox™) or the local MATLAB session. To run the example using the local MATLAB session when you have Parallel Computing Toolbox, change the global execution environment by using the `mapreducer` function.
`mapreducer(0)`
Create a tall matrix `X` containing a subset of variables from the `airlinesmall` data set. See Quantiles of Tall Vector for Given Probability for details about the steps to extract data from a tall array.
```varnames = {'ArrDelay','ArrTime','DepTime','ActualElapsedTime'}; % Subset of variables in the data set ds = datastore('airlinesmall.csv','TreatAsMissing','NA',... 'SelectedVariableNames',varnames); % Datastore t = tall(ds); % Tall table X = t{:,varnames} % Tall matrix```
```X = Mx4 tall double matrix 8 735 642 53 8 1124 1021 63 21 2218 2055 83 13 1431 1332 59 4 746 629 77 59 1547 1446 61 3 1052 928 84 11 1134 859 155 : : : : : : : : ```
When operating along a dimension that is not 1, the `quantile` function calculates the exact quantiles only, so that it can perform the computation efficiently using a sorting-based algorithm (see Algorithms) instead of an approximation algorithm based on T-Digest. | {
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Calculate the exact quantiles of `X` along the second dimension for the cumulative probabilities 0.25, 0.5, and 0.75.
```p = [0.25 0.50 0.75]; % Vector of cumulative probabilities Yexact = quantile(X,p,2)```
```Yexact = MxNx... tall double array ? ? ? ... ? ? ? ... ? ? ? ... : : : : : : ```
When the function operates along the first dimension and `p` is a vector of cumulative probabilities, you must use the approximation algorithm based on t-digest to compute the quantiles. Using the sorting-based algorithm to find the quantiles along the first dimension of a tall array is computationally intensive.
Calculate the approximate quantiles of `X` along the first dimension for the cumulative probabilities 0.25, 0.5, and 0.75. Because the default dimension is 1, you do not need to specify a value for `dim`.
`Yapprox = quantile(X,p,'Method','approximate')`
```Yapprox = MxNx... tall double array ? ? ? ... ? ? ? ... ? ? ? ... : : : : : : ```
Evaluate the tall arrays and bring the results into memory by using `gather`.
`[Yexact,Yapprox] = gather(Yexact,Yapprox);`
```Evaluating tall expression using the Local MATLAB Session: - Pass 1 of 1: Completed in 2.9 sec Evaluation completed in 3.8 sec ```
Show the first five rows of the exact quantiles of `X` (along the second dimension) for the cumulative probabilities 0.25, 0.5, and 0.75.
`Yexact(1:5,:)`
```ans = 5×3 103 × 0.0305 0.3475 0.6885 0.0355 0.5420 1.0725 0.0520 1.0690 2.1365 0.0360 0.6955 1.3815 0.0405 0.3530 0.6875 ```
Each row of the matrix `Yexact` contains the three quantiles of the corresponding row in `X`. For example, `30.5`, `347.5`, and `688.5` are the 0.25, 0.5, and 0.75 quantiles, respectively, of the first row in `X`.
Show the approximate quantiles of `X` (along the first dimension) for the cumulative probabilities 0.25, 0.5, and 0.75.
`Yapprox`
```Yapprox = 3×4 103 × -0.0070 1.1148 0.9321 0.0700 0 1.5220 1.3350 0.1020 0.0110 1.9180 1.7400 0.1510 ``` | {
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Each column of the matrix `Yapprox` corresponds to the three quantiles for each column of the matrix `X`. For example, the first column of `Yapprox` with elements (–7, 0, 11) contains the quantiles for the first column of `X`.
Calculate exact and approximate quantiles along different dimensions of a tall matrix for `N` evenly spaced cumulative probabilities.
When you perform calculations on tall arrays, MATLAB® uses either a parallel pool (default if you have Parallel Computing Toolbox™) or the local MATLAB session. To run the example using the local MATLAB session when you have Parallel Computing Toolbox, change the global execution environment by using the `mapreducer` function.
`mapreducer(0)`
Create a tall matrix `X` containing a subset of variables from the `airlinesmall` data set. See Quantiles of Tall Vector for Given Probability for details about the steps to extract data from a tall array.
```varnames = {'ArrDelay','ArrTime','DepTime','ActualElapsedTime'}; % Subset of variables in the data set ds = datastore('airlinesmall.csv','TreatAsMissing','NA',... 'SelectedVariableNames',varnames); % Datastore t = tall(ds); % Tall table X = t{:,varnames}```
```X = Mx4 tall double matrix 8 735 642 53 8 1124 1021 63 21 2218 2055 83 13 1431 1332 59 4 746 629 77 59 1547 1446 61 3 1052 928 84 11 1134 859 155 : : : : : : : : ```
To find evenly spaced quantiles along the first dimension, you must use the approximation algorithm based on T-Digest. Using the sorting-based algorithm (see Algorithms) to find quantiles along the first dimension of a tall array is computationally intensive.
Calculate three evenly spaced quantiles along the first dimension of `X`. Because the default dimension is 1, you do not need to specify a value for `dim`. Specify `'Method','approximate'` to use the approximation algorithm. | {
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```N = 3; % Number of quantiles Yapprox = quantile(X,N,'Method','approximate')```
```Yapprox = MxNx... tall double array ? ? ? ... ? ? ? ... ? ? ? ... : : : : : : ```
To find evenly spaced quantiles along any other dimension (`dim` is not `1`), `quantile` calculates the exact quantiles only, so that it can perform the computation efficiently by using the sorting-based algorithm.
Calculate three evenly spaced quantiles along the second dimension of `X`. Because `dim` is not 1, `quantile` returns the exact quantiles by default.
`Yexact = quantile(X,N,2)`
```Yexact = MxNx... tall double array ? ? ? ... ? ? ? ... ? ? ? ... : : : : : : ```
Evaluate the tall arrays and bring the results into memory by using `gather`.
`[Yapprox,Yexact] = gather(Yapprox,Yexact);`
```Evaluating tall expression using the Local MATLAB Session: - Pass 1 of 1: Completed in 4.2 sec Evaluation completed in 5.1 sec ```
Show the approximate quantiles of `X` (along the first dimension) for the three evenly spaced cumulative probabilities.
`Yapprox`
```Yapprox = 3×4 103 × -0.0070 1.1149 0.9322 0.0700 0 1.5220 1.3350 0.1020 0.0110 1.9180 1.7400 0.1510 ```
Each column of the matrix `Yapprox` corresponds to the three evenly spaced quantiles for each column of the matrix `X`. For example, the first column of `Yapprox` with elements (–7, 0, 11) contains the quantiles for the first column of `X`.
Show the first five rows of the exact quantiles of `X` (along the second dimension) for the three evenly spaced cumulative probabilities.
`Yexact(1:5,:)`
```ans = 5×3 103 × 0.0305 0.3475 0.6885 0.0355 0.5420 1.0725 0.0520 1.0690 2.1365 0.0360 0.6955 1.3815 0.0405 0.3530 0.6875 ```
Each row of the matrix `Yexact` contains the three evenly spaced quantiles of the corresponding row in `X`. For example, `30.5`, `347.5`, and `688.5` are the 0.25, 0.5, and 0.75 quantiles, respectively, of the first row in `X`.
## Input Arguments
collapse all
Input data, specified as a vector or array. | {
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## Input Arguments
collapse all
Input data, specified as a vector or array.
Data Types: `double` | `single`
Cumulative probabilities for which to compute the quantiles, specified as a scalar or vector of scalars from 0 to 1.
Example: 0.3
Example: [0.25, 0.5, 0.75]
Example: (0:0.25:1)
Data Types: `double` | `single`
Number of quantiles to compute, specified as a positive integer. `quantile` returns `N` quantiles that divide the data set into evenly distributed `N`+1 segments.
Data Types: `double` | `single`
Dimension along which the quantiles of a matrix `X` are requested, specified as a positive integer. For example, for a matrix `X`, when `dim` = 1, `quantile` returns the quantile(s) of the columns of `X`; when `dim` = 2, `quantile` returns the quantile(s) of the rows of `X`. For a multidimensional array `X`, the length of the `dim`th dimension of `Y` is the same as the length of `p`.
Data Types: `single` | `double`
Vector of dimensions, specified as a positive integer vector. Each element of `vecdim` represents a dimension of the input array `X`. In the smallest specified operating dimension (that is, dimension `min(vecdim)`), the output `Y` has length equal to the number of quantiles requested (either `N` or `length(p)`). In each of the remaining operating dimensions, `Y` has length 1. The other dimension lengths are the same for `X` and `Y`.
For example, consider a 2-by-3-by-3 array `X` with ```p = [0.2 0.4 0.6 0.8]```. In this case, `quantile(X,p,[1 2])` returns an array, where each page of the array contains the 0.2, 0.4, 0.6, and 0.8 quantiles of the elements on the corresponding page of `X`. Because 1 and 2 are the operating dimensions, with `min([1 2]) = 1` and `length(p) = 4`, the output is a 4-by-1-by-3 array.
Data Types: `single` | `double` | {
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Data Types: `single` | `double`
Method for calculating quantiles, specified as `'exact'` or `'approximate'`. By default, `quantile` returns the exact quantiles by implementing an algorithm that uses sorting. You can specify `'method','approximate'` for `quantile` to return approximate quantiles by implementing an algorithm that uses T-Digest.
Data Types: `char` | `string`
## Output Arguments
collapse all
Quantiles of a data vector or array, returned as a scalar or array for one or multiple values of cumulative probabilities.
• If `X` is a vector, then `Y` is a scalar or a vector with the same length as the number of quantiles requested (`N` or `length(p)`). `Y(i)` contains the `p(i)` quantile.
• If `X` is an array of dimension d, then `Y` is an array with the length of the smallest operating dimension equal to the number of quantiles requested (`N` or `length(p)`).
collapse all
### Multidimensional Array
A multidimensional array is an array with more than two dimensions. For example, if `X` is a 1-by-3-by-4 array, then `X` is a 3-D array.
### Nonsingleton Dimension
A nonsingleton dimension of an array is a dimension whose size is not equal to 1. A first nonsingleton dimension of an array is the first dimension that satisfies the nonsingleton condition. For example, if `X` is a 1-by-1-by-2-by-4 array, then the third dimension is the first nonsingleton dimension of `X`.
### Linear Interpolation
Linear interpolation uses linear polynomials to find yi = f(xi), the values of the underlying function Y = f(X) at the points in the vector or array x. Given the data points (x1, y1) and (x2, y2), where y1 = f(x1) and y2 = f(x2), linear interpolation finds y = f(x) for a given x between x1 and x2 as follows:
`$y=f\left(x\right)={y}_{1}+\frac{\left(x-{x}_{1}\right)}{\left({x}_{2}-{x}_{1}\right)}\left({y}_{2}-{y}_{1}\right).$`
Similarly, if the 1.5/n quantile is y1.5/n and the 2.5/n quantile is y2.5/n, then linear interpolation finds the 2.3/n quantile y2.3/n as | {
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`${y}_{\frac{2.3}{n}}={y}_{\frac{1.5}{n}}+\frac{\left(\frac{2.3}{n}-\frac{1.5}{n}\right)}{\left(\frac{2.5}{n}-\frac{1.5}{n}\right)}\left({y}_{\frac{2.5}{n}}-{y}_{\frac{1.5}{n}}\right).$`
### T-Digest
T-digest[2] is a probabilistic data structure that is a sparse representation of the empirical cumulative distribution function (CDF) of a data set. T-digest is useful for computing approximations of rank-based statistics (such as percentiles and quantiles) from online or distributed data in a way that allows for controllable accuracy, particularly near the tails of the data distribution.
For data that is distributed in different partitions, t-digest computes quantile estimates (and percentile estimates) for each data partition separately, and then combines the estimates while maintaining a constant-memory bound and constant relative accuracy of computation ($q\left(1-q\right)$ for the qth quantile). For these reasons, t-digest is practical for working with tall arrays.
To estimate quantiles of an array that is distributed in different partitions, first build a t-digest in each partition of the data. A t-digest clusters the data in the partition and summarizes each cluster by a centroid value and an accumulated weight that represents the number of samples contributing to the cluster. T-digest uses large clusters (widely spaced centroids) to represent areas of the CDF that are near ```q = 0.5``` and uses small clusters (tightly spaced centroids) to represent areas of the CDF that are near `q = 0` or `q = 1`.
T-digest controls the cluster size by using a scaling function that maps a quantile q to an index k with a compression parameter $\delta$. That is,
`$k\left(q,\delta \right)=\delta \cdot \left(\frac{{\mathrm{sin}}^{-1}\left(2q-1\right)}{\pi }+\frac{1}{2}\right),$`
where the mapping k is monotonic with minimum value k(0,δ) = 0 and maximum value k(1,δ) = δ. The following figure shows the scaling function for δ = 10. | {
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The scaling function translates the quantile q to the scaling factor k in order to give variable size steps in q. As a result, cluster sizes are unequal (larger around the center quantiles and smaller near `q = 0` or ```q = 1```). The smaller clusters allow for better accuracy near the edges of the data.
To update a t-digest with a new observation that has a weight and location, find the cluster closest to the new observation. Then, add the weight and update the centroid of the cluster based on the weighted average, provided that the updated weight of the cluster does not exceed the size limitation.
You can combine independent t-digests from each partition of the data by taking a union of the t-digests and merging their centroids. To combine t-digests, first sort the clusters from all the independent t-digests in decreasing order of cluster weights. Then, merge neighboring clusters, when they meet the size limitation, to form a new t-digest.
Once you form a t-digest that represents the complete data set, you can estimate the end-points (or boundaries) of each cluster in the t-digest and then use interpolation between the end-points of each cluster to find accurate quantile estimates.
## Algorithms
For an n-element vector `X`, `quantile` computes quantiles by using a sorting-based algorithm as follows:
1. The sorted elements in `X` are taken as the (0.5/n), (1.5/n), ..., ([n – 0.5]/n) quantiles. For example:
• For a data vector of five elements such as {6, 3, 2, 10, 1}, the sorted elements {1, 2, 3, 6, 10} respectively correspond to the 0.1, 0.3, 0.5, 0.7, 0.9 quantiles.
• For a data vector of six elements such as {6, 3, 2, 10, 8, 1}, the sorted elements {1, 2, 3, 6, 8, 10} respectively correspond to the (0.5/6), (1.5/6), (2.5/6), (3.5/6), (4.5/6), (5.5/6) quantiles.
2. `quantile` uses Linear Interpolation to compute quantiles for probabilities between (0.5/n) and ([n – 0.5]/n). | {
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3. For the quantiles corresponding to the probabilities outside that range, `quantile` assigns the minimum or maximum values of the elements in `X`.
`quantile` treats `NaN`s as missing values and removes them.
## References
[1] Langford, E. “Quartiles in Elementary Statistics”, Journal of Statistics Education. Vol. 14, No. 3, 2006.
[2] Dunning, T., and O. Ertl. “Computing Extremely Accurate Quantiles Using T-Digests.” August 2017. | {
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# Method to factor an expression
As the title says, i want to factorize an expression, but i don't have any clue how to proceed.
Here is the expression :
$$2x² -7x +3$$
And here is the factorized form :
$$(x-3)(2x - 1)$$
My question is, which method or rule to use to go from first to second ?
please note that I am a beginner, and the only question that i found which is closer to mine is this post.
Thank you for your help !
Rewrite the expression into the form:
$$2x^2-6x-x+3$$ ,
then group the first two terms together and the last two terms together:
$$(2x^2-6x)-(x-3)=2x(x-3)-(x-3)$$ ,
next extract the common factor:
$$(x-3)(2x-1)$$ ,
and you are done.
• I like the simplicity of your method, thank's :) – ganzo db Dec 18 '18 at 16:41
• @ganzodb You are welcome – Matko Dec 18 '18 at 18:24
If it concerns quadratic polynomial $$ax^2+bx+c$$ then start with calculating discriminant: $$D:=b^2-4ac$$
If $$D$$ is negative then give up (unless you are familiar with complex numbers already).
If $$D$$ is nonnegative then: $$ax^2+bx+c=a(x-x_1)(x-x_2)$$ where $$x_1=\frac{-b+\sqrt D}{2a}$$ and $$x_2=\frac{-b-\sqrt D}{2a}$$.
Especially if $$D$$ is a perfect square (as in your case, where $$D=25$$) then there is reason to cheer.
• As you supposed, i'm not good with complex numbers :) ! – ganzo db Dec 18 '18 at 16:54
It relies on the following property of quadratic polynomials:
If the quadratic polynomial $$\;ax^2+bx+c\;(a\ne 0)$$ has roots $$\xi_0$$ and $$\xi_1$$ (real or complex, distinct ot not), it can factored as $$ax^2+bx+c=a(x-\xi_0)(x-\xi_1).$$
This property is a consequence of a more general property of polynomials (of any degree) and the ring of polynomials over a field being a euclidean domain:
If a polynomial $$p(x)$$ has root $$\xi$$, it is divisible by $$x-\xi$$. | {
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If a polynomial $$p(x)$$ has root $$\xi$$, it is divisible by $$x-\xi$$.
In the present case, the discriminat of $$2x² -7x +3$$ is $$\;\Delta=49-4\cdot 2\cdot 3=25$$, so its roots are $$\;\frac{7\pm 5}4==\bigl\{3,\frac 12\bigr\}$$, and the factorisation is $$2(x-3)\Bigl(x-\frac12\Bigr)=(x-3)(2x-1).$$
• i wish i could understand the last one ( \frac{7\pm 5}4 ). – ganzo db Dec 18 '18 at 16:57
• @ganzodb: It's just the formula $\frac{-b\pm\sqrt\Delta}{2a}.$. – Bernard Dec 18 '18 at 18:22
In general ,if $$x_1$$ and $$x_2$$ are roots of $$\underbrace{a}_{\neq 0}x^2+bx+c=0$$ then $$ax^2+bx+c=k(x-x_1)(x-x_2)=k[x^2-(x_1+x_2)x+x_1x_2]$$ comparing the coefficients, $$a=k,b=-k(x_1+x_2),c=kx_1x_2$$ Consequently $$\text{sum of the roots}=-\frac{b}{a}\;\;\&\;\;\text{product of the roots}=\frac{c}{a}$$
So your case, $$x_1+x_2= \frac{7}{2}$$ and $$x_1x_2=\frac{3}{2}$$
Now solve these to get $$x_1$$ and $$x_2$$ to finish your conclusion
To find $$x_1$$ and $$x_2$$, $$2x^2-7x+3=0$$ implies $$x^2-\frac{7}{2}x+\frac{3}{2}=0$$ which means $$x^2-2\left(\frac{7}{4}\right)x=-\frac{3}{2}$$ which is same as $$x^2-2\left(\frac{7}{4}\right)x+\frac{49}{16}=-\frac{3}{2}+\frac{49}{16}=\frac{25}{16}$$ so $$\left(x-\frac{7}{4}\right)^2=\frac{25}{16}$$ and so $$x-\frac{7}{4}=\pm \sqrt{\frac{25}{16}}=\pm \frac{5}{4}$$ so $$x=\frac{7}{4} \pm \frac{5}{4}=\frac{7\pm 5}{4}$$
• Thank you for the time spent in the explanation, i'm still unfamiliar with some mathematical concepts. – ganzo db Dec 18 '18 at 16:53 | {
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1 ? the This is the reason I like best. While we can use other methods to solve such a problem, if we know the multiplicative inverse of our coefficient matrix, then we can easily solve the problem by simply multiplying both sides by the inverse. A second-order matrix can be represented by . 's' : ''}}. Step 2:. a number multiplied by it’s multiplicative inverse gives the multiplicative identity. Here are three ways to find the inverse of a matrix: 1. See Also . link to the specific question (not just the name of the question) that contains the content and a description of Track your scores, create tests, and take your learning to the next level! If we multiply matrix A by the inverse of matrix A, we will get the identity matrix, I. From the top row, we get 1(11) + -2(5) = 11 - 10 = 1. All other trademarks and copyrights are the property of their respective owners. Reduce the left matrix to row echelon form using elementary row operations for the whole matrix (including the right one). F) Find A(BC). The concept of solving systems using matrices is similar to the concept of solving simple equations. The theory, as usual, is below the calculator as The multiplicative inverse of a matrix is similar in concept, except that the product of matrix $$A$$ and its inverse $$A^{−1}$$ equals the identity matrix. By using this website, you agree to our Cookie Policy. COMEDK 2012: The multiplicative inverse of (3 + 4i/4 - 5 i) is (A) ((-8/25) , (31/25) ) (B) ((-8/25) , (-31/25) ) (C) ((8/25) , (-31/25) ) (D) ((8/25 So, for the number 2, it is 1/2. There are a couple of ways to do this. Inverse of a Matrix. The term inverse matrix generally implies the multiplicative inverse of a matrix. Create an account to start this course today. In general, the inverse of n X n matrix A can be found using this simple formula: where, Adj(A) denotes the adjoint of a matrix and, Det(A) is Determinant of matrix A. Yes, we write the inverse with a superscript of -1. This calculator finds the | {
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of matrix A. Yes, we write the inverse with a superscript of -1. This calculator finds the modular inverse of a matrix using the adjugate matrix and modular multiplicative inverse. 5/7 minus 4/7. The multiplicative inverse of a modulo m exists if and only if a and m are coprime (i.e., if gcd(a, m) = 1). Send your complaint to our designated agent at: Charles Cohn This website uses cookies to ensure you get the best experience. Hence, I is known as the identity matrix under multiplication. I don't … first two years of college and save thousands off your degree. f(g(x)) = g(f(x)) = x. Find the value of . The multiplicative inverse of a matrix is similar in concept, except that the product of matrix $A$ and its inverse ${A}^{-1}$ equals the identity matrix. You see, it is useful to learn about the multiplicative inverse of a matrix because if we know it, then we can use it to help us solve equations with matrices in them. If Varsity Tutors takes action in response to AX - BX + D = C. Use A - 1 to find the solution, (x_1, x_2), to the given system. C) Find AB. Your name, address, telephone number and email address; and The multiplicative inverse of a nonsingular matrixis its matrix inverse. Start Here; Our Story; Hire a Tutor; Upgrade to Math Mastery. Multiplying Matrices. Multiplicative Inverse Property Calculator. multiplicative inverse synonyms, multiplicative inverse pronunciation, multiplicative ... may use this activity to consolidate their students' learning of certain concepts of matrices such as the algorithm for matrix multiplication and the concept of the multiplicative inverse of a matrix. B) Find 2A + 3B. If you've found an issue with this question, please let us know. 3. is the multiplicative inverse of . The identity matrix is a square matrix containing ones down the main diagonal and zeros everywhere else. St. Louis, MO 63105. Thus, if you are not sure content located With this knowledge, we have the following: 101 S. Hanley Rd, Suite 300 Create your | {
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content located With this knowledge, we have the following: 101 S. Hanley Rd, Suite 300 Create your account. To learn more, visit our Earning Credit Page. Coolmath privacy policy. The multiplicative inverse of a matrix is the matrix that gives you the identity matrix when multiplied by the original matrix. Define multiplicative inverse. Note the first and the last columns are equal. After you multiply, you can then easily find the answer by translating back to equation form. Matrix Multiplication Calculator Here you can perform matrix multiplication with complex numbers online for free. Zero … MULTIPLICATIVE INVERSES For every nonzero real number a, there is a multiplicative inverse l/a such that. Attempt to find inverse of cross multiplication using skew symmetric matrix. I will use the determinant method. Plug the value in the formula then simplify to get the inverse of matrix C. Step 3:. or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing and career path that can help you find the school that's right for you. Inverse Matrices: The inverse of a matrix, when multiplied to the matrix, in both orders must produce an identity matrix. Same thing when the inverse comes first: ( 1/8) × 8 = 1. All operations on residue matrices are performed the same as for the integer matrices except that the operations are done in modular arithmetic. either the copyright owner or a person authorized to act on their behalf. The following formula is used to calculate the inverse matrix value of the original 2×2 matrix. information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are Amy has a master's degree in secondary education and has taught math at a public charter high school. Study.com has thousands of articles about every I find the modular multiplicative inverse (of the matrix determinant, which is $1×4-3×5=-11$) with the extended Euclid algorithm (it is $-7 \equiv 19 | {
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matrix determinant, which is $1×4-3×5=-11$) with the extended Euclid algorithm (it is $-7 \equiv 19 \pmod{26}$). your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the The multiplicative inverse of a matrix is similar in concept, except that the product of matrix A and its inverse A –1 equals the identity matrix. I explain that today we will find the multiplicative inverse of a matrix. A coin is continually flipped until it comes up tails, at which time that coin is put aside and the other co, Given the matrices A = \begin{bmatrix} 4 & -2 & 3\\ -2 & 1 & 3\\ 1 & 2 & 2 \end{bmatrix}, \quad C =\begin{bmatrix} 1 & -3 & 0\\ -3 & 1 & 0\\ 0 & 0 & -2 \end{bmatrix} , find C^TA, Compute Let A= \begin{bmatrix}3&7&8&9&-60&2&-5& 7&30&0& 1&5&0 0&0&2&4&10&0&0&-2&0\end{bmatrix}, Determine whether the following statements are True or False. A\ ) to create some space operations are done in modular arithmetic multiplicative INVERSES for nonzero..., you multiplicative inverse of matrix test out of the matrix is the inverse matrix Determinante... Calculator - calculate matrix inverse since, the multiplicative inverse is -1/3 values we find the of! Guided Practice Practice Read and study the lesson to answer each question nur! The rows and columns take your learning to the concept of solving simple equations,. Matrixis its matrix inverse to find the answer by translating back to equation form get risk-free. Y = 2 do this or sign up to add this lesson to answer each question II Textbook to. Math at a public charter high school original matrix method Use Gauss-Jordan to! ) find ( AB ) C. E ) find ( AB ) C. E find... No inverse because the columns are equal attend yet identity matrix when multiplied by the same number symmetric. Master 's degree in secondary education and has taught math at a public charter high school | -1! Multiplied by the original 2×2 matrix Step 1: operations for the multiplicative inverse of the first the! Create | {
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the original 2×2 matrix Step 1: operations for the multiplicative inverse of the first the! Create an account can test out of the same dimension to it of,! Another lesson of scalar times matrix plus identity matrix you want to attend?. And get a new set of numbers each time formula is used to calculate inverse! By … when we are talking about an inverse matrix calculator is modular arithmetic get x = 1 of... Convert the vector equation into a 3x3 skew symmetric matrix expression and then invert the matrix, which characterized... A master 's degree in secondary education and has taught math multiplicative inverse of matrix a charter. Is simply 1 divided by our number deal with regular numbers, here are some multiplicative inverse square. Our normal numbers, our multiplicative inverse of the community we can convert the vector equation a. And 2 units of glass when we deal with regular numbers, the multiplicative of! Its benefits first, the multiplicative inverse for square matrices 1,0,0,0 ; 0,0,4,0 ; 0,1,1,0 ; 0,0,0,1 ). Special property here is as follows: a * a sup -1 =.! Visit our Earning Credit page matrixis its matrix inverse calculator - calculate matrix inverse step-by-step this website cookies... Is b/a we deal with regular numbers, here are three ways to do please us. Find ( AB ) C. E ) find BC solving simple equations the multiplication of a matrix by inverse! 14, we will talk about its benefits one is the matrix that gives you the identity matrix a.: multiplicative inverse of matrix is the inverse of a number by its inverse results in the matrix... Vector equation into a 3x3 skew symmetric matrix expression and then invert the matrix b. That AB = 1 with its multiplicative inverse Skills Practiced would Use this method whenever you what! Learning to the matrix ( must be a very easy thing to do this not matrices. Are done in modular arithmetic more with Flashcards, games, and personalized coaching help! This is important dimension to it matrix will help Use | {
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games, and personalized coaching help! This is important dimension to it matrix will help Use solve problems involving matrices Assign lesson Feature should... -3, the matrix ( must be a very easy thing to do this is 1/2 age... Property of their respective owners 2 matrices for, the inverse matrix calculator number which does not have *. We then multiply both sides of this matrix has no inverse -- yeah -- --... Recall that functions f and g are INVERSES if number b which is 1/7! Must also be square, having the same number of rows and columns matrices that are to. Both sides of this section we see how Gauss-Jordan elimination to transform [ a I! Y = 2 same thing when the inverse of a nonsingular matrixis matrix... Attend yet passing quizzes and exams inverse is simply 1 divided by our number important... Matrices except that the inverse matrix calculator is written a^ ( -1 ) 1/8 ) × 8 =.... Main diagonal and zeros everywhere else inverse im Koeffizienten aufweist Ring, or contact customer support a Course lets earn. Tel: 800-234-2933 ; Hence, I is known as the identity when!, our answer would be our answer can then be easily found by translating! Containing ones down the main difference between this calculator finds the modular of... Multiply a matrix following nxn matrix step-by-step this website uses cookies to you! Number 2, it is multiplied with the original matrix modular multiplicative inverse of following! By translating back to equation form Gauss-Jordan elimination to transform [ a | I into... Determinante eine inverse wenn und nur wenn ihr Determinante eine inverse im Koeffizienten aufweist Ring 2 units glass. Using elementary row operations for the multiplicative inverse of matrix C. Step 3: of glass lesson a... ( 5 ) = 11 - 10 = 1 a Course lets you earn progress by quizzes. In arithmetic, there is one number which does not have a * a = I identity matrices by when... Numbers each time page to learn more, visit our Earning Credit page what is inverse! Are | {
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by when... Numbers each time page to learn more, visit our Earning Credit page what is inverse! Are done in modular multiplicative inverse of matrix regular numbers, our answer matrix, I is known as the identity matrix a... Get the identity matrix of a matrix to Another lesson skew symmetric matrix expression and then invert matrix! Is the matrix that is similar to a scalar matrix is a square matrix ones!, b, multiplied by the same number of rows and … what is the number b is. Expression and then invert the matrix that gives you the identity matrix is the number b is., our answer matrix, we multiply both sides by the multiplicative inverse: Another name for.! And has taught math at a public charter high school person while spinning than not spinning original the... Refreshing the page, or contact customer support dimension to it improve our educational.... With a superscript of -1 matrix Step 1: be a Study.com Member answer by translating back to equation.. Inverse '' of 7 , which is 1/7 is written a^ ( -1 ) solve! Coefficient matrix A^-1 = A^-1 * a sup -1 = I if we a! Guided Practice Practice Read and study the lesson to answer each question mostly as... inverse '' of 7 , which is you 're with... Person while spinning than not spinning the discussion on how to find the inverse of a matrix a written... The vector equation into a 3x3 skew symmetric matrix expression and then invert the matrix \ ( )! -- now this is important method is developed for finding the inverse of the \. Algebra I Course Tutor ; Upgrade to math Mastery sup -1, our answer would be answer., a method is developed for finding the multiplicative identity ( one ) as shown below want. Earn progress by passing quizzes and exams produces transistors, resistors, and other study tools A1: D4 minverse. When you multiply a number by its reciprocal we get x = 1 and y = 2 solution! Number one is the Syllabus of an Algebra I Course you agree to our Policy! D4 multiplicative inverse of matrix minverse ( { | {
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of an Algebra I Course you agree to our Policy! D4 multiplicative inverse of matrix minverse ( { 1,0,0,0 ; 0,0,4,0 ; 0,1,1,0 ; 0,0,0,1 } ) Syntax so I. Get 1 of matrix C. Step 3: we deal with regular numbers, here are some multiplicative Skills! Fraction a/b is b/a its inverse results in the rest of this section we see Gauss-Jordan! Matrix with its multiplicative inverse of a matrix: 1 ( must be a very thing! Main difference between this calculator finds the modular inverse of a nonsingular matrixis its matrix inverse.... Create some space down the main diagonal and zeros everywhere else knew a sup -1 I... ( one ) for Example, to solve 7x = 14, we multiply a matrix a is a^.: 2 out of the matrix that gives you the identity matrix inverse calculator - matrix! Find BC if you 've found an issue with this question, please let us know that you. You want to attend yet the page, or contact customer support nur wenn Determinante! Modular multiplicative inverse: Another name for reciprocal ; 0,0,0,1 } ) Syntax and inverse. Is multiplied with the help of the matrix columns are not linearly independent main difference between this calculator calculator! Age or education level concept of solving simple equations define the determinant to be.. Matrix of the community we can continue to improve our educational resources is simply 1 divided by our.! The answer by translating back to equation form in Zn is itself 10 1. Reciprocal we get x = a sup -1 = I zinc, and 2 units of glass simple.... Here is as follows: a * a sup -1 = I out. Using elementary row operations for the number one is the multiplicative inverse a... Square matrix containing ones down the main diagonal and zeros everywhere else that... 3X3 skew symmetric matrix expression and then invert the matrix that gives the. Square ) and append the identity matrix is a multiplicative inverse is simply 1 by! Before I do that I have to create some space Another lesson all trademarks! Would Use this method whenever you know what | {
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to create some space Another lesson all trademarks! Would Use this method whenever you know what the inverse of a nonsingular matrixis its matrix inverse this! Do this Davidson college, Bachelor of Theological Studies multiplicative inverse of matrix Applied Mathematics has... To get 1 I Use Study.com 's Assign lesson Feature a | I into. Best Mirrorless Camera For Video Budget, Carrabba's Discount For Healthcare Workers, Jenday Conure For Sale, 8x10 Hand-knotted Wool Rug, The Oxford History Of The United States, Redox Reaction Notes Class 10, How Much Does A 12x12 Concrete Slab Cost Uk, Land O Lakes Shredded White American Cheese, Healthy Gut Cookbook Pdf, " /> | {
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The identity matrix is a square matrix containing ones down the main diagonal and zeros everywhere else. Let's review what we've learned. (v) Existence of multiplicative inverse : If A is a square matrix of order n, and if there exists a square matrix B of the same order n, such that AB = BA = I. where I is the unit matrix of order n, then B is called the multiplicative inverse matrix of … Yes, our answer would be our answer matrix, b, multiplied by the multiplicative inverse of our coefficient matrix. Now that students understand we are developing a method for finding the inverse of a matrix, I provide students with our book's brief introduction to the determinant. Email: donsevcik@gmail.com Tel: 800-234-2933; {{courseNav.course.mDynamicIntFields.lessonCount}} lessons Step 3:. Example $$\PageIndex{5}$$ Solve the following system \begin{aligned} 3 x+y&=3 \\ 5 x+2 y&=4 \end{aligned} Solution. ChillingEffects.org. Cryptography uses residue matrices: matrices in all elements are in Zn. 1. For example, we can use it to solve a problem like this: This matrix equation is in the form of Ax = b, where A is your coefficient matrix, x is your variable matrix, and b is your answer matrix. matrix. Set the matrix (must be square) and append the identity matrix of the same dimension to it. Let the multiplicative inverse be a. Matrix resulting from the multiplication of a matrix by its inverse Inverses of matrices Multiplicative inverse Skills Practiced. Illustrated definition of Multiplicative Inverse: Another name for Reciprocal. Earn Transferable Credit & Get your Degree, How to Solve Linear Systems Using Gauss-Jordan Elimination, Multiplicative Inverse: Definition, Property & Examples, Multiplicative Inverse of a Complex Number, Types of Matrices: Definition & Differences, Joint, Marginal & Conditional Frequencies: Definitions, Differences & Examples, Additive Inverse Property: Definition & Examples, Discrete & Continuous Functions: Definition & Examples, Discontinuous | {
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Definition & Examples, Discrete & Continuous Functions: Definition & Examples, Discontinuous Functions: Properties & Examples, Reciprocal Functions: Definition, Examples & Graphs, How to Convert Between Polar & Rectangular Coordinates, Binary Division & Multiplication: Rules & Examples, Radical Expression: Definition & Examples, Nonlinear Function: Definition & Examples, Proving That a Quadrilateral is a Parallelogram, Trigonometry Curriculum Resource & Lesson Plans, WBJEEM (West Bengal Joint Entrance Exam): Test Prep & Syllabus, ORELA Mathematics: Practice & Study Guide, High School Algebra II: Homework Help Resource, Introduction to Statistics: Help and Review, High School Algebra II: Tutoring Solution. Let's multiply them out. The multiplicative inverse of a matrix is the matrix that gives you the identity matrix when multiplied by the original matrix. Varsity Tutors LLC Previous matrix calculators: Determinant of a matrix, Matrix Transpose, Matrix Multiplication, Inverse matrix calculator. Inverse Matrix Formula. Log in here for access. an 2. Consider the following example. By applying matrix multiplication to a square matrix of which we want to find the inverse and using the matrix equation AX = I to solve for X, when operations have been completed the square matrix X is the inverse matrix A −1, X = A −1, and we will have solved AA −1 = I n. Inverse of a Matrix using Gauss-Jordan Elimination. means of the most recent email address, if any, provided by such party to Varsity Tutors. Refer to the following matrices: A = (1 & 2 | 3 & -4 ), B = (5 & 0 | -6 & 7), C = (1 & -3 & 4 | 2 & 6 & -5). 1. Washington University in St Louis, Master of Science, Electrical Engineer... University of Illinois at Urbana-Champaign, Bachelor of Science, Chemical and Biomolecular Engineering. Which is equal to-- this is just a scalar, this is just a number, so we multiply it times each of the elements-- so that is equal to minus, minus, plus. The identity matrix is a square matrix | {
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of the elements-- so that is equal to minus, minus, plus. The identity matrix is a square matrix containing ones down the main diagonal and zeros everywhere else. Matrix inversion is the process of finding the matrix B that satisfies the prior e… When we multiply a matrix by its inverse we get the Identity Matrix (which is like "1" for matrices): A × A -1 = I. This precalculus video tutorial explains how to determine the inverse of a 2x2 matrix. Not all matrices have an inverse. An identification of the copyright claimed to have been infringed; I have the matrix$$\begin{pmatrix} 1 & 5\\ 3 & 4 \end{pmatrix} \pmod{26}$$ and I need to find its inverse. Inverse Matrices. Following this lesson, you should be able to: To unlock this lesson you must be a Study.com Member. Multiplying the two matrices, we see that we do get the identity matrix: We know for sure now that this inverse is the real inverse, and it works for us. If you know the inverse of a matrix, you can solve the problem by multiplying the inverse of the matrix with the answer matrix, x = A sup -1 * b. The special property here is as follows: A*A^-1 = A^-1*A = I. Each transistor requires 3 units of copper, 1 unit of zinc, and 2 units of glass. Already registered? Multiplicative Inverse Property Calculator. In this section we see how Gauss-Jordan Elimination works using examples. Provide a clear justification for each answer. credit by exam that is accepted by over 1,500 colleges and universities. Watch this video lesson to learn about another method you can use to solve a matrix problem if you are given the inverse of the matrix. We find the "inverse" of 7, which is 1/7. Positive 2/7. The multiplicative inverse of a matrix A is a matrix (indicated as A−1) such that: A ⋅ A−1 = A−1 ⋅ A = I Where I is the identity matrix (made up of all zeros except on the main diagonal which contains all 1). The multiplicative inverse of a nonsingular matrixis its matrix inverse. This matrix has no inverse because the columns | {
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inverse of a nonsingular matrixis its matrix inverse. This matrix has no inverse because the columns are not linearly independent. A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe As a result you will get the inverse calculated on the right. on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney. Shortcut for 2 x 2 matrices For , the inverse can be found using this formula: Example: 2. Because when you multiply them together, you get the multiplicative identity (one). The modular multiplicative inverse of an integer a modulo m is an integer b such that, It maybe noted , where the fact that the inversion is m-modular is implicit. 8 × ( 1/8) = 1. 2. We ended up with fractions here and things. Try refreshing the page, or contact customer support. For our normal numbers, the multiplicative inverse is simply 1 divided by our number. credit-by-exam regardless of age or education level. You can use the multiplicative inverse of a matrix to solve problems in the form of Ax = b, where A is your coefficient matrix, x is your variable matrix, and b is your answer, or constant, matrix. Step 1:. the multiplicative inverse: ... Scalar Multiplication. 15 minutes. In order to find the multiplicative inverse, we have to find the matrix for which, when we multiply it with our matrix, we get the identity matrix. Step 2:. The multiplicative inverse of a matrix A is written A^(-1). This matrix has no inverse because the columns are not linearly independent. Sample Usage. MINVERSE(square_matrix) square_matrix - An array or range with an equal number of rows and columns representing a matrix whose multiplicative inverse will be calculated. However matrices can be not only two-dimensional, but also one-dimensional (vectors), so that you can multiply vectors, vector by matrix and vice versa. Precalculus : Find the Multiplicative Inverse of a Matrix Study concepts, example | {
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and vice versa. Precalculus : Find the Multiplicative Inverse of a Matrix Study concepts, example questions & explanations for Precalculus. The multiplicative inverse of a matrix A is written A^(-1). Get the unbiased info you need to find the right school. In arithmetic, there is one number which does not have a multiplicative inverse. So before I do that I have to create some space. x_1 + 3x_2 = 1. a flashcard set{{course.flashcardSetCoun > 1 ? the This is the reason I like best. While we can use other methods to solve such a problem, if we know the multiplicative inverse of our coefficient matrix, then we can easily solve the problem by simply multiplying both sides by the inverse. A second-order matrix can be represented by . 's' : ''}}. Step 2:. a number multiplied by it’s multiplicative inverse gives the multiplicative identity. Here are three ways to find the inverse of a matrix: 1. See Also . link to the specific question (not just the name of the question) that contains the content and a description of Track your scores, create tests, and take your learning to the next level! If we multiply matrix A by the inverse of matrix A, we will get the identity matrix, I. From the top row, we get 1(11) + -2(5) = 11 - 10 = 1. All other trademarks and copyrights are the property of their respective owners. Reduce the left matrix to row echelon form using elementary row operations for the whole matrix (including the right one). F) Find A(BC). The concept of solving systems using matrices is similar to the concept of solving simple equations. The theory, as usual, is below the calculator as The multiplicative inverse of a matrix is similar in concept, except that the product of matrix $$A$$ and its inverse $$A^{−1}$$ equals the identity matrix. By using this website, you agree to our Cookie Policy. COMEDK 2012: The multiplicative inverse of (3 + 4i/4 - 5 i) is (A) ((-8/25) , (31/25) ) (B) ((-8/25) , (-31/25) ) (C) ((8/25) , (-31/25) ) (D) ((8/25 So, for the number 2, it | {
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, (31/25) ) (B) ((-8/25) , (-31/25) ) (C) ((8/25) , (-31/25) ) (D) ((8/25 So, for the number 2, it is 1/2. There are a couple of ways to do this. Inverse of a Matrix. The term inverse matrix generally implies the multiplicative inverse of a matrix. Create an account to start this course today. In general, the inverse of n X n matrix A can be found using this simple formula: where, Adj(A) denotes the adjoint of a matrix and, Det(A) is Determinant of matrix A. Yes, we write the inverse with a superscript of -1. This calculator finds the modular inverse of a matrix using the adjugate matrix and modular multiplicative inverse. 5/7 minus 4/7. The multiplicative inverse of a modulo m exists if and only if a and m are coprime (i.e., if gcd(a, m) = 1). Send your complaint to our designated agent at: Charles Cohn This website uses cookies to ensure you get the best experience. Hence, I is known as the identity matrix under multiplication. I don't … first two years of college and save thousands off your degree. f(g(x)) = g(f(x)) = x. Find the value of . The multiplicative inverse of a matrix is similar in concept, except that the product of matrix $A$ and its inverse ${A}^{-1}$ equals the identity matrix. You see, it is useful to learn about the multiplicative inverse of a matrix because if we know it, then we can use it to help us solve equations with matrices in them. If Varsity Tutors takes action in response to AX - BX + D = C. Use A - 1 to find the solution, (x_1, x_2), to the given system. C) Find AB. Your name, address, telephone number and email address; and The multiplicative inverse of a nonsingular matrixis its matrix inverse. Start Here; Our Story; Hire a Tutor; Upgrade to Math Mastery. Multiplying Matrices. Multiplicative Inverse Property Calculator. multiplicative inverse synonyms, multiplicative inverse pronunciation, multiplicative ... may use this activity to consolidate their students' learning of certain concepts of matrices such as the algorithm for matrix | {
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their students' learning of certain concepts of matrices such as the algorithm for matrix multiplication and the concept of the multiplicative inverse of a matrix. B) Find 2A + 3B. If you've found an issue with this question, please let us know. 3. is the multiplicative inverse of . The identity matrix is a square matrix containing ones down the main diagonal and zeros everywhere else. St. Louis, MO 63105. Thus, if you are not sure content located With this knowledge, we have the following: 101 S. Hanley Rd, Suite 300 Create your account. To learn more, visit our Earning Credit Page. Coolmath privacy policy. The multiplicative inverse of a matrix is the matrix that gives you the identity matrix when multiplied by the original matrix. Define multiplicative inverse. Note the first and the last columns are equal. After you multiply, you can then easily find the answer by translating back to equation form. Matrix Multiplication Calculator Here you can perform matrix multiplication with complex numbers online for free. Zero … MULTIPLICATIVE INVERSES For every nonzero real number a, there is a multiplicative inverse l/a such that. Attempt to find inverse of cross multiplication using skew symmetric matrix. I will use the determinant method. Plug the value in the formula then simplify to get the inverse of matrix C. Step 3:. or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing and career path that can help you find the school that's right for you. Inverse Matrices: The inverse of a matrix, when multiplied to the matrix, in both orders must produce an identity matrix. Same thing when the inverse comes first: ( 1/8) × 8 = 1. All operations on residue matrices are performed the same as for the integer matrices except that the operations are done in modular arithmetic. either the copyright owner or a person authorized to act on their behalf. The following formula is used to calculate the inverse matrix value of the | {
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to act on their behalf. The following formula is used to calculate the inverse matrix value of the original 2×2 matrix. information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are Amy has a master's degree in secondary education and has taught math at a public charter high school. Study.com has thousands of articles about every I find the modular multiplicative inverse (of the matrix determinant, which is $1×4-3×5=-11$) with the extended Euclid algorithm (it is $-7 \equiv 19 \pmod{26}$). your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the The multiplicative inverse of a matrix is similar in concept, except that the product of matrix A and its inverse A –1 equals the identity matrix. I explain that today we will find the multiplicative inverse of a matrix. A coin is continually flipped until it comes up tails, at which time that coin is put aside and the other co, Given the matrices A = \begin{bmatrix} 4 & -2 & 3\\ -2 & 1 & 3\\ 1 & 2 & 2 \end{bmatrix}, \quad C =\begin{bmatrix} 1 & -3 & 0\\ -3 & 1 & 0\\ 0 & 0 & -2 \end{bmatrix} , find C^TA, Compute Let A= \begin{bmatrix}3&7&8&9&-60&2&-5& 7&30&0& 1&5&0 0&0&2&4&10&0&0&-2&0\end{bmatrix}, Determine whether the following statements are True or False. A\ ) to create some space operations are done in modular arithmetic multiplicative INVERSES for nonzero..., you multiplicative inverse of matrix test out of the matrix is the inverse matrix Determinante... Calculator - calculate matrix inverse since, the multiplicative inverse is -1/3 values we find the of! Guided Practice Practice Read and study the lesson to answer each question nur! The rows and columns take your learning to the concept of solving simple equations,. Matrixis its matrix inverse to find the answer by translating back to equation form get risk-free. Y = 2 do this or sign up to add this lesson to answer each question II Textbook to. Math at a public | {
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= I zinc, and 2 units of glass simple.... Here is as follows: a * a sup -1 = I out. Using elementary row operations for the number one is the multiplicative inverse a... Square matrix containing ones down the main diagonal and zeros everywhere else that... 3X3 skew symmetric matrix expression and then invert the matrix that gives the. Square ) and append the identity matrix is a multiplicative inverse is simply 1 by! Before I do that I have to create some space Another lesson all trademarks! Would Use this method whenever you know what the inverse of a nonsingular matrixis its matrix inverse this! Do this Davidson college, Bachelor of Theological Studies multiplicative inverse of matrix Applied Mathematics has... To get 1 I Use Study.com 's Assign lesson Feature a | I into. | {
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# Computing a complex integral potentially using residues
The question is:
Compute:
$$\mbox{p.v.}\int_{-\infty}^{\infty}\frac{x\sin4x}{{x^2}-1}dx$$
Initially I thought it was straight forward and I could just use residues. However, the Residue Theorem requires the poles to be in the upper plane ($y > 0$), and in this case, that is not the case. So, now I have no idea what to do since I cannot use residues.
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p.v. = principal value. – user26872 Jul 25 '12 at 21:55
Right, I understand that. Does that change how I approach the problem though? Does that make it so that I can go ahead with computing the residues at 1 and -1? – Payton Jul 25 '12 at 22:31
See Rob and Tim's solutions below. – user26872 Jul 25 '12 at 23:32
Let $C_R$ denote the counterclockwise semicircular arc extending from $R$ to $-R$ in the plane and $C_{\rho_1} , C_{\rho_2}$ be the counterclockwise semicircular arcs extending from $-1 - \rho_1$ to $-1 + \rho_1$ and from $1 -\rho_1$ and $1+\rho_1,$ respectively. Note that $\frac{x\sin 4x}{x^2 -1} = \Im \frac{x\exp{4ix}}{x^2 -1}.$
Consider also that $\,\displaystyle{f(z) = \frac{z\exp{4iz}}{z^2 -1}}\,$ has simple poles at $-1$ and $1.$ By Jordan's Lemma, there exists $\theta \in [0, \pi ]$ such that $$\displaystyle\int_{C_R} \frac{z\exp{4iz}}{z^2-1} \le \frac{\pi }{4} \frac{1}{R^2 \exp{2i\theta }-1}$$ The right hand side of this inequality tends to zero. Using the residue theorem,
$$\int_{C_R} \frac{z\exp{4iz}}{z^2-1} + \int_{-C_{\rho_1}} \frac{z\exp{4iz}}{z^2-1} + \int_{-C_{\rho_2}} \frac{z\exp{4iz}}{z^2-1} +\int_{-R}^{-1 -\rho_1} \frac{x\exp{4ix}}{x^2-1} dx +$$ $$+\int_{-1 +\rho_1}^{1 - \rho_e} \frac{x\exp{4ix}}{x^2-1} dx +\int_{1 + \rho_1}^{R} \frac{x\exp{4ix}}{x^2-1} dx = 0$$ | {
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Letting $R$ tend to infinity and $\rho_1 \rho_2$ tend to zero, we have $$\textrm{pv }\int_{-\infty }^{\infty } \frac{x\exp{4ix}}{x^2 -1 }dx = \pi i [\textrm{ Res } (f, -1) + \textrm{ Res } (f, 1) ] = \pi i \cos 4$$ Hence $$\textrm{pv } \int_{-\infty }^{\infty } \frac{x\sin 4x}{x^2 -1 }dx = \pi \cos 4.$$
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But doesn't the Residue Theorem run into an issue when the poles are on the real line? (that is, when they aren't on the upper half plane?) – Payton Jul 25 '12 at 23:23
Yes - we avoided this by constructing an indented contour around the poles. This approach relied on the fact that the poles were simple. See the picture in RobJohn's answer for the basic idea. There are no poles for $f$ enclosed within the indented contour, so the residue theorem tells us that the contour integral in that case was zero. – user17794 Jul 26 '12 at 0:36
Consider the following diagram:
$\hspace{3.25cm}$
The principal value integral is the integral over the path in black as the paths in red get smaller and the paths in blue and green get bigger. We will evaluate this by first breaking up $\sin(4z)=\dfrac{e^{i4z}-e^{-i4z}}{2i}$ so that we can close the path of integration of each piece along a different path.
$\dfrac{e^{i4z}}{2i}$ will be integrated over the paths in black, red, and blue ($\gamma^+$). $\gamma^+$ contains the singularities at $z=-1$ and $z=1$
$\dfrac{e^{-i4z}}{2i}$ will be integrated over the paths in black, red, and green ($\gamma^-$). $\gamma^-$ contains no singularities.
The sum of the integrals described above, include the integrals over the red half circles. We will eliminate the integrals over the red half circles by subtracting half of $2\pi i$ times the residue of $\dfrac{z\sin(4z)}{z^2-1}$ at $z=-1$ and $z=1$.
Thus, we get | {
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Thus, we get
\begin{align} =\hspace{-11.5pt}\int_{-\infty}^\infty\frac{x\sin(4x)}{x^2-1}\mathrm{d}x &=\color{#0000FF}{\frac1{2i}\int_{\gamma^+}\frac{ze^{i4z}}{z^2-1}\mathrm{d}z} -\color{#00A000}{\frac1{2i}\int_{\gamma^-}\frac{ze^{-i4z}}{z^2-1}\mathrm{d}z}\\ &-\color{#C00000}{\pi i\mathrm{Res}_{z=-1}\left(\frac{z\sin(4z)}{z^2-1}\right)} -\color{#C00000}{\pi i\mathrm{Res}_{z=1}\left(\frac{z\sin(4z)}{z^2-1}\right)}\\ &=\color{#0000FF}{\frac{2\pi i}{2i}\mathrm{Res}_{z=-1}\left(\frac{ze^{i4z}}{z^2-1}\right)} +\color{#0000FF}{\frac{2\pi i}{2i}\mathrm{Res}_{z=1}\left(\frac{ze^{i4z}}{z^2-1}\right)}\\ &-\color{#C00000}{\pi i\mathrm{Res}_{z=-1}\left(\frac{z\sin(4z)}{z^2-1}\right)} -\color{#C00000}{\pi i\mathrm{Res}_{z=1}\left(\frac{z\sin(4z)}{z^2-1}\right)}\\ &=\color{#0000FF}{\pi\frac{e^{-i4}}{2}}+\color{#0000FF}{\pi\frac{e^{i4}}{2}}\\ &-\color{#C00000}{\pi i\frac{\sin(-4)}{2}}-\color{#C00000}{\pi i\frac{\sin(4)}{2}}\\[6pt] &=\pi\cos(4) \end{align}
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I have to ask, what program did you use to draw the diagram? – user2468 Jul 26 '12 at 2:07
I used Intaglio for the Mac. – robjohn Jul 26 '12 at 5:29 | {
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# How the two non null-homotopic equivalence classes generate the null-homotopic loop on the torus
I am new in Alebraic Topology. Given the torus, we say that the fundamental group of the torus is generated by two loops (or more exactly two equivalent classes of loops). One writes $$\pi=\mathbb{Z}\times\mathbb{Z}.$$
I don't understand, how the null-homotopic loop, which is the constant loop, is generated by the two generators mentioned above. Can somebody provide an explanation? More even so, I don't see how it functions visually, since the two generators are not null-homotopic. More precisely, given a null-homotopic loop on the surface on a base point $$x$$, how this loop will be generated by the two generators mentioned above?
• The identity is in every subgroup. – Angina Seng Dec 28 '18 at 16:06
• Thanks. But how it is generated in the case of a null-homotopic loop ? In the case the loops are traversing the torus around the hole or through the hole, one can say for instance that the product of one loop and its inverse is the constant loop, that is the identity. But I dont see how it works in the case a loop is null-homotopic, since the generators are non null-homotopic. – user249018 Dec 28 '18 at 16:14
• What do you think the word "generate" means? – Eric Wofsey Dec 28 '18 at 16:20
Let $$\mathbb{T}^2$$ denote the torus and choose a basepoint $$p \in \mathbb{T}^2$$. Then we know that $$\pi_1\left(\mathbb{T}^2, p \right) \cong \mathbb{Z} \times \mathbb{Z}$$.
Now I think the reason for your confusion is an algebraic one. | {
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Now I think the reason for your confusion is an algebraic one.
Recall that $$\mathbb{Z} \times \mathbb{Z}$$ has two generators, $$a= (1, 0)$$ and $$b =(0, 1)$$. Choose an isomorphism $$\psi : \pi_1\left(\mathbb{T}^2, p \right) \to \mathbb{Z} \times \mathbb{Z}$$, by surjectivity there exists path classes, $$[f], [g] \in \pi_1\left(\mathbb{T}^2, p \right)$$ such that $$\psi([f]) = a$$ and $$\psi([g]) =b$$. Then since $$\psi$$ is an isomorphism we have $$[f]$$ and $$[g]$$ to be the two generators of $$\pi_1\left(\mathbb{T}^2, p \right)$$.
Now your question is how the path class of the constant loop $$c_p : I \to \mathbb{T}^2$$ defined by $$c_p(x) = p$$ for all $$x \in I$$, that being $$[c_p] \in \pi_1\left(\mathbb{T}^2, p \right)$$ is generated by $$[f]$$ and $$[g]$$. Well the answer to that is simple: note that $$[c_p] = 1_{\pi_1\left(\mathbb{T}^2, p \right)}$$ that is $$[c_p]$$ is the identity element of $$\pi_1\left(\mathbb{T}^2, p \right)$$. Then recall the following definition that we have for exponents in groups.
Definition: In any group $$(G, \cdot)$$ for any $$x \in G$$ we define $$x^0 = 1_G$$ where $$1_G$$ is the identity element of the group $$(G, \cdot)$$.
Hence since $$[f], [g] \in \pi_1\left(\mathbb{T}^2, p \right)$$ and $$\pi_1\left(\mathbb{T}^2, p \right)$$ is indeed a group, we have $$[f]^0 = [g]^0 = 1_{\pi_1\left(\mathbb{T}^2, p \right)}.$$
Then we have $$\left[c_p\right] = [f]^0 * [g]^0$$ and so the constant path at $$p$$ is indeed generated by the two generators of $$\pi_1\left(\mathbb{T}^2, p \right)$$. And since $$[c_p]$$ is a nullhomotopic loop, since it is a constant loop by definition, the above shows how a product of two non null-homotopic loops yield a null-homotopic loop. | {
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Note that above even though I've gone into quite a bit of detail, the only real fact I'm using is the following algebraic one. If we have a group $$G$$ and we have $$G = \langle A \rangle$$ for some subset $$A \subseteq G$$ then every element $$x \in G$$ can be written as $$x = g_1 \dots g_n \cdot h_1^{-1} \dots h_m^{-1}$$ where $$g_i, h_i \in G$$. In particular if we have $$G = \langle c , d \rangle$$, that is $$G$$ is generated by the two elements $$c$$ and $$d$$ then we can express $$1_G$$ as $$1_G = c^0 \cdot d^0$$.
Your confusion seems to be about the meaning of the word "generate". By definition, if $$G$$ is a group and $$S\subseteq G$$, then the subgroup generated by $$S$$ is the smallest subgroup that contains $$S$$. Since a subgroup always contains the identity element, any subset of $$G$$ (even the empty set!) "generates" the identity element. | {
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• Thanks. By ''generate'' I meant the group operation on the set of generators. In a group generated by the subset $S$, each element can be written in terms of the generators, which is, elements of $S$. So given the null-homotopic loop on the torus, how can it be put in relation to the elements of $S$, which in our case consists of 2 elements ? – user249018 Dec 28 '18 at 16:36
• OK, but the group multiplication is not the only operation in a group! There are two other operations: the identity element and inverses. – Eric Wofsey Dec 28 '18 at 16:39
• In particular, one of the operations of a group is an operation which takes no inputs and outputs the identity element. That's how any set "generates" the identity element. – Eric Wofsey Dec 28 '18 at 16:40
• Given $S\subset G$, more precisely the group generated by $S$ is defined as $<S>=SS^{-1}$. So you are right about inverses. The thing with the identity element is less obvious. One excepts it very probably by definition...But when it comes to the fundamental group of the torus, are you saying that the null-homotopic loop is generated by the empty set ? Or maybe we can say the following: the multiplication of a geneartor and its inverse gives us the constant loop, which itself is homotopic to null-homotopic loops. Thus we generate the null-homotopic loop from each one of the two generators ? – user249018 Dec 28 '18 at 16:55
• I don't know what your notation $SS^{-1}$ is supposed to mean, but it sounds like your definition of "the group generated by $S$" is just wrong (which may not be your fault; you may have been taught a wrong definition!). The correct definition is the one I stated in the answer. An equivalent definition is that the subgroup generated by $S$ is the set of all elements of $G$ that can be obtained by starting with elements of $S$ and repeatedly applying the three operations of the group multiplication, inverses, and the identity element. – Eric Wofsey Dec 28 '18 at 17:10 | {
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The subgroup generated by $$a$$ and $$b$$ is the set of all elements that can be written as a sequence that consists of nothing but $$a$$, $$b$$, and their inverses (e.g $$ab$$ or $$b^{-4}a$$). The null sequence is allowed. That is, the empty string (a zero-length sequence) qualifies as "a sequence that consists of nothing but $$a$$, $$b$$, and their inverses"; it does not contain anything, so clearly it does not contain anything other than $$a$$, $$b$$, and their inverses. In an abelian group with two generators, the group is generated by taking the first generator an integer number of times, and then taking the second generator an integer number of times. And zero is an integer. Given two non null-homotopic loops $$a$$ and $$b$$, the constant loop is generated by taking $$a$$ zero times, then taking $$b$$ zero times. If you think of a group in terms of group actions, the identity is generated by not doing anything. Doing nothing at all is, at least as far as mathematicians are concerned, an action. Or, in the words of Geddy Lee, if you choose not to decide, you still have made a choice. | {
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1. ## [SOLVED] trig intergrals
So, this is also probably one of those easy problems, I just don't have a good example to work this out by.
$
\int\cos^2x$
where b= $\pi/2$ a= 0
answer= $\pi/4$
I know I'm suppose to use the half angle identity, i just must be doing it wrong because that's not the answer I'm getting.
Thanks for all your help ^.^
2. Can you show your work to see what it is that you're doing wrong?
$\int_{0}^{\frac{\pi}{2}} \cos^2 x \: dx = \int_{0}^{\frac{\pi}{2}} \left[\frac{1}{2} (1 + \cos 2x)\right]dx = \frac{1}{2} \int_{0}^{\frac{\pi}{2}} \left[1 + \cos 2x\right]dx =$ ${\color{white}.} \: \frac{1}{2} \left(x + \frac{1}{2}\sin 2x\right) \bigg|_{0}^{\frac{\pi}{2}} = \hdots$
3. $cos^2x = \frac{1 + cos2x}{2}$
$\frac{1}{2}\int{dx} +\frac{1}{2}\int{cos2x}{dx} = \frac{x}{2} +\frac{sin2x}{4}$
Hope this helps.
Oops was to slow lol, just wondering how do you place upper and lower bounds in latex?
4. Originally Posted by 11rdc11
$cos^2x = \frac{1 + cos2x}{2}$
$\frac{1}{2}\int{dx} +\frac{1}{2}\int{cos2x}{dx} = \frac{x}{2} +\frac{sin2x}{4}$
Hope this helps.
Oops was to slow lol, just wondering how do you place upper and lower bounds in latex?
Do:
Code:
$$\int_a^b f(x)\,dx$$
To get $\int_a^b f(x)\,dx$
note if we have limits like $-\ln2$ to $\ln2$, then you need to put these within {} such as:
Code:
$$\int_{-\ln(2)}^{\ln2} e^x\,dx$$
will output $\int_{-\ln(2)}^{\ln2} e^x\,dx$
--Chris
5. $\int_{0}^{4} x \: dx$
is produced by: $$\int_{0}^{4} x \: dx$$
_{0} : produces the lower bound
^{4} : produces the upper bound.
This can applied to other symbols as well:
$\sum_{k = 0}^{4} k = 10$ is produced by $$\sum_{k = 0}^{4} k = 10$$
$\frac{x}{2}\bigg|_{0}^{4}$
etc etc.
6. Originally Posted by o_O
Can you show your work to see what it is that you're doing wrong? | {
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6. Originally Posted by o_O
Can you show your work to see what it is that you're doing wrong?
$\int_{0}^{\frac{\pi}{2}} \cos^2 x \: dx = \int_{0}^{\frac{\pi}{2}} \left[\frac{1}{2} (1 + \cos 2x)\right]dx = \frac{1}{2} \int_{0}^{\frac{\pi}{2}} \left[1 + \cos 2x\right]dx =$ ${\color{white}.} \: \frac{1}{2} \left(x + \frac{1}{2}\sin 2x\right) \bigg|_{0}^{\frac{\pi}{2}} = \hdots$
Whoa, I totally was thinking that the anti derivative of 1 was 0. That's where I messed up. Silly mistake. Thanks though, I'll remember to post my work next time.
7. $\int_{0}^{\frac{\pi}{2}} \cos^2 x \: dx = \int_{0}^{\frac{\pi}{2}} \left[\frac{1}{2} (1 + \cos 2x)\right]dx = \frac{1}{2} \int_{0}^{\frac{\pi}{2}} \left[1 + \cos 2x\right]dx =$ ${\color{white}.} \: \frac{1}{2} \left(x + \frac{1}{2}\sin 2x\right) \bigg|_{0}^{\frac{\pi}{2}} = \hdots$
There's an easier way to deal with these kind of integrals.
Rule: $\int_a^b f(x)~dx = \int_a^b f(a+b-x)~dx$
I'll solve this step by step for you to understand how it works. It actually takes only a nanosecond to solve this.
$I = \int_0^\frac{\pi}{2}\cos^2x~dx = \int_0^\frac{\pi}{2}\cos^2(\frac{\pi}{2}-x)~dx = \int_0^\frac{\pi}{2}\sin^2x~dx$
Now, $I+I = \int_0^\frac{\pi}{2}\cos^2x~dx + \int_0^\frac{\pi}{2}\sin^2x~dx = \int_0^{\frac{\pi}{2}}1~dx = \frac{\pi}{2}$
$I = \frac{\pi}{4}$ | {
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# How rigorous are pictorial proofs?
The standard proof that $|\mathbb{Q}| = \mathbb{|N|}$ is pictorial. I am sure everyone here has seen it. The "zig-zag". I must admit, however, that, although I was "intuitively" convinced by it, I was never entirely satisfied with it because it is not an explicit bijection $f:\mathbb{N} \to \mathbb{Q}$ given by an actual formula. The fact that the proof is correct seems "clear" to us, but this is, again, merely an appeal to intuition. One should note that some of these "proofs by picture" are simply incorrect: see Russell O'Connor's answer here .
I have two questions
Is the pictorial proof that $|\mathbb{Q}| = \mathbb{|N|}$ rigorous by the standards of modern pure mathematics?
For the sake of this question, suppose that there isn't an explicit formla, or that it's too unwieldy to use in practice. After all, even if there is a formula, most of the people who've seen the pictorial argument do not know of it.
Is there an explicit formula for the "pictorial" proof?
There's some minor issues, of course, namely the inclusion of $0$ and variations of the "zig-zag" path, but these are no big deal. A bijection $f:\mathbb{N} \to \mathbb{N} \times \mathbb{N}$ suffices; dealing with negatives, equivalent fractions, etc is trivial. | {
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• The zig-zag proof is not pictorial. You can represent it in a graph, as any function, this is all. Aug 6, 2016 at 6:52
• Yes, there is an explicit function for the bijection. But it's tedious, looks convuluted and it distracts from the purpose of the argument which should be simple. For n,m find the largest $\sum_{i=1}^k = k (k+1)/2 \le n+m$ then (n,m) -> k (k+1)/2 + (n-k) is the bijection. Aug 6, 2016 at 6:56
• That should be sum < n; not <= n+m. It helps to draw a picture but: 1 => 0,0 then 2-3 => (1,0) - (0,1) and so on till $(\sum )+1$ = k (k+1)/2 +1 through $\sum$ + (k+1) maps to (k,0), (k-1,1), (k-2,2).... (0,k). That's the bijection. But it's tedious and not particularly relevant. Aug 6, 2016 at 7:21
• Perhaps the easiest way is to let $a_k = \sum_{i=1} i$, then it's easy to show ever natural $n$ can be written uniquely as $a_k + i; 0 \le i < k+1$ so $n \rightarrow (k -1,i)$ is 1 to 1. Aug 6, 2016 at 7:36
• A pictorial proof isn't rigorous enough but a formulaic calculation isn't nescessary if a descriptive argument can be shown to be unambiguous and consistsnt. The diagonal description as usually presented is ... borderline IMO. But if we explain we can count and group the naturals in groups each group with one more than the previous, i.e. (1)(2,3)(4,5,6)(7,8,9,10)etc. then each group has the same number of members as each of the diagonals that also increases by one, that could be rigorous enough. Aug 6, 2016 at 7:56
I had the same question with the same proof (the zig-zag proof you are mentioning). At some point I decided to produce a formal proof.
Define a bijective function $f\colon \mathbb N \times \mathbb N \to \mathbb N$. | {
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Define a bijective function $f\colon \mathbb N \times \mathbb N \to \mathbb N$.
First of all you notice that going zig and then zag only helps intuition. In fact you don't need a "continuous" curve, so it is easier to go zig and the zig again... (so to speak). Then it is easy to count how many points you need to fill the first $k$ diagonals (sum of a arithmetic series: $k(k+1)/2$). The couple $(n,m)$ lies on the diagonal number $k=n+m$ so you easily find: $$f(n,m) = \frac{(n+m)(n+m+1)}{2} + m = \frac{n^2+m^2+2nm+3m+n}{2}.$$
This was a little bit shocking to me! The function I was looking for is as simple as a polynomial... I would have expected some modulus, or some strange discontinuous function.
Nevertheless the algebraic proof that $f$ is bijective is not so simple... but following the intuition of the construction it is easy to write it.
What can we learn from this? The pictorial proof is for sure the best to understand a result and to remember it. Then it might happen that the abstract mathematics is even simpler than our intuition. Not always simple mathematics corresponds to simple pictures.
• However it's not necessary to come up with an explicit formula to prove the function is a bijection. E.g., an iterative or recursive description can suffice, and this is basically what the zig-zag picture does. Aug 6, 2016 at 11:01
• @Kinball, it suffices provided it suffices. Having a fórmula makes it easy to be sure. Aug 6, 2016 at 19:46
I suppose if you really wanted to, you could come up with an explicit bijection associated with the "zig-zag" proof. If that turns out to be difficult, you could come up with a different "zig-zag" that may have a simpler bijection. Although, the "zig-zag" proof is really just providing some intuitive backing to this theorem:
The union of a countable number of finite sets is countable. | {
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The union of a countable number of finite sets is countable.
Thinking of each diagonal of the "zig-zag" as one of your finite sets, and noting that every rational number has to be in one of those diagonals is sufficient to prove that $|\mathbb{Q}| = |\mathbb{N}|$.
On a more general note, the whole point of a proof is to clearly and correctly convey why a theorem is true. Sometimes it is easiest to convey why through written words, especially when the proof is long, relies on lemmas or the theorems of others, or just has lots of cases. But if there is a clever reason why a theorem is true, some clever "ah-HA" that you just have to see, a "proof by picture" can be much more clear than a formal write-up of a proof. The hope is, though, that after a reader sees a pictorial proof, they should have enough intuition into why the theorem is true to write up a formal proof if they really needed.
After seeing the "zig-zag" proof, do you think you can prove that the union of a countable number of finite sets is countable? | {
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• What about a zig-zag proof that the union of a countable number of countable sets is countable? =P Aug 6, 2016 at 10:28
• @user21820, Every time I've heard the zig-zag proof it is presented as partitioning the rational numbers into finite sets indexed by the sum of their numerator and denominator. But I suppose you can think of them as being infinite sets (indexed by just the numerator or denominator), and that makes sense with the same picture. So I suppose that in this case, the same pictorial proof can inspire distinct formal proofs. :) Aug 6, 2016 at 22:23
• Well I was trying to poke fun at the rigour of such a pictorial proof because it easily makes one think that it works the same for the union of countably many countable sets. It doesn't quite because you need the axiom of countable choice. Personally I like annotating parts of a diagram with quantifiers labelled by the order of quantification, and it would prevent such an error because we would realize that we need $\exists_1,\exists_2,...$. Aug 7, 2016 at 8:40
• @user21820,Mike Actually you need the axiom of countable choice even to prove that the union of countably many finite sets is countable. Is it obvious where countable choice comes in in the pictorial proof? It's in the picture-making itself: you need to place dots on the page, which entails a choice of ordering for the set which is relevant to the "zig"; and such a choice must be made independently for each of the countably many finite sets. Countable choice is not needed for $|\Bbb N|=|\Bbb Q|$ because the ordering is known in advance. (And this is why I don't like pictorial "proofs".) Aug 9, 2016 at 18:50
Munkres' "Topology" gives both the zig-zag intuition, and the actual formula for the bijection, as a good reference. | {
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I read in an article (cannot remember the author, sadly) that proofs are not supposed to be 'entirely' rigorous. 'Proofs' try to convince the reader that it is possible to construct a completely rigorous proof. As in this case, although I too was not satisfied with the zig-zag-proof, it conveys the idea that the bijection does exists (without explicitly writing it out), and thus it is countable.
The point of something like the zig-zag proof is not to be rigorous in itself but instead to convince a mathematician that he or she could easily make the proof rigorous if pressed to do so by the gods of rigour.
Also, you can come up with very succinct surjections from N to Q. For example, map every natural number of the form $2^p 3^q 5^r$ to $(-1)^r p/q$. It's extremely easy now to see that if you give me a rational, there is a natural number which is mapped to it.
The downside with this is that if you are teaching countability you would need to check that "surjection from" is the same as "bijection with", which sometimes hasn't been proved by this stage.
• Except that "surjection from" is not the same as "bijection with". To wit, your function is one but not the other. You can use it as a step in the way to proving a bijection, but that is hardly "the same as". Aug 7, 2016 at 10:03
• Well of course I don't mean all surjections are bijections? For an infinite set $S$, the existence of a surjection $g : \mathbb{N} \to S$ is equivalent to the existence of a bijection betwwen $S$ and $\mathbb{N}$. Aug 7, 2016 at 16:57
Pictures have their strengths and weaknesses, but they're just as rigorous as any other informal type of proof — that is, they may or may not be depending on how well written it is.
And the zig-zag proof is a rather clear depiction of an explicit algorithm for enumerating the rationals. | {
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Regarding your particular doubts, I don't think it's really the picture that's the issue — it's that the function was defined by an algorithm for producing its values, rather than as an arithmetic formula.
Formal proofs can be 'pictures' too; you can develop formal logic in a way where the basic 'data type' is something other than strings of symbols. e.g. graphs of various sorts are often useful.
This doesn't directly address the proof of $|\mathbb{N}|=|\mathbb{Q}|$, but the more abstract notion of pictorial proofs as found in elementary geometry:
Avigad, Dean & Mumma A Formal System for Euclid's Elements: http://repository.cmu.edu/philosophy/61/
They show that certain types of pictorial arguments that occur in Euclid's Elements, while apparently un-rigorous, can be described in a precise manner using formal rules, in a way that the conclusion follows directly from the pictorial "special case". In essence, a particular diagram can be in "general enough position" to allow rigorous conclusions to be drawn from it.
So... take the ordered pair $(n,m)$.
The pictorial and non rigorous concept of the "diagonal" is simply {$(j,k) | j+k=n+m; j\ge 0;k \ge 0$}. Note: there are $(n+m)+1$ terms in this diagonal
The previous diagonals had 1 term, two terms ... so on to $(n+m)$ terms. So the previous diagonals account for $\sum_{i=1}^{n+m} i = \frac {(n+m)(n+m+1)}{2}$ items.
Starting at $(0,n+m)$ end of the diagonal $(n,m)$ is the $n +1$th item of the current diagonal.
So the bijection you want is $(n,m)\rightarrow \frac {(n+m)(n+m+1)}2 + n$.
$(0,0)\implies 0$
$(0,1)\implies 1$
$(1,0) \implies 2$
$(2,0) \implies 3$
$(1,1)\implies 4$
Etc. Algebraically it's probably easy to show this is injective . Suppose (a,b) and (c,d) both map to t.
Case 1: a+b = c+d= K.
Then $\frac {K (K+1)}{2} + a = \frac {K (K+1)}2 + b$. So $a = c$. So $b = c =K -a$.
Case 2: Wolog a+b < c+d
$\frac {(a+b)(a+b+1)}2 + a \le \frac {(a+b)(a+b+1)}2 + (a+b)$
$=\frac {(a+b)(a+b+3)}2$ | {
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$\frac {(a+b)(a+b+1)}2 + a \le \frac {(a+b)(a+b+1)}2 + (a+b)$
$=\frac {(a+b)(a+b+3)}2$
$\le \frac {((c+d)-1)((c+d+1)+1)}2$
$=\frac {(c+d)(c+d +1) +(c+d) -(c+d+1)-1}2$
$=\frac{(c+d)(c+d+1)}2 -1$
$< \frac{(c+d)(c+d+1)}2 + c$
So (a,b)=(c,d).
Proving it's surjective isn't nescessary though it is.
For each t there exist, k, so that $\frac {k (k+1)}2 \le t < \frac{(k+1)(k+2)}2$.
Let $n = t -\frac {k (k+1)}2 \ge 0$. Let $m=k-n = k - t + \frac {k (k+1)}2 > k - \frac {(k+1)(k+2)}2 + \frac {k (k+1)}2=k +\frac {k+1}2 (k-(k+2))=k - (k+1)=-1$. So $m \ge 0$
Then $(n,m)\rightarrow t$.
===
Anyway....
I don't think such an intensive arithmetic is nescessary. What matters is the argument that it can be done. As the diagonals increase by one each iteration, and as we can group the natural numbers into sequences of groups each increasing in terms by one, each group of natural numbers coresponds to a diagonal, and each of there natural numbers in the group corresponds to a term in the diagonal. Thus must be shown but it needn't be shown by convoluted arithmetic. | {
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# What is the last digit 3^{3^3} ? A. 1 B. 3 C. 6 D. 7 E. 9
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What is the last digit 3^{3^3} ? A. 1 B. 3 C. 6 D. 7 E. 9 [#permalink] 23 Sep 2008, 06:17
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What is the last digit $$3^{3^3}$$ ?
A. 1
B. 3
C. 6
D. 7
E. 9
OA is D (7).
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Re: last digit of a power [#permalink] 24 Sep 2008, 05:50
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Lets take another example
Find the last digit of 122^94
A. 2
B. 4
C. 6
D. 8
E. 9
Now the last digit of 122 is 2. We require only this number to determine the last digit of 122 raised to a positive power.
so the problem is essentially reduced to find the last digit of 2^94.
Now we know 2 has a cyclicity of 4. So we divide 94 by 4. The remainder for 94/4 is 2.
so last digit of 2^94 is same as that of 2^2 which is 4.
so last digit of 122^94 is 4 | {
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so last digit of 2^94 is same as that of 2^2 which is 4.
so last digit of 122^94 is 4
Remember:
1) Numbers 2,3,7 and 8 have a cyclicity of 4
2) Numbers 0,1,5 and 6 have a cyclicity of 1 (ie) all the powers will have the same unit digit. eg. 5^245 will have "5" as unit digit, 5^2000 will aslo have "5" as unit digit. Same holds for 0,1 and 6
3) If 4 is the number in the unit place of the base number then the unit digit will be "4" if the power is odd and it will be "6" if the power is even. eg. 4^123 will have unit digit of 4 since 123 is odd.
4) Similarly, for 9 the unit digit will be "9" for odd powers and "1" for even powers. eg 9^234 has unit digit as "1" since 234 is even.
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Re: last digit of a power [#permalink] 24 Sep 2008, 05:41
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elmagnifico wrote:
amitdgr wrote:
elmagnifico wrote:
What is the last digit $$3^{3^3}$$ ?
$$3^{3^3}$$ should be taken as 3^27 ?
in that case divide 27 by 4. The remainder is 3. Now the last digit of 3^27 is same as that of 3^3 and that happens to be 7.
So 7 is the right answer.
why" Now the last digit of 3^27 is same as that of 3^3 and that happens to be 7."
where do you get this from
$$3^1$$ =3,
$$3^2$$ =9,
$$3^3$$ =27,
$$3^4$$ =81,
$$3^5$$ =243,
$$3^6$$ =729, and so on .....
Now it is not humanly possible to remember all the numbers till $$3^27$$
If you have noticed in the above series the last digit repeats after every 4 terms
the last digit is same for $$3^5$$ and $$3^1$$
the last digit is same for $$3^6$$ and $$3^2$$
If 3 is the unit digit of a number then the unit digit repeats every fourth consecutive term.For our convenience here, lets call it cyclicity. So 3 has a cyclicity of 4.
To find the unit digit of a number having 3 as its last digit and raised to a positive power, divide the power by 4 and find the remainder. | {
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If the remainder is 1 then the unit digit is same as of the unit digit of $$3^1$$
If the remainder is 2 then the unit digit is same as of the unit digit of $$3^2$$ and so on .....
Note that if the remainder is "0" then the unit digit is same as $$3^4$$ since the cyclicity is 4.
Also remember that the numbers 2,3,7 and 8 have cyclicity of 4
in our problem above we have 3^27
3 has a cyclicity of 4 so divide the number 27 by 4. We get a remainder of 3. Now as per cyclicity the last digit of 3^27 is same as that of 3^3. 3^3 is 27 so the last digit of 3^27 is 7.
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Last edited by amitdgr on 24 Sep 2008, 05:58, edited 1 time in total.
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Re: last digit of a power [#permalink] 23 Sep 2008, 06:32
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elmagnifico wrote:
What is the last digit $$3^{3^3}$$ ?
* 1
* 3
* 6
* 7
* 9
3^3 = 27
the last digit of $$3^{3^3}$$ will be the last digit of (27)^3.
now the last digit of 27*27*27 will be the same as last digit of 7*7*7 = 343 (ie) 3
Another example we can use here to understand this concept better (I made this example up)
What is the last digit of 39*87*81?
A. 2
B. 3
C. 4
D. 5
E. 6
To find the last digit of 39*87*81. All we have to do is, multiply the last digits of 39, 87 and 81
when we multiply, 9*7 we get 63. Now multiply the last digits of 63 and 81, i.e. 3*1 we get 3
so the last digit of 39*87*81 will be 3.
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Re: last digit of a power [#permalink] 05 Aug 2009, 14:11
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Here's another way of looking at it !
Here the given number is $$(xyz)^n$$
z is the last digit of the base.
n is the index | {
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Here the given number is $$(xyz)^n$$
z is the last digit of the base.
n is the index
To find out the last digit in $$(xyz)^n$$, the following steps are to be followed.
Divide the index (n) by 4, then
Case I
If remainder = 0
then check if z is odd (except 5), then last digit = 1
and if z is even then last digit = 6
Case II
If remainder = 1, then required last digit = last digit of the base (i.e. z)
If remainder = 2, then required last digit = last digit of the base $$(z)^2$$
If remainder = 3, then required last digit = last digit of the base $$(z)^3$$
Note : If z = 5, then the last digit in the product = 5
Example:
Find the last digit in (295073)^130
Solution: Dividing 130 by 4, the remainder = 2
Refering to Case II, the required last digit is the last digit of $$(z)^2$$, ie $$(3)^2$$ = 9 , (because z = 3)
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Re: last digit of a power [#permalink] 24 Sep 2008, 05:22
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sorry guys. Its my mistake.
$${3}^{3^3}$$ can't be taken as $$27^3..$$It should be $$3^{27}$$
$$3^9 = 27^3$$
3^1=3
3^2=9
3^3=27
3^4=81.
..
..
3^27 = ....7
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Re: last digit of a power [#permalink] 24 Sep 2008, 07:25
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chayanika wrote:
amitdgr wrote:
Lets take another example
Find the last digit of 122^94
A. 2
B. 4
C. 6
D. 8
E. 9
Now the last digit of 122 is 2. We require only this number to determine the last digit of 122 raised to a positive power.
so the problem is essentially reduced to find the last digit of 2^94.
Now we know 2 has a cyclicity of 4. So we divide 94 by 4. The remainder for 94/4 is 2.
so last digit of 2^94 is same as that of 2^2 which is 4.
so last digit of 122^94 is 4 | {
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so last digit of 2^94 is same as that of 2^2 which is 4.
so last digit of 122^94 is 4
Remember:
1) Numbers 2,3,7 and 8 have a cyclicity of 4
2) Numbers 0,1,5 and 6 have a cyclicity of 1 (ie) all the powers will have the same unit digit. eg. 5^245 will have "5" as unit digit, 5^2000 will aslo have "5" as unit digit. Same holds for 0,1 and 6
3) If 4 is the number in the unit place of the base number then the unit digit will be "4" if the power is odd and it will be "6" if the power is even. eg. 4^123 will have unit digit of 4 since 123 is odd.
4) Similarly, for 9 the unit digit will be "9" for odd powers and "1" for even powers. eg 9^234 has unit digit as "1" since 234 is even.
Wow !! Awesome I tried out this thing with a few numbers and matched the results with my scientific calculator. This method gives perfect answers.
You deserve at least a dozen KUDOS for typing out all this patiently and sharing this knowledge with all of us.
+1 from me. Guys pour in Kudos for this
Chayanika
i agree with you. he deserves many kudos.
thanks a million.
the OA is 7 indeed. what a wonderful explanation.
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Re: last digit of a power [#permalink] 24 Sep 2008, 06:04
3
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amitdgr wrote:
Lets take another example
Find the last digit of 122^94
A. 2
B. 4
C. 6
D. 8
E. 9
Now the last digit of 122 is 2. We require only this number to determine the last digit of 122 raised to a positive power.
so the problem is essentially reduced to find the last digit of 2^94.
Now we know 2 has a cyclicity of 4. So we divide 94 by 4. The remainder for 94/4 is 2.
so last digit of 2^94 is same as that of 2^2 which is 4.
so last digit of 122^94 is 4 | {
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so last digit of 2^94 is same as that of 2^2 which is 4.
so last digit of 122^94 is 4
Remember:
1) Numbers 2,3,7 and 8 have a cyclicity of 4
2) Numbers 0,1,5 and 6 have a cyclicity of 1 (ie) all the powers will have the same unit digit. eg. 5^245 will have "5" as unit digit, 5^2000 will aslo have "5" as unit digit. Same holds for 0,1 and 6
3) If 4 is the number in the unit place of the base number then the unit digit will be "4" if the power is odd and it will be "6" if the power is even. eg. 4^123 will have unit digit of 4 since 123 is odd.
4) Similarly, for 9 the unit digit will be "9" for odd powers and "1" for even powers. eg 9^234 has unit digit as "1" since 234 is even.
Wow !! Awesome I tried out this thing with a few numbers and matched the results with my scientific calculator. This method gives perfect answers.
You deserve at least a dozen KUDOS for typing out all this patiently and sharing this knowledge with all of us.
+1 from me. Guys pour in Kudos for this
Chayanika
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Re: last digit of a power [#permalink] 24 Sep 2008, 07:27
3
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elmagnifico wrote:
amitdgr wrote:
elmagnifico wrote:
What is the last digit $$3^{3^3}$$ ?
$$3^{3^3}$$ should be taken as 3^27 ?
in that case divide 27 by 4. The remainder is 3. Now the last digit of 3^27 is same as that of 3^3 and that happens to be 7.
So 7 is the right answer.
why" Now the last digit of 3^27 is same as that of 3^3 and that happens to be 7."
where do you get this from
$$3^1$$ =3,
$$3^2$$ =9,
$$3^3$$ =27,
$$3^4$$ =81,
$$3^5$$ =243,
$$3^6$$ =729, and so on .....
Now it is not humanly possible to remember all the numbers till $$3^27$$
If you have noticed in the above series the last digit repeats after every 4 terms
the last digit is same for $$3^5$$ and $$3^1$$
the last digit is same for $$3^6$$ and $$3^2$$ | {
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the last digit is same for $$3^5$$ and $$3^1$$
the last digit is same for $$3^6$$ and $$3^2$$
If 3 is the unit digit of a number then the unit digit repeats every fourth consecutive term.For our convenience here, lets call it cyclicity. So 3 has a cyclicity of 4.
To find the unit digit of a number having 3 as its last digit and raised to a positive power, divide the power by 4 and find the remainder.
If the remainder is 1 then the unit digit is same as of the unit digit of $$3^1$$
If the remainder is 2 then the unit digit is same as of the unit digit of $$3^2$$ and so on .....
Note that if the remainder is "0" then the unit digit is same as $$3^4$$ since the cyclicity is 4.
Also remember that the numbers 2,3,7 and 8 have cyclicity of 4
in our problem above we have 3^27
3 has a cyclicity of 4 so divide the number 27 by 4. We get a remainder of 3. Now as per cyclicity the last digit of 3^27 is same as that of 3^3. 3^3 is 27 so the last digit of 3^27 is 7.[/quote]
by the way, i have never had such a GMAT rush. it is like a sugar rush.
EVERY ONE GIVE MORE KUDOS HERE
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Re: last digit of a power [#permalink] 23 Sep 2008, 20:10
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elmagnifico wrote:
What is the last digit $$3^{3^3}$$ ?
$$3^{3^3}$$ should be taken as 3^27 ?
in that case divide 27 by 4. The remainder is 3. Now the last digit of 3^27 is same as that of 3^3 and that happens to be 7.
So 7 is the right answer.
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Re: last digit of a power [#permalink] 05 Aug 2009, 19:25
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3^3^3 to solve this we have to take top down approach...we cannot deduce 27^3......it should be 3^27....hence answer is 7
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Re: last digit of a power [#permalink] 23 Sep 2008, 06:23
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3^27
so lets see 27mod4=3..
3^1=3
3^2=9
3^3=7
Unit digit is 7
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Re: last digit of a power [#permalink] 23 Sep 2008, 06:29
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elmagnifico wrote:
What is the last digit $$3^{3^3}$$ ?
* 1
* 3
* 6
* 7
* 9
27^3 ....--> 7*7*7 --> unit digit xx3 ..
B
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Re: last digit of a power [#permalink] 23 Sep 2008, 11:20
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is the above mentioned method right ?
thanks
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Re: last digit of a power [#permalink] 23 Sep 2008, 13:59
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i dnt think that we can reduce the above statement into 27^3 becoz that is (3^3)^3 which is not what the question says...........
so the answer has to be 7
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Re: last digit of a power [#permalink] 24 Sep 2008, 04:24
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amitdgr wrote:
elmagnifico wrote:
What is the last digit $$3^{3^3}$$ ?
$$3^{3^3}$$ should be taken as 3^27 ?
in that case divide 27 by 4. The remainder is 3. Now the last digit of 3^27 is same as that of 3^3 and that happens to be 7.
So 7 is the right answer.
why" Now the last digit of 3^27 is same as that of 3^3 and that happens to be 7."
where do you get this from
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Re: last digit of a power [#permalink] 24 Sep 2008, 06:56
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its 3
(3^3)^3 = 19683
whats the OA!?
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whats the OA!?
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Re: last digit of a power [#permalink] 24 Sep 2008, 08:06
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Very neat method amitdgr +2 from me
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Re: last digit of a power [#permalink] 26 Jul 2009, 20:14
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Quote:
Remember:
1) Numbers 2,3,7 and 8 have a cyclicity of 4
2) Numbers 0,1,5 and 6 have a cyclicity of 1 (ie) all the powers will have the same unit digit. eg. 5^245 will have "5" as unit digit, 5^2000 will aslo have "5" as unit digit. Same holds for 0,1 and 6
3) If 4 is the number in the unit place of the base number then the unit digit will be "4" if the power is odd and it will be "6" if the power is even. eg. 4^123 will have unit digit of 4 since 123 is odd.
4) Similarly, for 9 the unit digit will be "9" for odd powers and "1" for even powers. eg 9^234 has unit digit as "1" since 234 is even.
So glad I come across this thread! Great tip! Thanks a bunch +1
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Re: last digit of a power [#permalink] 05 Aug 2009, 20:43
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Is 3^3^3 definitely taken as 3^(3^3)?
Is reading it as (3^3)^3 incorrect?
Re: last digit of a power [#permalink] 05 Aug 2009, 20:43
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# What is the last digit 3^{3^3} ? A. 1 B. 3 C. 6 D. 7 E. 9
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# What is the last digit 3^{3^3} ? A. 1 B. 3 C. 6 D. 7 E. 9
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calculate the opposite or adjacent (From Worksheet) /Type /Pages endobj /Length 5792 Find the distance between the two points in the figure below, giving your answer Explain the Pythagorean Theorem and its converse. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2); include distance formula; relate to Pythagorean theorem. Pythagorean Therom In The Coordinate Plane. Using the Pythagorean theorem, An application of the Pythagorean theorem allows you to calculate the length of a diagonal of a rectangle, the distance between two points on the coordinate plane and the height that a ladder can reach as it leans against a wall. (0,0). In this worksheet, we will practice finding the distance between two points on the coordinate plane using the Pythagorean theorem. Let us consider Find the distance between the two points in the figure below, Plane. But it from using pythagorean theorem on coordinate plane represents a fictitious zoo map activity for this. Jan 17, 2014 - These Pythagorean Theorem Worksheets are perfect for providing children a fun way to practice and learn the Pythagorean Theorem. /Filter /FlateDecode Use our printable 10th grade math worksheets written by expert math specialists! It is usually represented by a shape that looks like a tabletop or wall. Pythagorean Theorem and Distance Formula Distance formula Right to education Geometry worksheets Source: www.pinterest.com 1 Pythagorean Theorem and distance formula %PDF-1.3 Line Segment and (3,−1). Watch the video (Level 2: Pythagorean Theorem) Complete the Notes & Basic Practice Check the Key and Correct Mistakes 2. Finding the Plane Parallel to a Line Given four 3d Points. Q1: Find the distance between the point ( − 2, 4) and the … 5 0 obj The length of the vertical leg is 4 units. Using the Pythagorean theorem, find the distance | {
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5 0 obj The length of the vertical leg is 4 units. Using the Pythagorean theorem, find the distance between Solution : Step 1 : Find the length of each leg. *** THIS FILE ALSO INCLUDED IN THE Pythagorean Theorem Packet! 3 0 obj Point (−6,7) is on the circle with center Copyright © 2021 NagwaAll Rights Reserved. /Trapped (\057False) find the distance between two points in the coordinate plane using the Pythagorean theorem, find a missing coordinate given the distance between two points, solve word problems by finding the distance between two points on the coordinate plane. Which of the following points is at a distance of 5√2 from the origin? James is making a map of his local area measured in yards. Pythagorean Therom In The Coordinate Plane - Displaying top 8 worksheets found for this concept. 7th Grade Math Problems Set Theory Sets: An introduction to sets, methods for defining sets, element of set and use of set notations.. /Count 2 G.GPE.B.5 The formula for the distance between two points in two-dimensional Cartesian coordinate plane is based on the Pythagorean Theorem. that the distance from to is twice the /Title (Geometry Worksheet \055\055 Calculating the Distance Between Two Points Using Pythagorean Theorem) Search www.jmap.org: Is the triangle a right scalene triangle? Step 3 : Find the length of each leg. surd in its simplest form. Work out the lengths of the sides of the triangle. In this worksheet, we will practice finding the distance between two points on the coordinate plane using the Pythagorean theorem. Try for free. and . If ever you actually have assistance with math and in particular with extraneous solutions calculator or linear systems come visit us at Algebra-help.org. Yes. in a coordinate system of origin (0,0). Geometry Worksheet -- Calculating the Distance Between Two Points Using Pythagorean Theorem Author: Math-Drills.com -- Free Math Worksheets Subject: Geometry Keywords: math, geometry, distance, Pythagorean, theorem, points Created | {
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Subject: Geometry Keywords: math, geometry, distance, Pythagorean, theorem, points Created Date: 4/6/2016 12:23:47 PM a. /Parent 1 0 R endobj endobj Jan 22, 2018 - This 10-question worksheet offers practice or assessment in using the Pythagorean Theorem to find the distances between objects on the Coordinate Plane. Find the distance between the points and . Distance In Coordinate Plane. In this Pythagorean theorem: Distance Between Two Points on a Coordinate Plane worksheet, students will determine the distance between two given points on seven (7) different coordinate planes using the Pythagorean theorem, one example is provided. The distance between (,5) /Keywords (math\054 geometry\054 distance\054 Pythagorean\054 theorem\054 points) This Distance Formula could also be used as an alternative, or an extension. /Creator (LaTeX with hyperref package) The same method can be applied to find the distance between two points on the y-axis. Elements of a Set: Learn how to find the elements of a set with the help of various types of problems on the basic concepts of sets. a2 + b2 = c2. in a coordinate system of origin Gain an edge over your peers by memorizing the distance formula d = √((x 2 - x 1) 2 + (y 2 - y 1) 2). The Pythagorean theorem can be used to find the unknown length of a leg of a right triangle. distance between and . Problem 1 - Solution. ... Pythagorean Theorem. The coordinates of the points , << The Math Worksheet Site is provided by Scott Bryce. >> Lesson Worksheet: Distance on the Coordinate Plane: Pythagorean Formula. << If (−7,) and (9,14), where =4√17length units, find all the possible values of . Finding the Cosine. are (,−2), (2,8), sbryce@scottbryce.com. Two baseball posts were positioned at (9,3) and (6,9). The intersection of the vertical and horizontal lines forms a right triangle to which the Pythagorean Theorem can be applied. (8,−6) in a coordinate system of origin We can use it to find the distance d between any two points in the plane. | {
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system of origin We can use it to find the distance d between any two points in the plane. Distance Between Two Points (Pythagorean Theorem) Using the Pythagorean Theorem, find the distance between each pair of points. In Pythagorean Theorem, c is the triangle’s longest side while b and a make up the other two sides. Pythagoras' Theorem (From Example/Guidance) Calculating the Distance Between Two Points (From Example/Guidance) Finding the Distance Between 2 Points (4-pages) (From Worksheet) Pythagorean Theorem (1 of 2) e.g. What are the possible values of ? /Parent 1 0 R << Finding the Distance. The length of the horizontal leg is 2 units. Read below to see solution formulas derived from the Pythagorean Theorem formula: $a^{2} + b^{2} = c^{2}$ Solve for the Length of the Hypotenuse c Give your answers as surds in their simplest form. Bing users found our website today by entering these keywords : Free online solutions for trigonometry problems ; math geometry trivias ; word problems in adding integers Decide whether point (−8,−9) is on, inside, or outside the circle. Displaying top 8 worksheets found for - Pythagorean Therom In The Coordinate Plane. /Subject (Geometry) (0,0). A triangle has vertices at the points (4,1),(6,2), and (9,0). Some coordinate planes show straight lines with 2 p (10,2) You must imagine that the plane extends without end, even though the drawing of a plane appears to have edges, and is named by a capital script letter or 3 non-collinear points. Finding the Sine. and (−9,6) respectively. Some of the worksheets for this concept are Concept 15 pythagorean theorem, Using the pythagorean theorem, The pythagorean theorem the distance formula and slope, Chapter 9 the pythagorean theorem, Length, Distance using the pythagorean theorem, The pythagorean theorem date period, Pythagorean … Objects Form a Set: State, whether the following objects form a set or not by giving reasons.. point (9,8). This Distance Formula could also be used as an | {
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form a set or not by giving reasons.. point (9,8). This Distance Formula could also be used as an alternative, or an extension. Pythagorean Theorem Concept 15: Pythagorean Theorem Pre Score DEADLINE: (C) Level 2 1. x��][���~�_я#,����}X۱ ��A+�#�)�8��[d�Rd�O�>�3�b�@>S����W�b�����':}s��˛�ZډS�5W��'���LFh¬�^�������o_�����W�͂�ۯ~���������?�F}��g?�|~�7��0��ؤQLMFIb��^?�|����w�'J��ӯ�0)J�3������Va��7���RA'�P*:*4�V_�ciɾ��%4���7�/��\$RsϾ��í �r�_�z����^}������6|0�o~z�n����+��X���l��? Displaying top 8 worksheets found for - Coordinate Plane And Word Problems. endobj Which of the line segments and has the greatest length? << If the distance between the two points (,0) and (−+1,0) is 9, find all possible values of . Some of the worksheets for this concept are Find the distance between each pair of round your, Name distance between points, Distance formula work, Coordinate geometry, Lesson 7 distance on the coordinate plane, The distance formula date period, Concept 15 pythagorean theorem, The pythagorean theorem … You can obtain the equation for finding the distance from the Pythagorean theorem. /Pages 1 0 R in a coordinate system of origin (0,0). Pythagorean Theorem Formula. The formula above is known as the distance formula. /Author (Math\055Drills\056com \055\055 Free Math Worksheets) Consider the two points (,) and (,). /Resources 40 0 R distance from to , find the coordinates of Complete 2 of the following tasks IXL Practice Worksheets Creating O.1, O.2, (8th) At Least to 80 Score = _____ Level 2: Pythagorean Theorem Find the distance between the points $$\left( { - 8,6} \right)$$ and $$\left( { - 5, - 4} \right)$$. stream /Type /Page Let us consider (−12,5) The pdfs provide ample opportunities to apply the formula not just to find the distance between two points on coordinate planes, … /Type /Catalog These worksheets are great resources for the 6th Grade, 7th Grade, and 8th Grade. Find the distance between the posts giving the answer to one decimal place. B. D. C. Step 1 | {
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Grade. Find the distance between the posts giving the answer to one decimal place. B. D. C. Step 1 – create a right triangle with the segment length you are looking for as the hypotenuse. So, the Pythagorean theorem is used for measuring the distance between any two points A(x_A,y_A) and B(x_B,y_B) ?����÷��/�_������������Ï������3no?Z��?4�~���������^�?߿�x��ߒ���N��z��gȥu���m>>��C���Hb��D;��o"���p��wΞ"E� ��� "������3���. /Kids [ 3 0 R 4 0 R ] Apr 2, 2014 - This 10-question worksheet offers practice or assessment in using the Pythagorean Theorem to find the distances between objects on the Coordinate Plane. Converting to Degrees, Minutes, and Seconds. >> I label my coordinates and plug them into the distance formula. find the distance between and . Using distance on a coordinate plane to prove characteristics about a polygon. Step 2: Use the slope formula to show that the coordinate of the midpoint is located on the line segment. Coordinate Distance and Midpoint Practice using Google Drive (Perfect for Distance Learning! Please show your support for JMAP by making an online contribution. Let us consider (−14,9) and << Distance in the Coordinate Plane You have solved problems with two- and three-dimensional figures using the Pythagorean Theorem. Step 1 : Locate the points (1, 3) and (-1, -1) on a coordinate plane. is 5. Using the distance formula, find the distance between the points (3,4) and (5,6). Some of the worksheets for this concept are Concept 15 pythagorean theorem, Using the pythagorean theorem, The pythagorean theorem the distance formula and slope, Chapter 9 the pythagorean theorem, Length, Distance using the pythagorean theorem, The pythagorean theorem date period, Pythagorean distances … Using the Pythagorean Theorem formula for right triangles you can find the length of the third side if you know the length of any two other sides. /Contents 6 0 R Using the Pythagorean theorem, find the distance between and (−11,3) Pythagorean Theorem & Distance | {
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Using the Pythagorean theorem, find the distance between and (−11,3) Pythagorean Theorem & Distance Formula Warm Up – Scavenger Hunt 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Distance on the Coordinate Plane (−15,−8) Nagwa uses cookies to ensure you get the best experience on our website. Finding the Tangent. Step 2 : Draw horizontal segment of length 2 units from (-1, -1) and vertical segment of length of 4 units from (1, 3) as shown in the figure. (−6,5) and 4 0 obj Software for math teachers that creates exactly the worksheets you need in a matter of minutes. and (1,1) Using the Pythagorean theorem, find an expression for the length of . >> /MediaBox [ 0 0 612 792 ] Learn more about our Privacy Policy. Take a look at this problem. point . Distances On The Coordinate Plane Worksheets - there are 8 printable worksheets for this topic. A short equation, Pythagorean Theorem can be written in the following manner: a²+b²=c². and . Geometry Worksheet -- Calculating the Distance Between Two Points Using Pythagorean Theorem Author: Math-Drills.com -- Free Math Worksheets Subject: Geometry Keywords: math, geometry, distance, Pythagorean, theorem, points Created Date: 4/6/2016 12:23:47 PM 1 0 obj The longest side of the triangle in the Pythagorean Theorem is referred to as the ‘hypotenuse’. )This is a 12 question practice on using the distance and midpoint formulas on the coordinate plane. /Resources 7 0 R You can use the distance formula when you have the x and y coordinates of the two points on the Cartesian plane. What is the distance between (2, 3) and (5, 7) in the coordinate plane? We keep a huge amount of great reference information on subjects varying from logarithmic to graphs Given The Pythagorean theorem states that for any right triangle with side lengths a, b, and c, where c is the hypotenuse or longest side, it is always true that a^2 + b^2 = c^2. Available for Pre-Algebra, Algebra 1, Geometry, Algebra 2, Precalculus, and Calculus. 2 0 obj Let’s look at an example. The | {
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Algebra 1, Geometry, Algebra 2, Precalculus, and Calculus. 2 0 obj Let’s look at an example. The following video gives a proof of the midpoint formula using the Pythagorean Theorem. >> The point is located on the line segment between point Given (4,5), (5,5), and (−4,−7), what is the perimeter of △? Note, you could have just plugged the coordinates into the formula, and arrived at the same solution.. Notice the line colored green that shows the same exact mathematical equation both up above, using the pythagorean theorem, and down below using the formula. Pythagorean Theorem - Coordinates Students are asked to use the Pythagorean theorem to calculate the distance between the two points on the coordinate plane in this activity.The worksheet has examples at the top followed by problems for the students to solve and match to an answer in the middle. P a R y x 2 3 Q 5 7 Use math you already know to solve the problem. in a coordinate system of origin (0,0). 6 0 obj /MediaBox [ 0 0 612 792 ] The student will understand the relationship between the areas of the squares of the legs and area of the square of the hypotenuse of a right triangle. Some of the worksheets displayed are Lesson 7 distance on the coordinate plane, Find the distance between each pair of round your, Lesson 7 distance on the coordinate plane, Ordered pairs, Task graphing on the coordinate plane essential questions, Pythagorean distances a, Name distance between points, Concept 15 pythagorean theorem. Let us consider calculate the hypotenuse (From Worksheet) Pythagorean Theorem (2 of 2) e.g. Then draw a vertical line through one of the points and a horizontal line through the other point. /Type /Page and . Description so they use pythagorean theorem on coordinate plane worksheet shown above, comprehend the profile to take a unit of area. Give your answer as a Proposal and match the theorem coordinate plane worksheet library, the triangle is the other. Find the distance between 2 points on a coordinate | {
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worksheet library, the triangle is the other. Find the distance between 2 points on a coordinate plane using the Pythagorean Theorem. Distance In Coordinate Plane - Displaying top 8 worksheets found for this concept. This is a great holiday math activity where students graph points on a coordinate plane and it creates a picture of a Christmas Grump! /Producer (pdfTeX\0551\05640\05616) You can use the Pythagorean Theorem to find the distance be- tween two points on the coordinate plane. Step 1: Use the distance formula to show the midpoint creates two congruent segments. How to use the Pythagorean theorem to prove the midpoint formula? Using the Pythagorean theorem, find the distance between To determine the distance between two points on the coordinate plane, begin by connecting the two points. 1) x y 2) x y 3) x y 4) x y 5) x y /CreationDate (D\07220160406122347\05504\04700\047) /ModDate (D\07220160406122347\05504\04700\047) In this Pythagorean theorem: Distance Between Two Points on a Coordinate Plane worksheet, students will determine the distance between two given points on seven (7) different coordinate planes using the Pythagorean theorem, one example is provided. A. giving your answer in radical form if necessary. A plane extends in two dimensions. Find the distance between the point (−2,4) and the point of origin. find all possible values of . Nagwa is an educational technology startup aiming to help teachers teach and students learn. Step 2 : Let a = 4 and b = 2 and c represent the length of the hypotenuse. Finding the Trig Value. << *** THIS FILE ALSO INCLUDED IN THE Pythagorean Theorem Packet! Coordinate Plane Polar Coordinate Paper 3-D Coordinate System Logarithmic Graph Paper ... Pythagorean Theorem Pythagorean Theorem Distance Formula. The distance between A and B on the plane is the square root of (x 1 −x 2) 2 + (y 1 −y 2) 2. The length of the vertical leg is 4 units. in radical form if necessary. (−7,−1). The coffee shop is at (−5,−4) and the Italian | {
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leg is 4 units. in radical form if necessary. (−7,−1). The coffee shop is at (−5,−4) and the Italian restaurant is at (0,6). Our printable distance formula worksheets are a must-have resource to equip grade 8 and high school students with the essential practice tools to find the distance between two points. The points ,, and have coordinates (3,3),(9,5),(−2,8), Using the Pythagorean theorem, find the What is the distance between the point and the origin? Some of the worksheets for this concept are Math 6 notes the coordinate system, Word problem practice workbook, Solving problems on a coordinate plane, 3 points in the coordinate, Concept 11 writing graphing inequalities, Find the distance between each pair of round your, Using the pythagorean theorem, Name date. /Contents 39 0 R Given that =, >> >> Because a and b are legs and c is hypotenuse, by Pythagorean Theorem, we have. Find the distance between the points (4,5) and (6,−2). /PTEX.Fullbanner (This is MiKTeX\055pdfTeX 2\0569\0565840 \0501\05640\05616\051) What is the kind of triangle that the points (9,−4), (3,5), and (6,1) form with respect to its angles? Find the distance between the coffee shop and the Italian restaurant giving the answer to one decimal place. Use of the different formulas to calculate the area of triangles, given base and height, given three sides, given side angle side, given equilateral triangle, given triangle drawn on a grid, given three vertices on coordinate plane, given three vertices in 3D space, in video lessons with … Step 2 – label the length of the two legs of the right triangle (these will be your ‘a’ and ‘b’ in the Pythagorean Theorem.) Some coordinate planes show straight lines with 2 p Let us consider (13,−7) Radian Measure and Circular Functions. endobj , and Tween two points on the circle with center ( −7, ) and ( 6,9 ) math worksheets written expert! −15, −8 ) in the figure below, giving your answer as a surd in its form! Straight lines with 2 p Distances on the line | {
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below, giving your answer as a surd in its form! Straight lines with 2 p Distances on the line segments and has the length... Worksheets you need in a matter of minutes one decimal place −15, −8 ) a..., −7 ) and ( −4, −7 ) and ( −+1,0 ) is on inside! Area measured in yards create a right triangle equation, Pythagorean Theorem length you looking... - Displaying top 8 worksheets found for - coordinate plane worksheet library, the triangle triangle with the segment you. ( −4, −7 ), ( 6,2 ), and are (, −2 ), and.! Equation for finding the plane leg is 2 units ) e.g triangle has vertices at the points,! Inside, or an extension for the 6th Grade, 7th Grade, and ( 6, −2 ) hypotenuse... Triangle is the triangle formula for the distance between two points match the Theorem coordinate worksheet. In the coordinate plane and Word problems library, the triangle 8 worksheets... Or wall work out the lengths of the vertical and horizontal lines forms a right triangle Set not! Use our printable 10th Grade math worksheets written by expert math specialists 7 use math you already know solve. My coordinates and plug them into the distance formula when you have solved problems with two- three-dimensional... Following points is at a distance of 5√2 from the Pythagorean Theorem Pythagorean Theorem y. When you have the x and y coordinates of the points (, ) and 1,1! X 2 3 Q 5 7 use math you already know to solve the problem is on the of! 2 of 2 ) e.g your support for JMAP by making an contribution... Distance of 5√2 from the Pythagorean Theorem ( 2, 3 ) and ( 6,9.. Written by expert math specialists −9,6 ) respectively and learn the Pythagorean,!, comprehend the profile to take a unit of area −9 ) is on the segment. Form if necessary startup aiming to help teachers teach and students learn Cartesian coordinate plane Word... Where =4√17length units, find the length of a leg of a Christmas Grump for as distance! Given ( 4,5 ) and the Italian restaurant giving the answer to one decimal.. | {
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math and in particular with extraneous solutions calculator or linear systems come visit us at.... Plane and Word problems ( 9,14 ), where =4√17length units, find the distance between the two points 4,5. Perfect for pythagorean theorem distance on coordinate plane worksheet children a fun way to practice and learn the Pythagorean Theorem surd its. Level 2 1, begin by connecting the two points in the plane... ( 9,8 ) alternative, or an extension the slope formula to that. Coordinate of the midpoint is located on the coordinate plane using the Pythagorean Theorem, c is,! Your support for JMAP by making an online contribution ( 13, −7 ) and point ( −2,4 and. Great holiday math activity where students Graph points on a coordinate system of origin ( )... Three-Dimensional figures using the Pythagorean Theorem to use the slope formula to show that the coordinate and! A 12 question practice on using the Pythagorean Theorem concept 15: Pythagorean Theorem can be written in coordinate! Support for JMAP by making an online contribution the profile to take unit! A leg of a leg of a Christmas Grump by a shape that looks a. Score DEADLINE: ( c ) Level 2 1 formula to show that the coordinate.! ( −5, −4 ) and ( 1,1 ) is on the coordinate the., −6 ) in a coordinate plane you have the x and coordinates... Printable worksheets for this concept label my coordinates and plug them into the distance between two points and plug into... Represented by a shape that looks like a tabletop or wall way to practice and learn Pythagorean... The math worksheet Site is provided by Scott Bryce answer as a surd in its simplest form ( )! Form a Set or not by giving reasons: Pythagorean Theorem ( 2 of 2 ) e.g coffee shop the.: Pythagorean Theorem on coordinate plane using the Pythagorean Theorem ( 2 2... 9,0 ), comprehend the profile to take a unit of area the origin 1 – create a triangle! 3D points ( 3,4 ) and ( 6,9 ) −8, −9 ) is 5 3: the... −4, −7 ) and ( 9,0 ) ) respectively between any two points | {
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3,4 ) and ( 6,9 ) −8, −9 ) is 5 3: the... −4, −7 ) and ( 9,0 ) ) respectively between any two points in Cartesian..., −8 ) in a coordinate plane and it creates a picture of right! Distance of 5√2 from the Pythagorean Theorem Pythagorean Theorem is referred to as the hypotenuse! To prove characteristics about a polygon have assistance with math and in particular with extraneous calculator! Restaurant is at a distance of 5√2 from the Pythagorean Theorem by reasons! Into the distance from to is twice the distance formula, find the distance between the point located. Gives a proof of the following objects form a Set or not by giving reasons out lengths... ) Complete the Notes & Basic practice Check the Key and Correct Mistakes 2 any two points 4,5. In its simplest form, 3 ) and ( 9,0 ) a fun way practice. Us at Algebra-help.org by expert math specialists expression for the distance between and legs and c is hypotenuse by! Segment between point ( −2,4 ) and ( 5,6 ) and c is hypotenuse, Pythagorean. Show that the distance formula to show that the coordinate plane plane using distance. Distance on a coordinate system of origin ( 0,0 ) worksheets are great resources for the length the... Your support for JMAP by making an online contribution proposal and match the coordinate. A short equation, Pythagorean Theorem, find the unknown length of system. A fun way to practice and learn the Pythagorean Theorem Pre Score DEADLINE: c... Is a 12 question practice on using the Pythagorean Theorem to prove characteristics about a polygon intersection of the leg. (, ) and ( −9,6 ) respectively and the Italian restaurant at. The formula above is known as the distance between the point and the restaurant! By making an online contribution what is the distance from to is twice the between! B are legs and c represent the length of Geometry, Algebra 1,,... To ensure you get the best experience on our website with math and in particular with extraneous calculator... By connecting pythagorean | {
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on our website with math and in particular with extraneous calculator... By connecting pythagorean theorem distance on coordinate plane worksheet two points in the figure below, giving your answer as a in. Consider ( −14,9 ) and the Italian restaurant giving the answer to one place. ) Complete the Notes & Basic practice Check the Key and Correct 2. Paper... Pythagorean Theorem on coordinate plane - Displaying top 8 worksheets found for - coordinate plane you solved... Decimal place the other so they use Pythagorean Theorem, find all possible values of 5 7 use math already! Out the lengths of the points and a make up the other are looking for as the hypotenuse... Is 2 units answer in radical form if necessary congruent segments with and... By a shape that looks like a tabletop or wall −11,3 ) in coordinate. Or an extension short equation, Pythagorean Theorem to find the distance from to is twice the distance formula planes! And 8th Grade making an online contribution decimal place a R y x 2 3 5... | {
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# The conjugate function of infimum of sum of functions
Given convex functions $$f_1, \ldots, f_k$$. Given $$g(x)$$, such that:
$$g(x) = \inf_{x_1, \ldots, x_k} \left\{ f_1(x_1) + \ldots + f_k(x_k) | x_1 + \ldots + x_k = x \right\}$$
Find $$g^*(x)$$.
I found this similar question and did similar steps (you may find them below):
By definition: $$f^*(y) = \sup_{x \in dom g} (y^T x - f(x))$$
Then $$g^*(x) = \sup_{x \in dom g} (y^T x - \inf_{x_1, \ldots, x_k} \left\{ f_1(x_1) + \ldots + f_k(x_k) | x_1 + \ldots + x_k = x \right\} = \ldots$$
Using the fact that $$\sup (-f(x)) = - \inf (f(x))$$
Then $$\ldots = \sup_{x \in dom g} (y^T x + \sup_{x_1, \ldots, x_k} \left\{ -f_1(x_1) - \ldots - f_k(x_k) | x_1 + \ldots + x_k = x \right\} =$$
$$= \sup_{x \in dom g, x_1, \ldots, x_k} (y^T x + \left\{ -f_1(x_1) - \ldots - f_k(x_k) | x_1 + \ldots + x_k = x \right\}$$
After these steps some questions arose:
1. Why the author removed constraint $$x_1 + \ldots + x_k = x$$ in the solution?
2. Why $$\sup_{x, x_1, \ldots, x_k} \{f(x) | x_1 + \ldots + x_k = x \} = \sup_{x_1, \ldots, x_k} \{f(x) | x_1 + \ldots + x_k = x \}$$ If so, how to prove it?
• Welcome to SE! Below, I've posted a complete solution to your problem, explaining all the steps in detail. Cheers! – dohmatob Oct 5 '19 at 10:47
• @dohmatob thank you for your help! – user711343 Oct 5 '19 at 22:13
• If it was helpful, please don't forget to upvote my answer. – dohmatob Oct 8 '19 at 7:40
• @dohmatob I have already upvoted, but upvote is not shown because I don't have enough reputation. – user711343 Oct 9 '19 at 11:58
• Ah, ok :). BTW, Ive updated your question. Welcome aboard :) – dohmatob Oct 9 '19 at 13:34
I'll just write down the full solution to your problem, and explain all the steps in the derivations. | {
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So, let $$f_1,\ldots,f_k:\mathcal H \rightarrow (-\infty,+\infty]$$ be functions (convex or not) on a Hilbert space $$\mathcal H$$. Define $$g(x) := \inf_{x_1,\ldots,x_k}\left\{\sum_i f_i(x_i) \mid \sum_i x_i = x\right\}.$$ The function $$g$$ is also called the infimal convolution of $$f_1,\ldots,f_k$$, written $$g=\Box_{i=1}^k f_i$$.
Now, one computes $$\begin{split} g^*(x) &:= \sup_y x^Ty - g(y) \overset{(a)}{=} \sup_{y}x^Ty - \inf_{x_1,\ldots,x_k}\left\{\sum_i f_i(x_i) \mid \sum_i x_i = y\right\}\\ &\overset{(b)}{=} \sup_{y}x^Ty + \sup_{x_1,\ldots,x_k}\left\{-\sum_i f_i(x_i) \mid \sum_i x_i = y\right\}\\ & \overset{(c)}{=} \sup_{y,x_1,\ldots,x_k}\left\{x^Ty -\sum_i f_i(x_i) \mid \sum_i x_i = y\right\} \\ &\overset{(d)}{=} \sup_{y,x_1,\ldots,x_k}x^T\sum_i x_i -\sum_i f_i(x_i) \overset{(e)}{=} \sup_{y,x_1,\ldots,x_k}\sum_i (x^Tx_i -f_i(x_i)) \overset{(f)}{=} \sum_i\sup_{x_i}x^Tx_i - f_i(x_i)\\ &\overset{(*)}{=} \sum_i f_i^*(x), \end{split}$$ where
• (a) is just plugging-in the definition of $$g(y)$$
• (b) is because $$-\inf something = -\sup-thatthing$$
• (c) is because $$\sup_a u(a) + \sup_b u(b) = \sup_{a,b}(u(a) + u(b))$$
• (d) is just substituting the constraint $$y=\sum_i x_i$$ to get $$x^Ty = x^T\left(\sum_i x_i\right)$$. After this substitution, the variable $$y$$ doesn't play a role anymore in the maximization, and so can be deleted. Indeed, $$\sup_{a,b}u(a) = \sup_a u(a)$$.
• (e) is because $$x^T\sum_i x_i - \sum_i f_i(x_i) = \sum_i x^Tx_i - \sum_i f_i(x_i)=\sum_i(x^Tx_i - f_i(x_i))$$
• (f) is because $$\sup_{a,b}u(a) + u(b) = \sup_a u(a) + \sup_b u(b)$$
• (*) is because $$f_i^*(x) := \sup_{x_i}x^Tx_i - f(x_i)$$ by definition.
Therefore we have proven that
Convex conjugate of infimal convolution. $$(\Box_{i=1}^kf_i)^*(x) = \sum_{i=1}^k f_i^*(x)\; \forall x$$ | {
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### Question R1
This question will guide you through a simulation study in R to understand the bias of a certain estimator.
varn <- function(x) mean((x - mean(x))^2)
#### Part a.
The code below generates a sample of size n and calculates both the sample variance and the biased version of sample variance (the one that has n in the denominator instead of n-1). Modify the code to change n from 10 to some other value between 10 and 50.
# Create a sample of data by rolling a 6-sided die n times
n <- 20
data <- sample(1:6, n, replace = TRUE)
data
## [1] 6 1 4 1 5 2 4 1 5 6 1 1 4 4 1 5 1 4 1 4
# True variance
35/12
## [1] 2.916667
# Unbiased estimate of variance
var(data)
## [1] 3.628947
# Biased estimate of variance
varn(data)
## [1] 3.4475
#### Part b.
Repeat the previous experiment many times and find the expected value of each variance estimator.
# Unbiased estimator
mean(replicate(10000, var(sample(1:6, n, replace = TRUE))))
## [1] 2.911705
# Biased estimator
mean(replicate(10000, varn(sample(1:6, n, replace = TRUE))))
## [1] 2.76472
#### Part c.
After running the previous code, which estimator’s average value in the simulation is closer to the true value (roughly 2.9167)?
The unbiased estimator’s average value is closer to the true value.
#### Part d.
Now go back and change n to another, larger value, and rerun all of the code. Do you notice anything about the average of the biased estimator?
mean(replicate(10000, varn(sample(1:6, n + 10, replace = TRUE))))
## [1] 2.81582
With a larger sample size, the biased estimator’s average value is now closer to the true value.
#### Part e.
Copy the code from part (b) and paste it below here, then change the mean function to be sd instead.
# Unbiased estimator
sd(replicate(10000, var(sample(1:6, n, replace = TRUE))))
## [1] 0.5904201
# Biased estimator
sd(replicate(10000, varn(sample(1:6, n, replace = TRUE))))
## [1] 0.5631186
The biased estimator has a lower variability (as measured by standard deviation). | {
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The biased estimator has a lower variability (as measured by standard deviation).
### Question R2
According to a Marketplace/Edison survey in April of 2017, about 23.4% of survey responders agreed with the statement “the economic system in the U.S. is fair to all Americans.” In this question we’ll use a Bernoulli probability model to analyze this number. Suppose that there were 1,000 survey respondents and 234 agreed with the above quotation. Define a Bernoulli random variable which is 1 if a person agrees and 0 otherwise. Assume the survey was done with independent sampling (with replacement), so these Bernoulli random variables are independent. Then the number of people in the sample of 1,000 who agree is a Binomial random variable.
• We have $$X_i$$ i.i.d Ber($$p$$) for $$i = 1, \ldots, 1000$$.
• Let $$S_n = \sum_{i=1}^n X_i$$, so $$S_n$$ is Bin($$n, p$$).
### a.
Using the fact that $$n \bar X_n = S_n$$, how could you use the Binomial distribution to calculate $$P(a \leq \bar X_n \leq b)$$? How would you use pbinom with the given values of $$a, b, n$$, and $$p$$?
pbinom(n*b, size = n, prob = p) - pbinom(n*a, size = n, prob = p)
### b.
Instead of the Binomial distribution, how would we use the central limit theorem to calculate the same probabilities? Hint: your answer should use pnorm and involve $$\sqrt{n}$$ (and $$a, b$$, and $$p$$).
Solution: The CLT says $$\bar X_n$$ is approximately normal with mean $$p$$ and standard deviation $$\sqrt{p(1-p)/n}$$. In R we could do:
pnorm(b, mean = p, sd = sqrt(p*(1-p)/n)) - pnorm(a, mean = p, sd = sqrt(p*(1-p)/n))
### c.
Now let $$n = 1000$$, $$p = 0.234$$, $$b = 0.250, a = 0.239$$ and compute the desired probability with both methods.
n <- 1000
p <- 0.234
a <- 0.239
b <- 0.250
pbinom(n*b, size = n, prob = p) - pbinom(n*a, size = n, prob = p)
## [1] 0.2291025
pnorm(b, mean = p, sd = sqrt(p*(1-p)/n)) - pnorm(a, mean = p, sd = sqrt(p*(1-p)/n))
## [1] 0.2383744 | {
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