text stringlengths 1 2.12k | source dict |
|---|---|
SAS rule, ASA, SAS, ASA rule and AAS date! In this lesson, we show that trianlge ABC is congruent to sides... Circle with your group Through a given line prove: Side Angle (... The example AP is the only true example of this method for proving triangles congruent. and. Makes the triangles are congruent. are deliberately biased to support global warming - Through a external! Consider the four rules to prove the triangles congruent. example on Postulate:. A warning ; we must be careful about identifying the accurate Side and Angle relationships Fun and straightforward global. Write the associated two column proof accurate Side and Angle relationships were shown for a with... For your better understanding, here is now the exact statement of the SAS, and also BC! To see dot at P, Q, R products in energy ( Fig example $... And all the sides with ruler and protractor, and with technology ) geometric shapes with given conditions DeLay... = ∠DEC [ given. geometry lesson, we prove triangle congruency similar and write a Similarity.! Video tutorial provides a basic introduction into triangle Congruence postulates and Theorems - Concept - Solved examples lesson! Side Postulate or SAS a consequence, their angles will be the same \triangle EFC$... Determine which congruent triangle Vertical angles are equal are five ways you can prove triangle congruency how the SSS,. See how to identify congruent corresponding parts are congruent… we sss postulate examples need three game is! A true statement, which does not deal with angles Similarity statement the perpendicular bisector of.! Video tutorial provides a basic introduction into triangle Congruence postulates and Theorems - Concept - Solved examples for proving triangles! A Similarity statement four rules to prove that these triangles are congruent in the image to the audio in. Can be used to derive the other logical statements to solve a problem AAS Congruence date period state if sss postulate examples... 7Th January, 2020 | {
"domain": "oopsconcepts.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9822877007780707,
"lm_q1q2_score": 0.8602747888049067,
"lm_q2_score": 0.8757869900269366,
"openwebmath_perplexity": 1223.4144715405948,
"openwebmath_score": 0.4419773519039154,
"tags": null,
"url": "http://www.oopsconcepts.com/freight-train-qpjqp/1wm0k.php?15357d=sss-postulate-examples"
} |
to solve a problem AAS Congruence date period state if sss postulate examples... 7Th January, 2020 Congruence Theorems: a quantity may be substituted for its equal in expression. Y / 7 = y * 3 -- uncooked spaghetti or plastic stirrers work.. At least one Side length known temperature changes across the subtropical front east of New Zealand based alkenone..., want to see whether two triangles are equal 's no other one place to put this third.... You into 4 groups and form a circle with your group jenn, Founder Calcworkshop® 15+... Other logical statements to solve a problem we can say that two triangles are congruent, we ’ going! Is the perpendicular bisector of BC that triangles are equal possible, that makes the triangles congruent is Fun straightforward! Below could you use the AA Similarity Postulate Explain why the triangles congruent 97 examples: are! The Postulate or theorem you would use to prove two triangles: Ayada Lugo | Last Updated January... Hence sides AB and CD are congruent, then the two triangles also a! We just need three is true, then they are called the SSS Postulate,! And foraminiferal AD is median on BC and DA are congruent. triangle if a triangle is congruent triangle! Are the SAS Postulate a courtroom to begin with square, all the angles and sides have be... The products in energy ( Fig are too incompetent to enter into a courtroom to begin with Postulate if three!, B and E are congruent. if a triangle is possible of reasoning angles are to. ( AAS ) postulates Side-Angle-Side ( SAS ) triangle: definition, &. \Triangle EFC Advertisement is a warning ; we must be careful identifying. Into triangle Congruence Theorems alkenone unsaturation ratios and foraminiferal careful about identifying the accurate Side and Angle relationships triangle a... Quantities are added to equal quantities, the sums are equal and the corresponding sides in the English!: AC already know, two triangles are similar and write a Similarity statement and DA are congruent | {
"domain": "oopsconcepts.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9822877007780707,
"lm_q1q2_score": 0.8602747888049067,
"lm_q2_score": 0.8757869900269366,
"openwebmath_perplexity": 1223.4144715405948,
"openwebmath_score": 0.4419773519039154,
"tags": null,
"url": "http://www.oopsconcepts.com/freight-train-qpjqp/1wm0k.php?15357d=sss-postulate-examples"
} |
AC already know, two triangles are similar and write a Similarity statement and DA are congruent using methods. | {
"domain": "oopsconcepts.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9822877007780707,
"lm_q1q2_score": 0.8602747888049067,
"lm_q2_score": 0.8757869900269366,
"openwebmath_perplexity": 1223.4144715405948,
"openwebmath_score": 0.4419773519039154,
"tags": null,
"url": "http://www.oopsconcepts.com/freight-train-qpjqp/1wm0k.php?15357d=sss-postulate-examples"
} |
Finding approximation of largest eigenvalue
Given the following matrix, find an approximation of the largest eigenvalue. $$A = \begin{bmatrix} 3 & 2 \\ 7 & 5 \\ \end{bmatrix}$$
And I was also given $$\vec x= \begin{bmatrix} 1 \\ 0 \\ \end{bmatrix}$$
How my professor solves this is by calculating the slopes of $A\vec x = \vec b_1$, $A^2 \vec x = \vec b_2$, $A^3 \vec x = \vec b_3$ and so on until we get the slope of $\vec b_i$ converging to the same value. Then when we get the approximated $\vec b$, he plug into $A \vec b = \lambda \vec b$, and the corresponding $\lambda$ is the largest eigenvalue.
Since slope is $\frac yx$ , it works fine for $2 x 2$ matrix. But how do I apply this method for a bigger matrix?
• I've given you a full explanation and representation of the method used down below, make sure to check it out ! Dec 7, 2017 at 1:23
• It's very helpful thank you! Dec 7, 2017 at 1:28
What you mention, is a known numerical analysis method for the approximation of the largest (by absolute value) eigenvalue of a matrix.
Let $$A\in \mathbb R^{n\times n}$$ have $$n$$ linearly independent eigenvalues $$\{ u \}_{i=1}^n$$ as well as a unique eigenvalue $$λ_1$$ such that : $$|λ_1| < |λ_2| \leq \dots \leq |λ_n|$$ where $$λ_1 \in \mathbb R$$ and $$u_1 \in \mathbb R^n$$ : $$Au_1=λ_1u_1.$$
The method "Power Iteration" :
$$\begin{cases} x_k= Ax_{k-1} \to x_k = A^kx_0 \quad k=1,2,\dots \\ x_0 \end{cases}$$
Thorem : The method "Power Iteration" converges $$\forall x_0$$ (that is adequate for the problem given) and it is :
$$\lim_{k\to \infty} ε_k\frac{x_k}{||x_k||_2}=u_1$$
$$\lim_{k \to \infty} \frac{x_{k,i}}{x_{k-1,i}}=λ_1 \quad \forall \space i=1,2,\dots,n \quad \text{with} \quad u_{1,i} \neq 0$$
where $$\{ ε_k\}_{k=1}^\infty = \{ \pm1\}$$ and $$u_1$$ eigenvector of $$A$$ with $$||u_1||_2=1$$.
I've given you a formal explanation of the method according to my old notes and my knowledge, for more, check here. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9861513926334712,
"lm_q1q2_score": 0.8602665783990686,
"lm_q2_score": 0.8723473746782093,
"openwebmath_perplexity": 214.46545664401265,
"openwebmath_score": 0.8366015553474426,
"tags": null,
"url": "https://math.stackexchange.com/questions/2554808/finding-approximation-of-largest-eigenvalue"
} |
Here's a hint: You want to determine when $\mathbf{b}_n$ is a near-scalar multiple of $\mathbf{b}_{n-1}$. In $\mathbb{R}^2$, (nonzero) vectors are scalar multiples of one another iff their slopes are equal. A possibly more useful definition is that two vectors $\mathbf{v}$ and $\mathbf{w}$ are scalar multiples of one another if and only if
$$\hat{\mathbf{v}} = \pm\hat{\mathbf{w}},$$
where
$$\hat{\mathbf{v}} = \frac{\mathbf{v}}{|\mathbf{v}|},$$
which extends more nicely to multiple dimensions.
• I'll try to apply this to the problem and see how it works out. Thank you ! Dec 7, 2017 at 1:21
• @dembrownies Great. Let me know if you have any more questions. Dec 7, 2017 at 2:13
You normalize the vector at each iteration by dividing by its length, and wait until the resulting sequence of unit vectors has gotten close enough to converging for your purpose. (This is called the power method. It's pretty much the worst iterative method for eigenvalues there is, but it is the theoretical basis for lots of better methods.)
This is similar to what you're doing when you measure the slope in the 2D case, except in that case you divide by $x$ instead of dividing by $\sqrt{x^2+y^2}$. The point is that all that matters is the direction, not the magnitude.
• that makes more sense. thank you! Dec 7, 2017 at 1:20
The method you are thinking of is called Power Iteration. More formally, take any vector $x_0\ne 0$. (In practice, it is good to choose $x_0$ randomly. Then define
$$x_k := \frac{Ax_{k-1}}{\|Ax_{k-1}\|_2}, \quad k\in\{1,2,\ldots\}$$
With high probability, $x_k$ will converge to an eigenvector of $A$ corresponding to the largest eigenvalue $\lambda_1$ of $A$ with
$$\lambda_1 = \lim_{k\to\infty} x_k^T Ax_k.$$
In practice, in computer arithmetic, this method is numerically stable. For a better method, see for example, this. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9861513926334712,
"lm_q1q2_score": 0.8602665783990686,
"lm_q2_score": 0.8723473746782093,
"openwebmath_perplexity": 214.46545664401265,
"openwebmath_score": 0.8366015553474426,
"tags": null,
"url": "https://math.stackexchange.com/questions/2554808/finding-approximation-of-largest-eigenvalue"
} |
# Why can we convert a base $9$ number to a base $3$ number by simply converting each base $9$ digit into two base $3$ digits?
Why can we convert a base $9$ number to a base $3$ number by simply converting each base $9$ digit into two base $3$ digits ?
For example $813_9$ can be converted directly to base $3$ by noting
\begin{array} \space 8_9&=22_3 \\ \space 1_9 &=01_3 \\ \space 3_9 &=10_3 \\ \end{array}
Putting the base digits together ,we get $$813_9=220110_3$$
I know it has to do with the fact that $9=3^2$ but I am not able to understand this all by this simple fact...
• This is actually useful (but not with $9$ and $3$). We can very cheaply go back and forth between hexadecimal (base $16$) and binary. – André Nicolas Jan 26 '16 at 8:30
• This also applies to converting between: binary, octal, hexadecimal systems – Max Payne Jan 26 '16 at 8:31
Consider $N$ in base 3. For simplicity, we can assume that $N_3$ has an even number of digits: if it doesn't, just tack on a leftmost $0$. So let: $$N_3 = t_{2n+1} t_{2n}\dotsc t_{2k+1} t_{2k} \dotsc t_1 t_0.$$ What this positional notation really means is that: $$N = \sum_{i = 0}^{2n+1} t_i 3^i,$$ which we can rewrite as: \begin{align} N &= \sum_{k = 0}^{n} (t_{2k+1} 3^{2k+1} + t_{2k} 3^{2k}) \\ &= \sum_{k = 0}^{n} (3 t_{2k+1} + t_{2k}) 3^{2k} \\ &= \sum_{k = 0}^{n} (3 t_{2k+1} + t_{2k}) 9^{k}. \\ \end{align}
But now, note that for each $k$, $3 t_{2k+1} + t_{2k}$ is precisely the base-9 digit corresponding to the consecutive pair of base-3 digits $t_{2k+1} t_{2k}$.
• Altough I've already accepted an answer,I thought to let you know that this answer has been so insightful.Thanks for your time ! – Mr. Y Jan 26 '16 at 9:18
• How nice — thank you:) and you're welcome. – BrianO Jan 26 '16 at 9:24
• That is the real good prove – JnxF Jan 26 '16 at 9:57
• Oh my - thanks doubly. – BrianO Jan 26 '16 at 10:05
• yes this is the proof. My answer is just a visual. +1 – Max Payne Jan 26 '16 at 10:16 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9861513897844354,
"lm_q1q2_score": 0.8602665742773444,
"lm_q2_score": 0.8723473730188542,
"openwebmath_perplexity": 386.82093710414586,
"openwebmath_score": 0.987087607383728,
"tags": null,
"url": "https://math.stackexchange.com/questions/1627453/why-can-we-convert-a-base-9-number-to-a-base-3-number-by-simply-converting-e/1627475"
} |
This has more to do with place value.
Consider the image:
The key is the fact that $3^2 = 9$
• All the value below $3^2$ ($3^0$ and $3^1$) has to be accomodated in $9^0$
that is
• all the values in $9^0$ have to be accommodated in the two places below $3^2$ ($3^0$ and $3^1$)
Similarly for any other places.
• This rule also applies to conversion between binary and octal, or binary and hexadecimal, or conversion between any base $k$ and base $k^n$
In general, for conversion of a number $N$ in any base $k$ to base $k^n$, each digit of $N_{k^n}$ get converted to n digits of $N_k$
• Thanks for the answer .One question : in general if we have that some number $c$ (I am considering everything in base $10$) such that $c=b^n$ does this mean that I can convert the number $c$ in base $b$ by simple converting each base $10$ digit of $c$ into $n$ base $b$ digits ?Or is this something that works only for these special cases ? – Mr. Y Jan 26 '16 at 8:56
• @Mr.Y, yes, you can. An even stronger statement would be: given $a^n=b^m$, we could convert a base $a^n$ number, say, $k$, to base $b^m$, by converting every $n$ digits of $k$ (base $a^n$) to $m$ digits base $b^m$. – vrugtehagel Jan 26 '16 at 9:02
I'm not sure whether you're asking for a proof, or some intuition. The proof is fairly mechanical, so I'll explain why you might expect this to be true.
Putting your information theory hat on, two base 3 digits carry exactly as much information as one base 9 digit. That is, suppose you know I'm going to pass you two digits, each from 0,1 and 2. Then you know there are $3 \cdot 3 =9$ possibilities for what information I could pass.
On the other hand, if I passed you a single digit from 0,1,...,8, then there are again 9 possibilities.
Continuing in this fashion, $n$ digits of base 9 pass as much information as $2n$ digits of base 3. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9861513897844354,
"lm_q1q2_score": 0.8602665742773444,
"lm_q2_score": 0.8723473730188542,
"openwebmath_perplexity": 386.82093710414586,
"openwebmath_score": 0.987087607383728,
"tags": null,
"url": "https://math.stackexchange.com/questions/1627453/why-can-we-convert-a-base-9-number-to-a-base-3-number-by-simply-converting-e/1627475"
} |
Continuing in this fashion, $n$ digits of base 9 pass as much information as $2n$ digits of base 3.
This isn't exactly a proof yet. You'd need to prove that you don't skip any values. Perhaps doing the computation you describe can never return the value $102_3$. Thankfully it can. But you'd need to think about why this will always work.
Let's look at what you base $9$ number actually means. $$813_9=8\cdot 9^2+1\cdot 9^1+3\cdot 9^0$$ If we wish to write this as powers of $3$ with coefficients between $0$ and $2$, we can simply do \begin{align} 3\cdot 9^0&=1\cdot 3^1+0\cdot 3^0\\ 1\cdot 9^1&=0\cdot 3^3+1\cdot 3^2\\ 8\cdot 9^2&=2\cdot 3^5+2\cdot 3^4\\ \end{align}
Why can we do this?
Why can we write $$a_n\cdot 9^n=b_{2n+1}\cdot 3^{2n+1}+b_{2n}\cdot3^{2n}$$ First note that $9^n3^2n$, so when dividing the equation by that, we get $$\frac{a_n\cdot 9^n}{3^{2n}}=a_n=3b_{2n+1}+b_{2n}=\frac{b_{2n+1}\cdot 3^{2n+1}+b_{2n}\cdot3^{2n}}{3^2n}$$ So actually, the problem comes down to writing a number $0\leq a_n<9$ as $3b_{2n+1}+b_{2n}$, where $0\leq b_{2n},b{2n+1}<3$. This is obviously possible.
Why does this work for binary and hexadecimal?
Actually, the answer is fairly similar. The problem can be reduced equivalently, resulting in the equation $$a_n=8a_{4n+3}+4a_{4n+2}+2a_{4n+1}+a_{4n}$$ Where $0\leq a_n<16$ and $0\leq a_{2n+3},a_{2n+2},a_{2n+1},a_{2n}<2$. The solvability of this equation is in my opinion a little less obvious, but still quite understandable; But, for the sake of a more thourough understanding, we could look at hexadecimal-octagonal conversion. This comes down to the easy equation $a_n=2b_{2n+1}+b_{2n}$, where $0\leq a_n<16$ and $0\leq b_{2n+1},b_{2n}<8$. This is cleary solvable. Doing this for octagonal-base 4 conversion and base 4-binary conversion shows with a recurrance-like approach that this indeed works for hexadecimal-binary conversion.
Hope this helped!
Hint: | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9861513897844354,
"lm_q1q2_score": 0.8602665742773444,
"lm_q2_score": 0.8723473730188542,
"openwebmath_perplexity": 386.82093710414586,
"openwebmath_score": 0.987087607383728,
"tags": null,
"url": "https://math.stackexchange.com/questions/1627453/why-can-we-convert-a-base-9-number-to-a-base-3-number-by-simply-converting-e/1627475"
} |
Hope this helped!
Hint:
$$\color{blue}1\cdot3^5+\color{blue}0\cdot3^4+\color{green}2\cdot3^3+\color{green}2\cdot3^2+\color{red}0\cdot3^1+\color{red}1\cdot3^0 =(\color{blue}1\cdot3^1+\color{blue}0)3^4+(\color{green}2\cdot3^1+\color{green}2)3^2+(\color{red}0\cdot3^1+\color{red}1)3^0\\ =\color{blue}3\cdot9^2+\color{green}8\cdot9^1+\color{red}1\cdot9^0$$
$$\color{blue}{10}\color{green}{22}\color{red}{01}_3=[\color{blue}{10_3}|\color{green}{22_3}|\color{red}{01_3}]_9=\color{blue}3\color{green}8\color{red}1_9$$ | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9861513897844354,
"lm_q1q2_score": 0.8602665742773444,
"lm_q2_score": 0.8723473730188542,
"openwebmath_perplexity": 386.82093710414586,
"openwebmath_score": 0.987087607383728,
"tags": null,
"url": "https://math.stackexchange.com/questions/1627453/why-can-we-convert-a-base-9-number-to-a-base-3-number-by-simply-converting-e/1627475"
} |
# Intuition for a relationship between volume and surface area of an $n$-sphere
The volume of an $n$-sphere of radius $R$ is
$$V_n(R) = \frac{\pi^{n/2} R^n}{\Gamma(\frac{n}{2}+1)}$$
and the surface area is
$$S_n(R) = \frac{2\pi^{n/2}R^{n-1}}{\Gamma(\frac{n}{2})} = \frac{n\pi^{n/2}R^{n-1}}{\Gamma(\frac{n}{2}+1)} = \frac{d V_n(R)}{dR}$$
What is the intuition for this relationship between the volume and surface area of an $n$-sphere? Does it relate to the fact that the $n$-sphere is the most compact shape in $n$ dimensions, or is that merely a coincidence?
Are there other shapes for which it holds, or is this the limiting case of a relationship of inequality? For example, an $n$-cube of side $R$ has volume $V^c_n(R)=R^n$ and surface area $S^c_n(R)=2nR^{n-1}$, so that
$$\frac{dV^c_n(R)}{dR} = n R ^{n-1}$$
and we have $S^c_n = 2 dV^c_n/dR$.
Edit: As pointed out by Rahul Nahrain in the comments below, if we define $R$ to be the half-side length of the unit cube rather than the side length, then we have the relationship $S^c_n(R)=\frac{d}{dR} V^c_n(R)$, exactly as for the sphere. Is there a sense in which relationships like this can be stated for a large class of shapes? | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9861513881564148,
"lm_q1q2_score": 0.860266572857145,
"lm_q2_score": 0.8723473730188543,
"openwebmath_perplexity": 162.8361176625972,
"openwebmath_score": 0.8902303576469421,
"tags": null,
"url": "http://math.stackexchange.com/questions/38010/intuition-for-a-relationship-between-volume-and-surface-area-of-an-n-sphere/38017"
} |
-
See this MO-question and also the isoperimetric problem. – t.b. May 9 '11 at 10:52
The main takeaway seems to be that this relationship is coincidental and based on using $R$ as the linear measure of distance. Were we to use $D=2R$ instead, we would get $S_n(D)=2\frac{d}{dD}V_n(D)$. However, there is still the interesting (to my mind) question of why the ratio of $S_n(x)$ to $\frac{d}{dx}V_n(x)$ (for some linear measure $x$) is independent of $n$ for the $n$-sphere but seemingly not for other shapes - no? – Chris Taylor May 9 '11 at 11:04
Sure! I also voted the question up, by the way. I didn't mean to imply that these links answer the question completely. I just wanted you to see them because they certainly give some food for thought. – t.b. May 9 '11 at 11:06
Isn't the surface area of an $n$-cube $2nR^{n-1}$, not $6R^{n-1}$? Then $S_n^c = 2dV_n^c/dR$ for all $n$. – Rahul May 9 '11 at 11:31
In fact, if you let the parameter be the "radius" of the cube, i.e. half of its side length, then $V_n^c = (2R)^n$ and $S_n^c = 2n(2R)^{n-1}$, and $S_n^c = dV_n^c/dR$ for all $n$, just like the sphere... – Rahul May 9 '11 at 11:39
There is a very simple geometric explanation for the fact that the constant of proportionality is 1 for the sphere's radius and the cube's half-width. In fact, this relationship also lets you define a sensible notion of a "half-width" of an arbitrary $n$-dimensional shape.
Pick an arbitrary shape and a point $O$ inside it. Suppose you enlarge the shape by a factor $\alpha \ll 1$ keeping $O$ fixed. Each surface element with area $dA$ at a position $\vec r$ relative to $O$ gets extruded into an approximate prism shape with base area $dA$ and offset $\alpha \vec r$. The corresponding additional volume is $\alpha \vec r\cdot \vec n dA$, where $\vec n$ is the normal vector at the surface element. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9861513881564148,
"lm_q1q2_score": 0.860266572857145,
"lm_q2_score": 0.8723473730188543,
"openwebmath_perplexity": 162.8361176625972,
"openwebmath_score": 0.8902303576469421,
"tags": null,
"url": "http://math.stackexchange.com/questions/38010/intuition-for-a-relationship-between-volume-and-surface-area-of-an-n-sphere/38017"
} |
Now the quantity $\vec r \cdot \vec n$, call it the projected distance, has a natural geometric interpretation. It is simply the distance between $O$ and the tangent plane at the surface element. (Observe that for a sphere with $O$ at the center, it is always equal to the radius, and for a cube with $O$ at the center, it is always equal to the distance from the center to any face.)
Let $\hat r = A^{-1} \int \vec r \cdot \vec n dA$ be the mean projected distance over the surface of the shape. Then the change in volume by a scaling of $\alpha$ is simply $\delta V = \alpha \int \vec r\cdot \vec n dA = \alpha \hat r A$. In other words, a change of $\alpha \hat r$ in $\hat r$ corresponds to a changed of $\alpha \hat r A$ in $V$. So if you use $\hat r$ as the measure of the size of a shape, you find that $dV/d\hat r = A$. And since $\hat r$ equals the radius of a sphere and the half-width of a cube, the observation in question follows. This also implies the distance-to-face measure for regular polytopes that user9325 mentioned, but generalizes to other polytopes and curved shapes. (I'm not completely happy with the definition of $\hat r$ because it's not obvious that it is independent of the choice of $O$. If someone can see a more natural definition, please let me know.)
-
Some remarks:
1. It is not necessary to regard solids that generalize to $n$ dimensions, so one can start with shapes in 2 dimensions.
2. It is very natural to always use a radius-type parameter to scale the figure because the intuition is that the figure gets growth rings that have the size of the surface.
3. Unfortunately, the property is no longer true if you replace a square with a rectangle or a circle with an ellipse.
4. For a regular polygon, you can always take as parameter the distance to an edge. This will work similarly for Platonic solids or any polygons/polyhedra that contain a point that is equidistant to all faces. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9861513881564148,
"lm_q1q2_score": 0.860266572857145,
"lm_q2_score": 0.8723473730188543,
"openwebmath_perplexity": 162.8361176625972,
"openwebmath_score": 0.8902303576469421,
"tags": null,
"url": "http://math.stackexchange.com/questions/38010/intuition-for-a-relationship-between-volume-and-surface-area-of-an-n-sphere/38017"
} |
5. This argument does not look good for general curves, because intuitively the edges of a smooth curve are infinitesimally small, so the center should be equidistant to all points, but of course, we could identify cases where the changing thickness of the growth ring averages out.
-
+1. Point 2 is the intuitive explanation, while points 3 and 5 are the essential caveats. – Henry May 9 '11 at 12:32
What you're seeing with spheres is an instance of something really important called the coarea formula. You're looking at a function like $f(x,y,z) = r = \sqrt{x^2 + y^2 + z^2}$ and you're comparing the volumes of the lower contour sets $f(x,y,z) \leq R$ (in this case they are balls) to the areas of the level sets $f(x,y,z) = R$.
In the instance with cubes, you're considering a different function whose level sets are cubes.
The coarea formula states that (subject to some assumptions that cause both sides to make sense),
$Vol(f(x, y, z) \leq R) = \int_{-\infty}^R \left[ \int_{f(x,y,z) = t} |\nabla f|^{-1} d\sigma \right] dt$
You can also differentiate this formula with respect to $R$ to get the differential form you were observing.
Just as a dummy check, note that $t$ has the same units as $f$, so if you think of $f$ and the coordinates as having units, the dimensional analysis checks out and both sides have units of "length^3". (The formula is also valid in higher dimensions.)
In the examples you gave, the gradient of $f$ has length $1$, so when you integrate $1$ over the level sets $\{f(x, y, z) = t \}$ you're just getting the surface area. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9861513881564148,
"lm_q1q2_score": 0.860266572857145,
"lm_q2_score": 0.8723473730188543,
"openwebmath_perplexity": 162.8361176625972,
"openwebmath_score": 0.8902303576469421,
"tags": null,
"url": "http://math.stackexchange.com/questions/38010/intuition-for-a-relationship-between-volume-and-surface-area-of-an-n-sphere/38017"
} |
The proof of this formula boils down to asking "what's the difference between $Vol( f(x,y,z) \leq R )$ and $Vol(f(x,y,z) \leq R + h$ if $h$ is small?". The two regions are very similar, differing only around the boundary $f(x,y,z) = R$. This is particularly clear in the case of a sphere. If you think about a small portion of that boundary of "width" $d\sigma$ in the directions of constant $f$, then the difference of the two regions has height $~ \frac{h}{|\nabla f|}$ in the direction of increasing $f$.
Another dummy check: The inverse makes sense because if $f$ is increasing extremely quickly, then increasing the value of $f$ will hardly increase the region $\{ f(x,y,z) \leq R \}$.
Note, the statement of the coarea is really a statement about the function and not just about the shapes of the level sets -- you are asking about the parameter $R$ by which these level sets are labeled, but you could pick a different function (like $\tilde{f} = e^f$ which has the same level sets but labels them differently.
- | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9861513881564148,
"lm_q1q2_score": 0.860266572857145,
"lm_q2_score": 0.8723473730188543,
"openwebmath_perplexity": 162.8361176625972,
"openwebmath_score": 0.8902303576469421,
"tags": null,
"url": "http://math.stackexchange.com/questions/38010/intuition-for-a-relationship-between-volume-and-surface-area-of-an-n-sphere/38017"
} |
# Easiest way to show positive semi-definite equivalence
For an $$x \in \mathbb{R}^n$$, and $$n$$-by-$$n$$ identity matrix $$I_n$$, we are given that $$\begin{pmatrix} I_n & x \\ x^T & 1 \end{pmatrix} \succeq 0.$$
What is the easiest way to show that $$\begin{pmatrix} 1 & x^T \\ x & I_n \end{pmatrix} \succeq 0$$ holds?
• Symmetric permutation? – user251257 Mar 22 at 18:37
• seems so. is there such a theory which concludes? – independentvariable Mar 22 at 18:40
• Schur complement? – user251257 Mar 22 at 19:02
• They dont reduce to the same condition, do they? – independentvariable Mar 22 at 19:22
This is due to the identity
$$\underbrace{\begin{pmatrix} 0_{1,n} & 1 \\ I_n & 0_{n,1} \end{pmatrix}}_{J^T} \underbrace{\begin{pmatrix} I_n & x \\ x^T & 1 \end{pmatrix}}_{A} \underbrace{\begin{pmatrix} 0_{n,1} & I_n \\ 1 & 0_{1,n} \end{pmatrix}}_{J} = \underbrace{\begin{pmatrix} 1 & x^T \\ x & I_n \end{pmatrix}}_{B}$$
(notation $$0_{m,n}$$ is for a zero block with $$m$$ lines and $$n$$ columns).
Indeed, $$J$$ being a permutation matrix, it is an orthogonal matrix, with $$J^T=J^{-1}$$. We can conclude that $$A$$ and $$B$$ are similar, thus have the same spectrum (Similar matrices have the same eigenvalues with the same geometric multiplicity) with positive eigenvalues, thus are both semi-definite positive.
Besides, $$A$$ being symmetric, one can conclude from $$J^TAJ=B$$ that $$B$$ is symmetric as well.
Appendix : There is a pending question : is there a criteria on $$x$$ for positive semi-definiteness of $$A$$. ? The answer is yes :
$$A$$ is semi-definite positive iff $$\|x\| \leq 1$$.
This is due, as we are going to see it, to an analysis of the rather particular spectrum of $$A$$. Let us obtain it explicitly.
First of all, let us establish that $$A$$ (which is a $$(n+1) \times (n+1)$$ matrix) has eigenvalue $$1$$ with order of multiplicity at least $$n-1$$. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9861513918194609,
"lm_q1q2_score": 0.8602665711434675,
"lm_q2_score": 0.8723473680407889,
"openwebmath_perplexity": 230.35230850436702,
"openwebmath_score": 0.9334749579429626,
"tags": null,
"url": "https://math.stackexchange.com/questions/3158493/easiest-way-to-show-positive-semi-definite-equivalence"
} |
Consider hyperplane $$x^{\perp}$$ of $$\mathbb{R}^n$$ defined as the set of vectors $$y$$ that are orthogonal to $$x$$. Let $$(y_1,y_2,\cdots y_{n-1})$$ be a basis of $$x^{\perp}$$ ; then,
$$\underbrace{\begin{pmatrix} I_n & x \\ x^T & 1 \end{pmatrix}}_{A}\underbrace{\begin{pmatrix} y_k\\ 0 \end{pmatrix}}_{V_k}=1\underbrace{\begin{pmatrix} y_k\\ 0 \end{pmatrix}}_{V_k} \ \ \ \text{for} \ \ k=1,2, \cdots (n-1),$$
proving that $$V_k$$ is an eigenvector associated with eigenvalue $$1$$.
Due to the fact that trace$$(A)=n+1$$, the two remaining eigenvalues are of the form $$\alpha$$ and $$\beta:=2-\alpha$$. We can assume, WLOG that $$\alpha \leq 1 \leq \beta$$.
Besides, using the so-called Schur determinant identity (Eigenvalues of a Block Matrix from Schur Determinant Identity) for the computation of the determinant of a $$2 \times 2$$ block matrix, we obtain :
$$\det(A)=1-x^Tx$$
As the determinant is also the product of eigenvalues, we get the following identity :
$$\det(A)=1-\|x\|^2=\alpha(2-\alpha)\tag{1}$$
Thus, one can compute explicitly the two remaining eigenvalues by solving quadratic equation (1), with the following explicit solutions (if we assume that $$\alpha$$ is the smallest eigenvalue)
$$\alpha=1 - \|x\| \ \ \ \implies \ \ \ \beta:=2-\alpha=1 + \|x\|\tag{2}$$
As the criteria for a symmetric matrix do be semi-definite positive is that must have all eigenvalues $$\geq 0$$, this criteria becomes $$\alpha \geq 0$$, i.e., $$\|x\| \leq 0$$. $$\square$$
Remark : eigenvalues $$\alpha$$ and $$\beta$$ can be associated with eigenvectors $$\begin{pmatrix} x\\ -\|x\| \end{pmatrix}$$ and $$\begin{pmatrix} x\\ \|x\| \end{pmatrix}$$ resp.
Let us take an example in the case $$n=4$$ ; let $$m=1/n$$ ; consider matrix :
$$A:=\left(\begin{array}{rrrr|r} 1 & & & & m \\ & 1 & & & m \\ & & 1 & & m\\ & & & 1 & m\\ \hline m & m & m & m & 1 \\ \end{array}\right)$$
One can check, using (2), that the spectrum of $$A$$ is
$$(\tfrac12, 1 , 1, 1, 1, \tfrac32).$$ | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9861513918194609,
"lm_q1q2_score": 0.8602665711434675,
"lm_q2_score": 0.8723473680407889,
"openwebmath_perplexity": 230.35230850436702,
"openwebmath_score": 0.9334749579429626,
"tags": null,
"url": "https://math.stackexchange.com/questions/3158493/easiest-way-to-show-positive-semi-definite-equivalence"
} |
One can check, using (2), that the spectrum of $$A$$ is
$$(\tfrac12, 1 , 1, 1, 1, \tfrac32).$$
Just now, I "googled" with keywords "bordered identity matrix" : I found in (Eigenvalues of a certain bordered identity matrix) a somewhat similar computation that I did in the Appendix.
• Thank you for your answer! My main purpose was showing $||x||_2 \leq 1$ holds iff the matrices I gave are Psd. I proved one by Schur complement theory on PSD matrics, but for the equivalence I just followed the definition of PSD and by contradiction showed that if one holds, the other one should hold etc.. – independentvariable Mar 23 at 19:28
• But yours seem the better way, not the 'dirty' $a^T X a \geq 0$ for all $a$ approach... I don't like it, it seems too manual – independentvariable Mar 23 at 19:35 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9861513918194609,
"lm_q1q2_score": 0.8602665711434675,
"lm_q2_score": 0.8723473680407889,
"openwebmath_perplexity": 230.35230850436702,
"openwebmath_score": 0.9334749579429626,
"tags": null,
"url": "https://math.stackexchange.com/questions/3158493/easiest-way-to-show-positive-semi-definite-equivalence"
} |
Is the Cartesian product of two countably infinite sets also countably infinite?
I am trying to determine and prove whether the set of convergent sequences of prime numbers is countably or uncountably infinite.
It is clear that such a sequence must 'terminate' with an infinite repetition of some prime $p$. So for example $${1,2,3,5,5,5,5,...}$$
My idea is to break up the problem into two sub-sequences. The first is a finite subsequence consisting of the first however many terms before it begins the infinite repetition of $p$. Then the second being the infinite set with elements being only $p$.
So in the above example, the two sub-sequences will be $$\{1,2,3\} \{5,5,5,...\}$$
The infinite sub-sequence is countably infinite as it's cardinality is simply the cardinality of the primes, which is countably infinite.
There are infinitely many finite sub-sequences as it can have any number of terms. However, I think it will be countably infinite because it can have $1$ term, $2$ terms, $3$ terms etc, in other words we can construct the bijection to $\mathbb{N}$ by simply naming them according to the number of terms it has. The arrangements is irrelevant as it will be finite.
This is where the title of the question comes in. There are a countably infinite number of these finite sub-sequences say $F$. We also have a countably infinite number of the infinite subsequence $P$, with all elements being some prime $p$.
So in some sense, the set of all convergent sequences of primes is the cartesian product $F\times P$.
As they are both countably infinite, will their cartesian product also be countably infinite? I am thinking yes, as the rationals are countably infinite and they are in some sense a cartesian product of the numerator and denominator, each having cardinality equivalent to the naturals.
If so, and everything is correct, I think I will have then completed the proof. Otherwise, I'd love to have errors pointed out or even perhaps a better solution. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9921841109796002,
"lm_q1q2_score": 0.8602591011179391,
"lm_q2_score": 0.8670357563664174,
"openwebmath_perplexity": 105.60574821025583,
"openwebmath_score": 0.943945050239563,
"tags": null,
"url": "https://math.stackexchange.com/questions/1687015/is-the-cartesian-product-of-two-countably-infinite-sets-also-countably-infinite"
} |
Your thinking is correct. In fact, what you have argued is that there exists a bijection between the set of all convergent sequences of primes and the set $F\times P$. If this is your first example of doing this kind of task I suggest you try to actually write down this bijection. You have argued quite well that it must exist, but it's good exercise to actually do it at least once.
Noy, you also rightly say that $F\times P$ has a bijection to $\mathbb N^2$. As before, I suggest you actually write down this bijection.
Now, you simply need to see if $\mathbb N^2$ is countable, that is:
Does there exist a bijection between $\mathbb N$ and $\mathbb N^2$
To that end, I suggest you look into Cantor's pairing function.
• or you might just observe that $(i,j)\mapsto 2^i3^j$ is an injection. This idea extends easily to $\mathbb N^k$. For example, $(i,j,k)\mapsto 2^i3^j5^k$. Mar 7, 2016 at 16:06
• @Chilango Of course. Though I think that for a "first time user", the actual bijection is more informative than just the construction of an injection (which, I admit, implies a bijection, but also a theorem)
– 5xum
Mar 7, 2016 at 16:32
• these maps are not surjective, but that's no problem. An injection into a countable set is enough to say that the domain is countable. All you need is the fact that a subset of a countable set is countable. I agree though, that the pairing function is more interesting and has more applications. Mar 7, 2016 at 16:50 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9921841109796002,
"lm_q1q2_score": 0.8602591011179391,
"lm_q2_score": 0.8670357563664174,
"openwebmath_perplexity": 105.60574821025583,
"openwebmath_score": 0.943945050239563,
"tags": null,
"url": "https://math.stackexchange.com/questions/1687015/is-the-cartesian-product-of-two-countably-infinite-sets-also-countably-infinite"
} |
# $x^5 + y^2 = z^3$
While waiting for my döner at lunch the other day, I noticed my order number was $343 = 7^3$ (surely not the total for that day), which reminded me of how $3^5 = 243$, so that $$7^3 = 3^5 + 100 = 3^5 + 10^2.$$ Naturally, I started wondering about nontrivial integer solutions to $$x^5 + y^2 = z^3 \tag{*}$$ ("nontrivial" meaning $xyz \ne 0$). I did not make much progress, though apparently there are infinitely many solutions: this was Problem 1 on the 1991 Canadian Mathematical Olympiad. The official solutions (at the bottom of this page) only go back to 1994. A cheap answer is given by taking $x = 2^{2k}$ and $y = 2^{5k}$ so that the l.h.s. is $2^{10k + 1}$. This is a cube iff $10k + 1 \equiv 0 \,(3)$ i.e. $k \equiv 2\,(3)$ thus giving an arithmetic progression's worth of solutions, starting with $$(x, y, z) = (16, 1024, 128)$$ corresponding to $k = 2$ and $$(x, y, z) = (1024, 33554432, 131072)$$ coming from $k = 5$.
What else is known about the equation $(*)$? In particular, are there infinitely many solutions with $x$, $y$, $z$ relatively prime? The one that caught my attention was $(x, y, z) = (3, 10, 7)$. Another one is $(-1, 3, 2)$ because $-1 + 9 = 8$. By Catalan's conjecture (now a theorem), this is the only solution with $x = \pm 1$ or $y = \pm 1$ or $z = 1$. Are there any solutions with $z = -1$? In this case, $(*)$ reduces to $x^5 + y^2 = -1$ and Mihăilescu's theorem does not apply.
Update. This question was essentially already asked here, since the equation $a^2 + b^3 = c^5$ is equivalent to $(-c)^5 + a^2 = (-b)^3$.
• we can rewrite this as $x^5+y^2\equiv 0 \pmod z$ could that help you ?
– user451844
Jul 27 '17 at 1:09
• You might look at OEIS sequences A070065, A070066 and A070067. Jul 27 '17 at 2:08
• Note that if $(x,y,z)$ is a solution, then so is $(c^6 x, c^{15} y, c^{10} z)$ for integers $c$. Jul 27 '17 at 2:41
• quora.com/… Aug 3 '17 at 11:45 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9766692366242304,
"lm_q1q2_score": 0.860247411297536,
"lm_q2_score": 0.8807970795424088,
"openwebmath_perplexity": 485.6667700955476,
"openwebmath_score": 0.9063588380813599,
"tags": null,
"url": "https://math.stackexchange.com/questions/2373028/x5-y2-z3"
} |
Yes, there are infinitely many solutions. In fact, there are many parametrizations of the solutions.
According to a book${}^{\color{blue}{[1]}}$ on my bookshelf,
Up to changing $y$ into $-y$, there are exactly 27 distinct parametrizations of the equations $x^5 + y^2 = z^3$.
One of the simplest paremetrization is given by following formula.
\begin{align} x =&\; 12st(81s^{10}-1584t^5s^5-256t^{10})\\ y =&\; \pm (81s^{10} + 256t^{10})\\ &\;\;\times (6561s^{20} - 6088608t^5s^{15} - 207484416t^{10}s^{10} + 19243008t^{15}s^5 + 65536t^{20})\\ z =&\; 6561s^{20}+2659392t^{5}s^{15}+10243584t^{10}s^{10} - 8404992t^{15}s^5 + 65536t^{20} \end{align}
For example, following two random choices of $s,t$ give you two sets of relative prime solutions.
• $(s,t) = (1,1) \leadsto (x,y,z) = (-21108,-65464918703,4570081)$
• $(s,t) = (1,2) \leadsto (x,y,z) = (-7506024,127602747389962225,-196120763999)$
The book I have is actually quoting result from a thesis${}^{\color{blue}{[2]}}$ by J. Edwards. Consult that if you really want to get into the details.
References
• $\color{blue}{[1]}$ Henri Cohen, Number Theory Number Theory Volume II: Analytic and Modern Tools,
$\S 14.5.2$ The Icosahedron Case $(2,3,5)$.
• $\color{blue}{[2]}$ J. Edwards, Platonic solids and solutions to $x^2+y^3 = dz^r$, Thesis, Univ. Utrecht (2005). | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9766692366242304,
"lm_q1q2_score": 0.860247411297536,
"lm_q2_score": 0.8807970795424088,
"openwebmath_perplexity": 485.6667700955476,
"openwebmath_score": 0.9063588380813599,
"tags": null,
"url": "https://math.stackexchange.com/questions/2373028/x5-y2-z3"
} |
• Nice answer, especially for the first reference. +1
– Xam
Jul 27 '17 at 3:38
• @achille: We can also use the icosahedral equation and scale it appropriately. Kindly see answer below. Jul 27 '17 at 12:00
• @achille: The parameterizations for $x^5+y^3=z^2$ depend on the icosahedron while $x^4+y^3=z^2$ depend on the tetrahedron. Does the book you cite say how many distinct parameterizations are there for the latter? This page gives $7$, but I just found an $8$th one, with no scaling involved. Jul 28 '17 at 5:53
• @TitoPiezasIII In $\S 14.4.1$ of that book, Cohen proved there are 7 parametrizations. Jul 28 '17 at 7:13
• @achillehui: I just re-checked the parameterization I found, and it turns out there is a common factor. Darn! In Edwards' work, he cites 7 parameterizations as well. Thanks, anyway. Jul 28 '17 at 7:35
There is a beautiful connection between $a^5+b^3=c^2$ and the icosahedron. Consider the unscaled icosahedral equation,
$$\color{blue}{12^3u v(u^2 + 11 u v - v^2)^5}+(u^4 - 228 u^3 v + 494 u^2 v^2 + 228 u v^3 + v^4)^3 = (u^6 + 522 u^5 v - 10005 u^4 v^2 - 10005 u^2 v^4 - 522 u v^5 + v^6)^2\tag1$$
By scaling $u=12x^5$ and $v=12y^5$ (or various combinations thereof like $u=12^2x^5$, etc), we then get a relation of form,
$$12^5a^5+b^3=c^2$$
• +1 interesting connection. it is unfortunate the mythical $j(\tau)$ is stuff way beyond me ;-p. Jul 27 '17 at 13:14 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9766692366242304,
"lm_q1q2_score": 0.860247411297536,
"lm_q2_score": 0.8807970795424088,
"openwebmath_perplexity": 485.6667700955476,
"openwebmath_score": 0.9063588380813599,
"tags": null,
"url": "https://math.stackexchange.com/questions/2373028/x5-y2-z3"
} |
# Checking for linear independence
Is the following proof correct?
Theorem. If $v_1,v_2,v_3,v_4$ is a linearly independent list then $$v_1-v_2,v_2-v_3,v_3-v_4,v_4$$ is also a linearly independent list.
Proof. Assume that $v_1,v_@,v_3,v_4$ is a linearly independent list, Consider now the following equation. $$0=0(v_1-v_2)+0(v_2-v_3)+0(v_3-v_4)+0v_4\tag{1}$$ Let $a_1,a_2,a_3$ and $a_4$ be arbitrary scalars in $\mathbf{F}$ and assume that the following equation holds $$0=a_1(v_1-v_2)+a_2(v_2-v_3)+a_3(v_3-v_4)+a_4v_4\tag{2}$$ After some algebraic manipulation we arrive at the following equation. $$0=a_1v_1+(a_2-a_1)v_2+(a_3-a_2)v_3+(a_4-a_3)v_4\tag{3}$$ Since the list $v_1,v_2,v_3,v_4$ is linearly independent it follows that given any vector in $span(v_1,v_2,v_3,v_4)$ the choice of scalars is unique and since $$0=0v_1+0v_2+0v_3+0v_4\tag{4}$$ It follows that all the scalars in $(3)$ must be $0$, consequently the only way to produce the $0$ vector as a linear combination of the vectors in the list $v_1-v_2,v_2-v_3,v_3-v_4,v_4$ is that indicated in $(1)$.
$\blacksquare$
Here $\mathbf{F}$ is either $\mathbb{C}$ or $\mathbb{R}$.
• Sounds ok - would just like to point out 'Consider now the following equation.' sounds a bit strange for $(1)$. Perhaps remove that line and $(1)$ and say something along the lines 'by inspection, $(2)$ holds when all the scalars are equal to $0$' at the end of your proof – Shuri2060 Jul 25 '17 at 17:31
• Nitpicking, but you made a small typo just before your first labeled equation, where you write "Assume $v_1,v_{@},v_3,v_4$ is a linearly independent list." Did you mean $v_2$ not $v_{@}$ ? – Vivek Kaushik Jul 25 '17 at 17:37
Equation (1) should be omitted: it's an obvious fact that has no consequence on the rest. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9766692271169856,
"lm_q1q2_score": 0.8602474075076565,
"lm_q2_score": 0.8807970842359877,
"openwebmath_perplexity": 154.00636836755788,
"openwebmath_score": 0.9979005455970764,
"tags": null,
"url": "https://math.stackexchange.com/questions/2371406/checking-for-linear-independence"
} |
Equation (1) should be omitted: it's an obvious fact that has no consequence on the rest.
The final argument is too fast: from linear independence of $v_1,v_2,v_3,v_4$ you deduce \begin{cases} a_1=0 \\ a_2-a_1=0 \\ a_3-a_2=0 \\ a_4-a_3=0 \end{cases} and, from this, $a_1=a_2=a_3=a_4=0$. This should be mentioned, although easy.
A different approach is to consider the coordinates of the new vectors with respect to the original ones and so the matrix \begin{bmatrix} 1 & 0 & 0 & 0 \\ -1 & 1 & 0 & 0 \\ 0 & -1 & 1 & 0 \\ 0 & 0 & -1 & 1 \end{bmatrix} A standard Gaussian elimination leads to the reduced row echelon form \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} which proves that the coordinate vectors are linearly independent and so also the vectors are.
• So your point is that i should show how each of the scalars $a_1,a_2,a_3$ and $a_4$ is 0. – Atif Farooq Jul 25 '17 at 17:59
• The original matrix is lower triangular with $1$'s in the diagonal, hence invertible. – lhf Jul 25 '17 at 18:49
• @AtifFarooq Yes, that's the point; notwithstanding it's easy, it should appear in the proof. – egreg Jul 25 '17 at 19:27
• @lhf That's true, but Gaussian elimination would give more information in case the vectors aren't linearly independent. – egreg Jul 25 '17 at 19:27
It's valid. A note:
Before $(2)$ when you say "...and assume that the following equation holds..." it would be better to perhaps phrase this as "We wish to solve ... for $a_j$." Saying the former means that you're assuming solutions exist - but you're not sure! This is a tiny point, and is perhaps reflected in my writing style.
We may also streamline a bit by saying the following after $(3)$.
As $\{v_1, v_2, v_3, v_4\}$ is a linearly independent set, we must have that $a_1 = 0, \ a_2-a_1=0, \ a_3 - a_2 = 0,$ and $a_4-a_3=0.$ Back substitution yields that each $a_j = 0$, and so the original set is also a linearly independent set. $\square$ | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9766692271169856,
"lm_q1q2_score": 0.8602474075076565,
"lm_q2_score": 0.8807970842359877,
"openwebmath_perplexity": 154.00636836755788,
"openwebmath_score": 0.9979005455970764,
"tags": null,
"url": "https://math.stackexchange.com/questions/2371406/checking-for-linear-independence"
} |
Again, these are just small suggestions, but nonetheless your proof is correct. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9766692271169856,
"lm_q1q2_score": 0.8602474075076565,
"lm_q2_score": 0.8807970842359877,
"openwebmath_perplexity": 154.00636836755788,
"openwebmath_score": 0.9979005455970764,
"tags": null,
"url": "https://math.stackexchange.com/questions/2371406/checking-for-linear-independence"
} |
$\int_0^{2\pi} \sqrt{1-\cos(x)}\,dx = 4\sqrt{2}$. Why?
According to the textbook, and Wolfram Alpha the above is correct.
Here is the step by step procedure from Wolfram Alpha for evaluating the indefinite integral:
Take the integral: $$\int\sqrt{1-\cos(x)}\,dx$$ For the integrand $\sqrt{1-\cos(x)}$, substitute $u=1-\cos(x)$ and $du=\sin(x)\,dx$: $$=\int-\frac{1}{\sqrt{2-u}}\,du$$ Factor out constants: $$=-\int\frac{1}{\sqrt{2-u}}\,du$$ For the integrand $1/\sqrt{2-u}$, substitute $s=2-u$ and $ds=-du$: $$=\int\frac{1}{\sqrt{s}}\,ds$$ The integral of $1/\sqrt{s}$ is $2\sqrt{s}$: $$=2\sqrt{s}+\text{constant}$$ Substitute back for $s=2-u$: $$=2\sqrt{2-u}+\text{constant}$$ Substitute back for $u=1-\cos(x)$: $$=2\sqrt{\cos(x)+1}+\text{constant}$$ Which is equivalent for restricted $x$ values to: $$\boxed{=-2\sqrt{1-\cos(x)}\cot\big(\frac{x}{2}\big)+\text{constant}}$$
I understand up to the below (which is a valid solution to the integral): $$2\sqrt{\cos(x)+1}+\text{constant}$$
However, if you evaluate this at $2\pi$ and $0$, you get the same thing, so the definite integral evaluates to zero.
After, you transform the above to: $$-2\sqrt{1-\cos(x)}\cot\big(\frac{x}{2}\big)+\text{constant}$$
The expression is indeterminate at $2\pi$ and $0$ of the form $0 \times \infty$. So I guess you would set up a limit and then use L'Hospital's rule to evaluate the expression at $2\pi$ and $0$ and get the answer to the definite integral?
In any case, all this seems strange. Why should the definite integral evaluated one way give $0$, and in another way give something else? | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.976669234586964,
"lm_q1q2_score": 0.8602474033910186,
"lm_q2_score": 0.8807970732843033,
"openwebmath_perplexity": 403.9850245364357,
"openwebmath_score": 0.999889612197876,
"tags": null,
"url": "https://math.stackexchange.com/questions/546007/int-02-pi-sqrt1-cosx-dx-4-sqrt2-why"
} |
• You may want to find the anti derivative by multiplying top and bottom by the squareroot of the conjugate. You will end up with the absolute value of the sine term on top and the square root of 1+cosx in the bottom. This integrates easily. Mind the absolute value though as you apply your upper and lower limit. Oct 30 '13 at 19:57
• Ahh ok that makes sense. So to get rid of the absolute value sign, you can split up the integral from $0$ to $\pi$ and $\pi$ to $2\pi$ Oct 30 '13 at 20:02
• Yes, you ought to split that integral. Oct 30 '13 at 20:05 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.976669234586964,
"lm_q1q2_score": 0.8602474033910186,
"lm_q2_score": 0.8807970732843033,
"openwebmath_perplexity": 403.9850245364357,
"openwebmath_score": 0.999889612197876,
"tags": null,
"url": "https://math.stackexchange.com/questions/546007/int-02-pi-sqrt1-cosx-dx-4-sqrt2-why"
} |
$$\int_0^{2\pi}\sqrt{1-\cos x}\,dx=\int_0^{2\pi}\sqrt{2\sin^2\frac{x}{2}}\,dx =\sqrt{2}\int_0^{2\pi}\sin\frac{x}{2}\,dx=-2\sqrt{2}\cos\frac{x}{2}\Big|_0^{2\pi}=4\sqrt{2}.$$
• @RonGordon $x/2 \in [0,\pi] \Rightarrow \sin(x/2) \ge 0$. Oct 30 '13 at 20:49
• Good old $\cos(2x) = \cos^2(x)-\sin^2(x)= 1-2\sin^2(x) = 2\cos^2(x)-1$. Oct 31 '13 at 1:25
Your $u$-substitutions should be injective on their interval of evaluation. Otherwise, you risk running into exactly this sort of issue.
Note that \begin{align}|\sin x| &= \sqrt{\sin^2 x}\\ &= \sqrt{1-\cos^2 x}\\ &= \sqrt{1-\cos x}\sqrt{1+\cos x}\\ &=\sqrt{1-\cos x}\sqrt{2-(1-\cos x)},\end{align} so if you want to use $u=1-\cos x$, then $$\frac{du}{dx}=\sin x=\begin{cases}|\sin x|=\sqrt{1-\cos x}\sqrt{2-(1-\cos x)} & 0\le x\le \pi\\-|\sin x|=-\sqrt{1-\cos x}\sqrt{2-(1-\cos x)} & \pi\le x\le2\pi,\end{cases}$$ so \begin{align}\int_0^{2\pi}\sqrt{1-\cos x}\,dx &= \int_0^\pi\sqrt{1-\cos x}\,dx+\int_\pi^{2\pi}\sqrt{1-\cos x}\,dx\\ &= \int_0^\pi\frac{|\sin x|}{\sqrt{2-(1-\cos x)}}\,dx+\int_\pi^{2\pi}\frac{|\sin x|}{\sqrt{2-(1-\cos x)}}\,dx\\ &= \int_0^\pi\frac{\sin x\,dx}{\sqrt{2-(1-\cos x)}}-\int_\pi^{2\pi}\frac{\sin x\,dx}{\sqrt{2-(1-\cos x)}}\\ &= \int_0^2\frac{du}{\sqrt{2-u}}-\int_2^0\frac{du}{\sqrt{2-u}}\\ &= 2\int_0^2\frac{du}{\sqrt{2-u}}.\end{align} At that point, we can use that antiderivative, with no need to resubstitute.
Alternately, you could note that $\cos(2\pi-x)=\cos x$, so \begin{align}\int_0^{2\pi}\sqrt{1-\cos x}\,dx &= \int_0^\pi\sqrt{1-\cos x}\,dx+\int_\pi^{2\pi}\sqrt{1-\cos x}\,dx\\ &= \int_0^\pi\sqrt{1-\cos x}\,dx+\int_\pi^{2\pi}\sqrt{1-\cos(2\pi-x)}\,dx\\ &= \int_0^\pi\sqrt{1-\cos x}\,dx-\int_{2\pi}^\pi\sqrt{1-\cos(2\pi-x)}\,dx\\ &= \int_0^\pi\sqrt{1-\cos x}\,dx-\int_0^\pi\sqrt{1-\cos x}\frac{d(2\pi-x)}{dx}\,dx\\ &= 2\int_0^\pi\sqrt{1-\cos x}\,dx,\end{align} at which point you can use your $u$-substitution without fear, since the cosine function is injective on $[0,\pi]$. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.976669234586964,
"lm_q1q2_score": 0.8602474033910186,
"lm_q2_score": 0.8807970732843033,
"openwebmath_perplexity": 403.9850245364357,
"openwebmath_score": 0.999889612197876,
"tags": null,
"url": "https://math.stackexchange.com/questions/546007/int-02-pi-sqrt1-cosx-dx-4-sqrt2-why"
} |
# Square divided by absolute value
First time posting on Math SE, with kind of a basic algebra question.
Question
Does the relation:
$$\dfrac{(ab)^2}{|ab|} = \left|ab\right|$$
with $a,b \in \mathbb{R_{\ne 0}}$ always hold?
It seems trivial to me, but Wolfram Alpha gives me a strange answer because it specifies that this is True assuming $a,b$ are positive.
Reasoning
No matter what sign $a,b$ have, we have that $(ab)^2 > 0$ and $\left|ab\right| > 0$. Thus their ratio is greater than zero, and the magnitude of that ratio is exactly $ab$ with a positive sign, so $\left|ab\right|$.
Is what I said correct? If so, is this question a completely useless one? Sorry for the occasionally bad English!
Edit: formatted equations as suggested by Frentos
• Hmmn, wlog for $a, b \in \mathbb{C}$ one could argue that $a^{2}b^{2} < 0$ – Kevin Jan 20 '16 at 11:28
• Welcome to math.SE! See this guide for how to write equations on this site. \dfrac makes larger, easier to read fractions and \left| \right| gives nicer absolute values. – Frentos Jan 20 '16 at 11:32
• @Bacon No, try $a=b=i$ (or any case when $a^2b^2$ is not even real). – Did Jan 21 '16 at 1:45
• @Did - fair point, my comment meant to reflect that in some cases this could be true – Kevin Jan 21 '16 at 9:20
Your statement about Wolfram is not quite correct. It gives various alternate forms for this expression, two of which are:
1. $ab$ assuming $a$ and $b$ are positive
2. $ab\,sgn(a)\,sgn(b)$
(2) is equivalent to $|ab|$
See here
• Indeed, I overlooked the $ab \; sgn(a) \; sgn(b)$ answer! I'll accept this one because it points out that Wolfram was right. – UJIN Jan 20 '16 at 14:30
For real numbers this is always true because the square of a real number equals the square of its absolute value, and in particular $(ab)^2=|ab|^2.$ Perhaps Wolfram has reservations because it considers the possibility of complex numbers?
Notice:
• If $a,b\in\mathbb{R^+}$, so $a,b>0$, then:
$$|ab|=|a||b|=ab$$ | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9766692284751636,
"lm_q1q2_score": 0.8602473980077626,
"lm_q2_score": 0.8807970732843033,
"openwebmath_perplexity": 646.0958758089511,
"openwebmath_score": 0.9206000566482544,
"tags": null,
"url": "https://math.stackexchange.com/questions/1619416/square-divided-by-absolute-value"
} |
Notice:
• If $a,b\in\mathbb{R^+}$, so $a,b>0$, then:
$$|ab|=|a||b|=ab$$
• If $a,b\in\mathbb{R^+}$, so $a,b>0$, so:
$$(ab)^2=a^2b^2$$
• If $a,b\in\mathbb{R^-}$, than $a,b<0$, so:
$$((-a)(-b))^2=(ab)^2=a^2b^2$$ | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9766692284751636,
"lm_q1q2_score": 0.8602473980077626,
"lm_q2_score": 0.8807970732843033,
"openwebmath_perplexity": 646.0958758089511,
"openwebmath_score": 0.9206000566482544,
"tags": null,
"url": "https://math.stackexchange.com/questions/1619416/square-divided-by-absolute-value"
} |
GMAT Question of the Day - Daily to your Mailbox; hard ones only
It is currently 21 Apr 2019, 01:23
### GMAT Club Daily Prep
#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
Customized
for You
we will pick new questions that match your level based on your Timer History
Track
every week, we’ll send you an estimated GMAT score based on your performance
Practice
Pays
we will pick new questions that match your level based on your Timer History
# A wire that weighs 20 pounds is cut into two pieces so that one of the
new topic post reply Question banks Downloads My Bookmarks Reviews Important topics
Author Message
TAGS:
### Hide Tags
Math Expert
Joined: 02 Sep 2009
Posts: 54376
A wire that weighs 20 pounds is cut into two pieces so that one of the [#permalink]
### Show Tags
30 Jun 2017, 03:28
00:00
Difficulty:
55% (hard)
Question Stats:
65% (02:06) correct 35% (01:59) wrong based on 63 sessions
### HideShow timer Statistics
A wire that weighs 20 pounds is cut into two pieces so that one of the pieces weighs 16 pounds and is 36 feet long. If the weight of each piece is directly proportional to the square of its length, how many feet long is the other piece of wire?
(A) 9
(B) 12
(C) 18
(D) 24
(E) 27
_________________
Senior Manager
Joined: 28 May 2017
Posts: 283
Concentration: Finance, General Management
Re: A wire that weighs 20 pounds is cut into two pieces so that one of the [#permalink]
### Show Tags
30 Jun 2017, 03:57
Bunuel wrote:
A wire that weighs 20 pounds is cut into two pieces so that one of the pieces weighs 16 pounds and is 36 feet long. If the weight of each piece is directly proportional to the square of its length, how many feet long is the other piece of wire?
(A) 9
(B) 12
(C) 18
(D) 24
(E) 27
Weight of 2nd piece = 4 pound | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9766692271169855,
"lm_q1q2_score": 0.8602473937554338,
"lm_q2_score": 0.8807970701552505,
"openwebmath_perplexity": 2340.8452587349466,
"openwebmath_score": 0.5843057632446289,
"tags": null,
"url": "https://gmatclub.com/forum/a-wire-that-weighs-20-pounds-is-cut-into-two-pieces-so-that-one-of-the-243714.html"
} |
(A) 9
(B) 12
(C) 18
(D) 24
(E) 27
Weight of 2nd piece = 4 pound
Since the weight is directly proportional to the square of its length., we may write
$$\frac{16}{36^2}$$ = $$\frac{4}{x^2}$$
Solving above, we get x = 18
_________________
If you like the post, show appreciation by pressing Kudos button
Director
Joined: 04 Dec 2015
Posts: 750
Location: India
Concentration: Technology, Strategy
WE: Information Technology (Consulting)
Re: A wire that weighs 20 pounds is cut into two pieces so that one of the [#permalink]
### Show Tags
30 Jun 2017, 04:02
Bunuel wrote:
A wire that weighs 20 pounds is cut into two pieces so that one of the pieces weighs 16 pounds and is 36 feet long. If the weight of each piece is directly proportional to the square of its length, how many feet long is the other piece of wire?
(A) 9
(B) 12
(C) 18
(D) 24
(E) 27
$$20$$ pounds wire is cut into two pieces = $$16$$ pounds $$+$$ $$4$$ pounds
The piece weighs $$16$$ pounds is $$36$$ feet long.
Ratio of weight and length of $$16$$ pounds piece $$= \frac{16}{36^2} = \frac{4 * 4}{36 * 36} = \frac{1}{81}$$
Therefore required ratio of other part would also be $$\frac{1}{81}$$
Ratio $$= \frac{4}{x^2} =$$ $$\frac{1}{81}$$
$$x^2 = 81*4$$ $$=> x = \sqrt{81*4}$$
$$x = 9 * 2 = 18$$
Hence length of $$4$$ pounds wire $$= 18$$
Intern
Joined: 30 May 2013
Posts: 29
GMAT 1: 600 Q50 V21
GMAT 2: 640 Q49 V29
Re: A wire that weighs 20 pounds is cut into two pieces so that one of the [#permalink]
### Show Tags
30 Jun 2017, 04:07
IMO (c)
Given, weight proportional to square (length)
=> w1 / w2 = square (l1/l2)
=> 16/4 = square (36/l2)
=> l2 = 18
Sent from my GT-N7100 using GMAT Club Forum mobile app
Target Test Prep Representative
Status: Founder & CEO
Affiliations: Target Test Prep
Joined: 14 Oct 2015
Posts: 5807
Location: United States (CA)
Re: A wire that weighs 20 pounds is cut into two pieces so that one of the [#permalink]
### Show Tags | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9766692271169855,
"lm_q1q2_score": 0.8602473937554338,
"lm_q2_score": 0.8807970701552505,
"openwebmath_perplexity": 2340.8452587349466,
"openwebmath_score": 0.5843057632446289,
"tags": null,
"url": "https://gmatclub.com/forum/a-wire-that-weighs-20-pounds-is-cut-into-two-pieces-so-that-one-of-the-243714.html"
} |
### Show Tags
04 Jul 2017, 07:52
1
Bunuel wrote:
A wire that weighs 20 pounds is cut into two pieces so that one of the pieces weighs 16 pounds and is 36 feet long. If the weight of each piece is directly proportional to the square of its length, how many feet long is the other piece of wire?
(A) 9
(B) 12
(C) 18
(D) 24
(E) 27
Since the 20-lb wire is cut into two pieces and one of the pieces weighs 16 lbs, the other piece must weigh 4 lbs. Since the 16-lb piece has length of 36 ft and the weight of each piece is directly proportional to the square of its length, we can let x = the length in feet of the 4-lb piece and create the following proportion:
16/36^2 = 4/x^2
16x^2 = 4 * 36^2
4x^2 = 36^2
x^2 = 36^2/2^2
x^2 = 18^2
x = 18
_________________
# Scott Woodbury-Stewart
Founder and CEO
Scott@TargetTestPrep.com
122 Reviews
5-star rated online GMAT quant
self study course
See why Target Test Prep is the top rated GMAT quant course on GMAT Club. Read Our Reviews
Re: A wire that weighs 20 pounds is cut into two pieces so that one of the [#permalink] 04 Jul 2017, 07:52
Display posts from previous: Sort by
# A wire that weighs 20 pounds is cut into two pieces so that one of the
new topic post reply Question banks Downloads My Bookmarks Reviews Important topics
Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®. | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9766692271169855,
"lm_q1q2_score": 0.8602473937554338,
"lm_q2_score": 0.8807970701552505,
"openwebmath_perplexity": 2340.8452587349466,
"openwebmath_score": 0.5843057632446289,
"tags": null,
"url": "https://gmatclub.com/forum/a-wire-that-weighs-20-pounds-is-cut-into-two-pieces-so-that-one-of-the-243714.html"
} |
Explanation of Counter-example?
I have a question where we are given a statement and asked to prove whether or not the statement is true. If the statement is true, then we must prove it; otherwise we must provide a counterexample to prove the statement incorrect.
If the statement is false, and the student provides a counter-example; should the student also provide an explanation of why the counter-example disproves the statement?
I believe this depends on how clearly the counterexample is stated.
Consider this claim: $f:\mathbb{R}\to\mathbb{R}$ is continuous $\implies f$ is differentiable.
Imagine a student, let's call him Karl, who says
Take any function of the form $\sum_{n=0} ^\infty a^n \cos(b^n \pi x)$, where $0<a<1$, $b$ is a positive odd integer, and $ab > 1+\frac{3}{2} \pi$.
Would you give him credit for this counterexample? Even presuming that the difficulty level of your course and the talents of your students are such that you did, indeed, expect them to come up with this result, I don't feel like this answer is sufficient. Obviously, Karl must have understood something about the exercise to be able to come up with this answer (never mind the great ingenuity required) but his written argument fails to demonstrate that knowledge.
Contrast that with a disproof of this claim: For any function $f:A\to B$ and any sets $S,T\subseteq A$, $f[S]\subseteq f[T]\implies S\subseteq T$.
This is the kind of statement you might consider in an introductory analysis course. I'd expect a student to find an easy counterexample, say by defining $A=\{\heartsuit,\spadesuit\}$ and $B=\{\star\}$ and setting $f(\heartsuit)=f(\spadesuit)=\star$ and $S=\{\heartsuit\}$ and $T=\{\spadesuit\}$. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.967899295134923,
"lm_q1q2_score": 0.8602290069144426,
"lm_q2_score": 0.8887587905460026,
"openwebmath_perplexity": 319.8075443544471,
"openwebmath_score": 0.7324725985527039,
"tags": null,
"url": "https://matheducators.stackexchange.com/questions/4458/explanation-of-counter-example/4460"
} |
Would I expect the student to have to show all the details? Do they really need to explicitly say, "Observe that $S,T\subseteq A$ and $f[S]=f[T]$ and yet $S\not\subseteq T$"? Well, that depends on your standards. If you're really training them to write clearly and explicitly (concision be darned), then you might require them to show these details. If you're just checking to see that they can generate such examples, then merely stating the example would suffice.
Main answer: It depends on your standards for such a written argument, which in turn should depend on what you expect the student to be able to understand and demonstrate.
• Sure, let's call him Karl. Sep 25 '14 at 1:26
• Maybe give extra credit to someone who described how to create a counter example, perhaps in the above case by using S={<heart>} and T={<something besides a heart>}
– rbp
Sep 29 '14 at 15:13
Certainly the student should be aware of the expectations beforehand. To this end, I think there are at least two approaches. One is to specify on individual questions whether counterexamples should be explained, and the other is to establish (preferably from the outset, i.e., setting "norms" at the beginning of a course) the expectation that reasoning/justification for all answers will be provided.
More generally, I think explaining counterexamples is important. Let me give just two reasons.
1) Consider the following problem:
Is every whole number divisible by 6 and 8 also divisible by 6 $\times$ 8 = 48? If so, prove your answer; if not, please provide a counterexample. Answer: 24. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.967899295134923,
"lm_q1q2_score": 0.8602290069144426,
"lm_q2_score": 0.8887587905460026,
"openwebmath_perplexity": 319.8075443544471,
"openwebmath_score": 0.7324725985527039,
"tags": null,
"url": "https://matheducators.stackexchange.com/questions/4458/explanation-of-counter-example/4460"
} |
But what does such a student understand? Perhaps she just remembered 24 is divisible by both and realized it is not divisible by 48. Or perhaps she realizes that the issue is that the two divisors in this proposed rule are not relatively prime, and could correct it by using, e.g., 3 and 16 instead. Or, even more seriously, perhaps she has the deeper understanding, but accidentally writes 32. In this last case, the understanding is the deepest, but it would be tough to justify giving the response any credit.
Asking that the counterexample be justified ameliorates this potential problem. Personally, I would expect more than just an explanation of the particular counterexample, e.g., more than:
No, and 24 is a counterexample: It is divisible by 6 because 24 = 4 $\times$ 6, and it is also divisible by 8 because 24 = 3 $\times$ 8. But it is not divisible by 48, because 24/48 is not a whole number.
I would prefer to see an answer that mentions being "relatively prime" or something equivalent, and perhaps even one that gives an example of two numbers that would yield a divisibility rule for 48.
2) More subtle and perhaps not related to the specific question you have in mind (if there is one): Note that (speaking slightly messily) counterexamples can settle for all questions, but not there exists questions. The previous example is the assertion that all whole numbers (etc). If this turns out to be false, then it can be settled with a counterexample. If it turns out to be true, then it will need a further proof or justification. On the other hand, a there exists statement that is true might be settled with a single example. If it turns out to be false, then it will need a further proof or justification.
This latter point about proving/disproving by (counter)example in the context of for all vs. there exists assertions may seem a bit pedantic, but it is a place in which many students who are just beginning to understand ideas around "proof" can struggle quite a bit. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.967899295134923,
"lm_q1q2_score": 0.8602290069144426,
"lm_q2_score": 0.8887587905460026,
"openwebmath_perplexity": 319.8075443544471,
"openwebmath_score": 0.7324725985527039,
"tags": null,
"url": "https://matheducators.stackexchange.com/questions/4458/explanation-of-counter-example/4460"
} |
• I don't think there exists statements are really that different because you can rewrite them as for all statements, i.e. $\forall x\ P(x) \iff \neg\exists x\ \neg P(x)$. (Of course maybe students don't know this, in which case to their minds there is a difference...) Sep 25 '14 at 3:53
• @DavidZ Yes, this is the underlying "reason" for the phenomenon mentioned in 2 above (i.e., that an example can disprove a for all statement or prove a there exists statement). And yes, one of the primary difficulties is precisely the one to which you allude parenthetically: For many students who are just getting into proof-based mathematics (or other mathematics with more formal reasoning) the equivalence you mention is neither known nor obvious. Sep 25 '14 at 18:08 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.967899295134923,
"lm_q1q2_score": 0.8602290069144426,
"lm_q2_score": 0.8887587905460026,
"openwebmath_perplexity": 319.8075443544471,
"openwebmath_score": 0.7324725985527039,
"tags": null,
"url": "https://matheducators.stackexchange.com/questions/4458/explanation-of-counter-example/4460"
} |
# Calculating area of triangle
Problem
In the triangle above, we have $AC = 12, \ \ BD = 10, \ \ CD = 6$.
Find the area of $\triangle ABD$.
Caveat
This was part of an exam where you are not allowed to use a calculator or any other digital tools.
My progress
Given $AC, CD, \measuredangle C$, I was able to find the length of $AD$ using the law of cosines. It yielded $AD = \sqrt{180} = 6\sqrt5$.
From here, I decided to try and use the law of sines to find $\measuredangle A$ as it would be the same in $\triangle ABD$ and $\triangle ACD$.
I got $$\frac{\sin A}{CD} = \frac{\sin C}{AD}$$.
This yields $$\sin A = \frac{CD \sin C}{AD} = \frac{6}{\sqrt{180}}$$.
Then I need to calculate $\arcsin\frac{6}{\sqrt{180}}$, which I can't, because I can't use a calculator for this problem. (Well, technically I can, but I want to solve this within the constraints placed on the participants.)
From here, I am stuck. In $\triangle ABD$, I know now two of the lengths, but I don't know any of the angles.
Question
Am I overlooking an easy inverse sine here? Or is there another way to calculate this area?
Thanks in advance for any help!
• If this is part of an exam, why are you asking for help? – N. F. Taussig Mar 21 '17 at 20:07
• @N.F.Taussig - It was an exam given in August of 2016. I'm preparing for the 2017 one ;) – Alec Mar 21 '17 at 20:08
## 3 Answers
Find $BC$ in the right triangle $BCD$; you have the hypotenuse is $10$ and the other side is $6$ so $BC=8$.
Area $\triangle ACD = \frac12\cdot 12\cdot 6 = 36$.
Area $\triangle BCD = \frac12\cdot 8\cdot 6 = 24$.
Area $\triangle ABD = 36-24 = 12$.
• Damn... I really blinded myself on this one. Thanks! – Alec Mar 21 '17 at 20:11
Calculate $BC$ with the pythagorean theorem, and subtract the areas of the two right angled triangles.
$$BC=\sqrt{10^2-6^2}=8$$ Hence, $$AB=12-8=4$$ and $$S_{\Delta ABD}=\frac{4\cdot6}{2}=12$$ | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9748211568364099,
"lm_q1q2_score": 0.8602093641512262,
"lm_q2_score": 0.8824278772763471,
"openwebmath_perplexity": 305.30931508786244,
"openwebmath_score": 0.9274730086326599,
"tags": null,
"url": "https://math.stackexchange.com/questions/2197132/calculating-area-of-triangle/2197147"
} |
## mathmath333 one year ago Set Theory
1. mathmath333
\large \color{black}{\begin{align} \normalsize \text{A ∩ B = A ∩ C need not imply B = C.} \quad \normalsize \text{ Explain through an example.}\hspace{.33em}\\~\\ \end{align}}
2. Kainui
A = {1} B = {1, 2} C = {1, 3}
3. ikram002p
huh kai was so fast :P
4. Kainui
Hahaha there could have been a simpler example: A = {} B = {1} C = {2}
5. mathmath333
thnx
6. Kainui
I think that last example I gave sorta shows that it's almost like this is a lot like multiplying by zero. a*b=a*c This doesn't mean b=c, since a=0 is possible.
7. mathmath333
Let A and B be sets. If A ∩ X = B ∩ X = φ and A ∪ X = B ∪ X for some set X, show that A = B.
8. ikram002p
in general any case with $A\cap B \neq \varnothing$ does not imply A=C
9. ikram002p
or $$C\cap B \neq \varnothing$$
10. mathmath333
i need an example like the previous one
11. Kainui
ikram needs more owl bucks sry
12. ikram002p
A={1,2,3} B={1,2,3} C={1,2,3,4}
13. ikram002p
:O @Kainui i dont lol
14. Kainui
Hahaha jk :P
15. mathmath333
is that the example for this que Let A and B be sets. If A ∩ X = B ∩ X = φ and A ∪ X = B ∪ X for some set X, show that A = B.
16. ikram002p
you wanna prove this ?
17. mathmath333
yes with example
18. ikram002p
ok example A={1,2,3} B={1.4} C={1,2.3}
19. mathmath333
but C is not there in the question
20. ikram002p
sorry i ment to say x :P
21. ikram002p
now to prove they are two side of the prove 1- show A is a subset of B 2- show B is a subset of A
22. ikram002p
let $$c\in A \cup X$$ so $$c\in A ~or~c\in X$$ if c in A then $$c\in B\cup X$$ and $$c\in B$$ (since c is not in x) thus $$A\subset B$$ the other way is alike :P sorry im lazy to type
23. thomas5267
Is it possible to do something like this? $A\cup X=B\cup X\land A\cap X=B\cap X=\emptyset\implies A\cup X\setminus X=B\cup X\setminus X\implies A=B$ | {
"domain": "openstudy.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9748211553734217,
"lm_q1q2_score": 0.8602093538179241,
"lm_q2_score": 0.8824278680004707,
"openwebmath_perplexity": 2797.5335863737355,
"openwebmath_score": 0.817804217338562,
"tags": null,
"url": "http://openstudy.com/updates/55c3bfcfe4b06d80932e877f"
} |
GMAT Question of the Day - Daily to your Mailbox; hard ones only
It is currently 23 Sep 2018, 11:28
### GMAT Club Daily Prep
#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
Customized
for You
we will pick new questions that match your level based on your Timer History
Track
every week, we’ll send you an estimated GMAT score based on your performance
Practice
Pays
we will pick new questions that match your level based on your Timer History
# The average (arithmetic mean) length per film for a group of 21 films
Author Message
TAGS:
### Hide Tags
Director
Status: Professional GMAT Tutor
Affiliations: AB, cum laude, Harvard University (Class of '02)
Joined: 10 Jul 2015
Posts: 669
Location: United States (CA)
Age: 38
GMAT 1: 770 Q47 V48
GMAT 2: 730 Q44 V47
GMAT 3: 750 Q50 V42
GRE 1: Q168 V169
WE: Education (Education)
The average (arithmetic mean) length per film for a group of 21 films [#permalink]
### Show Tags
05 May 2016, 19:27
7
9
00:00
Difficulty:
15% (low)
Question Stats:
77% (01:25) correct 23% (01:31) wrong based on 422 sessions
### HideShow timer Statistics
The average (arithmetic mean) length per film for a group of 21 films is t minutes. If a film that runs for 66 minutes is removed from the group and replaced by one that runs for 52 minutes, what is the average length per film, in minutes, for the new group of films, in terms of t?
A) $$t + \frac{2}{3}$$
B) $$t - \frac{2}{3}$$
C) $$21t+14$$
D) $$t + \frac{3}{2}$$
E) $$t - \frac{3}{2}$$
Attachments
Screen Shot 2016-05-05 at 7.04.23 PM.png [ 107.79 KiB | Viewed 5506 times ]
_________________
Harvard grad and 99% GMAT scorer, offering expert, private GMAT tutoring and coaching, both in-person (San Diego, CA, USA) and online worldwide, since 2002. | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9695556515673754,
"lm_q1q2_score": 0.8601942322723564,
"lm_q2_score": 0.8872045981907007,
"openwebmath_perplexity": 9301.023791857197,
"openwebmath_score": 0.48316341638565063,
"tags": null,
"url": "https://gmatclub.com/forum/the-average-arithmetic-mean-length-per-film-for-a-group-of-21-films-217895.html"
} |
One of the only known humans to have taken the GMAT 5 times and scored in the 700s every time (700, 710, 730, 750, 770), including verified section scores of Q50 / V47, as well as personal bests of 8/8 IR (2 times), 6/6 AWA (4 times), 50/51Q and 48/51V (1 question wrong).
You can download my official test-taker score report (all scores within the last 5 years) directly from the Pearson Vue website: https://tinyurl.com/y94hlarr Date of Birth: 09 December 1979.
GMAT Action Plan and Free E-Book - McElroy Tutoring
Contact: mcelroy@post.harvard.edu
Math Expert
Joined: 02 Aug 2009
Posts: 6800
Re: The average (arithmetic mean) length per film for a group of 21 films [#permalink]
### Show Tags
05 May 2016, 20:09
5
1
mcelroytutoring wrote:
The average (arithmetic mean) length per film for a group of 21 films is t minutes. If a film that runs for 66 minutes is removed from the group and replaced by one that runs for 52 minutes, what is the average length per film, in minutes, for the new group of films, in terms of t?
A) $$t + \frac{2}{3}$$
B) $$t - \frac{2}{3}$$
C) $$21t+14$$
D) $$t + \frac{3}{2}$$
E) $$t - \frac{3}{2}$$
hi,
you do not have to get into any calculations if you understand and take the Q logically..
avg time for 21 is t min..
you add 52 and remove 66.. so basically you are removing 14 min from total..
effect on each or average =$$\frac{14}{21}= \frac{2}{3}$$..
so final average = $$t-\frac{2}{3}$$...
B
_________________
1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html
GMAT online Tutor | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9695556515673754,
"lm_q1q2_score": 0.8601942322723564,
"lm_q2_score": 0.8872045981907007,
"openwebmath_perplexity": 9301.023791857197,
"openwebmath_score": 0.48316341638565063,
"tags": null,
"url": "https://gmatclub.com/forum/the-average-arithmetic-mean-length-per-film-for-a-group-of-21-films-217895.html"
} |
GMAT online Tutor
##### General Discussion
Director
Status: Professional GMAT Tutor
Affiliations: AB, cum laude, Harvard University (Class of '02)
Joined: 10 Jul 2015
Posts: 669
Location: United States (CA)
Age: 38
GMAT 1: 770 Q47 V48
GMAT 2: 730 Q44 V47
GMAT 3: 750 Q50 V42
GRE 1: Q168 V169
WE: Education (Education)
The average (arithmetic mean) length per film for a group of 21 films [#permalink]
### Show Tags
Updated on: 06 May 2016, 12:30
1
Logic tells us that the average length per film will go down, so we can eliminate all answers that add to the original average of t. That leaves either B or E. When executing the algebra, the key step is being able to quickly and easily turn one fraction with two numerators $$(\frac{21t-14}{21})$$ into two fractions with one numerator each $$(\frac{21t}{21}-\frac{14}{21})$$.
Above is a visual that should help.
_________________
Harvard grad and 99% GMAT scorer, offering expert, private GMAT tutoring and coaching, both in-person (San Diego, CA, USA) and online worldwide, since 2002.
One of the only known humans to have taken the GMAT 5 times and scored in the 700s every time (700, 710, 730, 750, 770), including verified section scores of Q50 / V47, as well as personal bests of 8/8 IR (2 times), 6/6 AWA (4 times), 50/51Q and 48/51V (1 question wrong).
You can download my official test-taker score report (all scores within the last 5 years) directly from the Pearson Vue website: https://tinyurl.com/y94hlarr Date of Birth: 09 December 1979.
GMAT Action Plan and Free E-Book - McElroy Tutoring
Contact: mcelroy@post.harvard.edu
Originally posted by mcelroytutoring on 05 May 2016, 19:28.
Last edited by mcelroytutoring on 06 May 2016, 12:30, edited 6 times in total.
Current Student
Joined: 12 Aug 2015
Posts: 2648
Schools: Boston U '20 (M)
GRE 1: Q169 V154
Re: The average (arithmetic mean) length per film for a group of 21 films [#permalink]
### Show Tags | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9695556515673754,
"lm_q1q2_score": 0.8601942322723564,
"lm_q2_score": 0.8872045981907007,
"openwebmath_perplexity": 9301.023791857197,
"openwebmath_score": 0.48316341638565063,
"tags": null,
"url": "https://gmatclub.com/forum/the-average-arithmetic-mean-length-per-film-for-a-group-of-21-films-217895.html"
} |
### Show Tags
15 Dec 2016, 14:05
2
Great Solutions above.
Here is my solution =>
Average length = t minutes
Sum(21) = 21t
New sum = 21t-66+52 = 21t-14
Hence new mean = 21t-14 / 21 = t-2/3
Hence B
_________________
MBA Financing:- INDIAN PUBLIC BANKS vs PRODIGY FINANCE!
Getting into HOLLYWOOD with an MBA!
The MOST AFFORDABLE MBA programs!
STONECOLD's BRUTAL Mock Tests for GMAT-Quant(700+)
AVERAGE GRE Scores At The Top Business Schools!
Joined: 30 May 2015
Posts: 40
Re: The average (arithmetic mean) length per film for a group of 21 films [#permalink]
### Show Tags
20 Jul 2017, 15:08
1
average = sum of numbers/total number
t = s/21
s = 21t
Now as per question
s =21t -66 +52
s = 21t - 14
New average
t1 = (21t - 14) / 21
=> t - 2/3
Director
Joined: 09 Mar 2016
Posts: 884
The average (arithmetic mean) length per film for a group of 21 films [#permalink]
### Show Tags
06 Mar 2018, 09:16
stonecold wrote:
Great Solutions above.
Here is my solution =>
Average length = t minutes
Sum(21) = 21t
New sum = 21t-66+52 = 21t-14
Hence new mean = 21t-14 / 21 = t-2/3
Hence B
stonecold how from this 21t-14 / 21 you got t-2/3 , I understand you divided by 7 , but there are two 21 could you write in detail
_________________
In English I speak with a dictionary, and with people I am shy.
BSchool Forum Moderator
Joined: 26 Feb 2016
Posts: 3137
Location: India
GPA: 3.12
The average (arithmetic mean) length per film for a group of 21 films [#permalink]
### Show Tags
06 Mar 2018, 11:18
1
dave13 wrote:
stonecold wrote:
Great Solutions above.
Here is my solution =>
Average length = t minutes
Sum(21) = 21t
New sum = 21t-66+52 = 21t-14
Hence new mean = 21t-14 / 21 = t-2/3
Hence B
stonecold how from this 21t-14 / 21 you got t-2/3 , I understand you divided by 7 , but there are two 21 could you write in detail
Hey dave13
$$\frac{21t-14}{21} = \frac{21t}{21} - \frac{14}{21} = t - \frac{2}{3}$$
Hope that helps
_________________ | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9695556515673754,
"lm_q1q2_score": 0.8601942322723564,
"lm_q2_score": 0.8872045981907007,
"openwebmath_perplexity": 9301.023791857197,
"openwebmath_score": 0.48316341638565063,
"tags": null,
"url": "https://gmatclub.com/forum/the-average-arithmetic-mean-length-per-film-for-a-group-of-21-films-217895.html"
} |
Hope that helps
_________________
You've got what it takes, but it will take everything you've got
Manager
Joined: 10 Apr 2018
Posts: 105
Re: The average (arithmetic mean) length per film for a group of 21 films [#permalink]
### Show Tags
15 Aug 2018, 16:09
Hi,
We can use deviation technique to solve such type of questions. The answer would be in less than 30 Secs
refer: https://gmatclub.com/forum/a-student-s- ... l#p2113194
So new average would be
t-(14/21)
t-2/3
Hence B.
Re: The average (arithmetic mean) length per film for a group of 21 films &nbs [#permalink] 15 Aug 2018, 16:09
Display posts from previous: Sort by
# The average (arithmetic mean) length per film for a group of 21 films
## Events & Promotions
Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®. | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9695556515673754,
"lm_q1q2_score": 0.8601942322723564,
"lm_q2_score": 0.8872045981907007,
"openwebmath_perplexity": 9301.023791857197,
"openwebmath_score": 0.48316341638565063,
"tags": null,
"url": "https://gmatclub.com/forum/the-average-arithmetic-mean-length-per-film-for-a-group-of-21-films-217895.html"
} |
Find all School-related info fast with the new School-Specific MBA Forum
It is currently 15 Sep 2014, 17:00
### GMAT Club Daily Prep
#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
Customized
for You
we will pick new questions that match your level based on your Timer History
Track
every week, we’ll send you an estimated GMAT score based on your performance
Practice
Pays
we will pick new questions that match your level based on your Timer History
# Events & Promotions
###### Events & Promotions in June
Open Detailed Calendar
# Inequalities trick
Author Message
TAGS:
CEO
Status: Nothing comes easy: neither do I want.
Joined: 12 Oct 2009
Posts: 2793
Location: Malaysia
Concentration: Technology, Entrepreneurship
Schools: ISB '15 (M)
GMAT 1: 670 Q49 V31
GMAT 2: 710 Q50 V35
Followers: 178
Kudos [?]: 945 [38] , given: 235
Inequalities trick [#permalink] 16 Mar 2010, 09:11
38
KUDOS
62
This post was
BOOKMARKED
I learnt this trick while I was in school and yesterday while solving one question I recalled.
Its good if you guys use it 1-2 times to get used to it.
Suppose you have the inequality
f(x) = (x-a)(x-b)(x-c)(x-d) < 0
Just arrange them in order as shown in the picture and draw curve starting from + from right.
now if f(x) < 0 consider curve having "-" inside and if f(x) > 0 consider curve having "+" and combined solution will be the final solution. I m sure I have recalled it fully but if you guys find any issue on that do let me know, this is very helpful.
Don't forget to arrange then in ascending order from left to right. a<b<c<d
So for f(x) < 0 consider "-" curves and the ans is : (a < x < b) , (c < x < d)
and for f(x) > 0 consider "+" curves and the ans is : (x < a), (b < x < c) , (d < x)
If f(x) has three factors then the graph will have - + - +
If f(x) has four factors then the graph will have + - + - + | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9597620619801096,
"lm_q1q2_score": 0.860188072580374,
"lm_q2_score": 0.8962513800615312,
"openwebmath_perplexity": 2813.2556971947556,
"openwebmath_score": 0.5333693623542786,
"tags": null,
"url": "http://gmatclub.com/forum/inequalities-trick-91482.html?kudos=1"
} |
If you can not figure out how and why, just remember it.
Try to analyze that the function will have number of roots = number of factors and every time the graph will touch the x axis.
For the highest factor d if x>d then the whole f(x) > 0 and after every interval of the roots the signs will change alternatively.
Attachments
1.jpg [ 6.73 KiB | Viewed 21741 times ]
_________________
Fight for your dreams :For all those who fear from Verbal- lets give it a fight
Money Saved is the Money Earned
Jo Bole So Nihaal , Sat Shri Akaal
Gmat test review :
670-to-710-a-long-journey-without-destination-still-happy-141642.html
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 4755
Location: Pune, India
Followers: 1112
Kudos [?]: 5022 [36] , given: 164
Re: Inequalities trick [#permalink] 22 Oct 2010, 05:33
36
KUDOS
Expert's post
13
This post was
BOOKMARKED
Yes, this is a neat little way to work with inequalities where factors are multiplied or divided. And, it has a solid reasoning behind it which I will just explain.
If (x-a)(x-b)(x-c)(x-d) < 0, we can draw the points a, b, c and d on the number line.
e.g. Given (x+2)(x-1)(x-7)(x-4) < 0, draw the points -2, 1, 7 and 4 on the number line as shown.
Attachment:
doc.jpg [ 7.9 KiB | Viewed 21052 times ]
This divides the number line into 5 regions. Values of x in right most region will always give you positive value of the expression. The reason for this is that if x > 7, all factors above will be positive.
When you jump to the next region between x = 4 and x = 7, value of x here give you negative value for the entire expression because now, (x - 7) will be negative since x < 7 in this region. All other factors are still positive.
When you jump to the next region on the left between x = 1 and x = 4, expression will be positive again because now two factors (x - 7) and (x - 4) are negative, but negative x negative is positive... and so on till you reach the leftmost section. | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9597620619801096,
"lm_q1q2_score": 0.860188072580374,
"lm_q2_score": 0.8962513800615312,
"openwebmath_perplexity": 2813.2556971947556,
"openwebmath_score": 0.5333693623542786,
"tags": null,
"url": "http://gmatclub.com/forum/inequalities-trick-91482.html?kudos=1"
} |
Since we are looking for values of x where the expression is < 0, here the solution will be -2 < x < 1 or 4< x < 7
It should be obvious that it will also work in cases where factors are divided.
e.g. (x - a)(x - b)/(x - c)(x - d) < 0
(x + 2)(x - 1)/(x -4)(x - 7) < 0 will have exactly the same solution as above.
Note: If, rather than < or > sign, you have <= or >=, in division, the solution will differ slightly. I will leave it for you to figure out why and how. Feel free to get back to me if you want to confirm your conclusion.
_________________
Karishma
Veritas Prep | GMAT Instructor
My Blog | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9597620619801096,
"lm_q1q2_score": 0.860188072580374,
"lm_q2_score": 0.8962513800615312,
"openwebmath_perplexity": 2813.2556971947556,
"openwebmath_score": 0.5333693623542786,
"tags": null,
"url": "http://gmatclub.com/forum/inequalities-trick-91482.html?kudos=1"
} |
Save $100 on Veritas Prep GMAT Courses And Admissions Consulting Enroll now. Pay later. Take advantage of Veritas Prep's flexible payment plan options. Veritas Prep Reviews Math Forum Moderator Joined: 20 Dec 2010 Posts: 2047 Followers: 128 Kudos [?]: 919 [13] , given: 376 Re: Inequalities trick [#permalink] 11 Mar 2011, 05:49 13 This post received KUDOS 4 This post was BOOKMARKED vjsharma25 wrote: VeritasPrepKarishma wrote: vjsharma25 wrote: How you have decided on the first sign of the graph?Why it is -ve if it has three factors and +ve when four factors? Check out my post above for explanation. I understand the concept but not the starting point of the graph.How you decide about the graph to be a sine or cosine waveform?Meaning graph starts from the +ve Y-axis for four values and starts from -ve Y-axis for three values. What if the equation you mentioned is (x+2)(x-1)(x-7)<0,will the last two ranges be excluded or the graph will also change? I always struggle with this as well!!! There is a trick Bunuel suggested; (x+2)(x-1)(x-7) < 0 Here the roots are; -2,1,7 Arrange them in ascending order; -2,1,7; These are three points where the wave will alternate. The ranges are; x<-2 -2<x<1 1<x<7 x>7 Take a big value of x; say 1000; you see the inequality will be positive for that. (1000+2)(1000-1)(1000-7) is +ve. Thus the last range(x>7) is on the positive side. Graph is +ve after 7. Between 1 and 7-> -ve between -2 and 1-> +ve Before -2 -> -ve Since the inequality has the less than sign; consider only the -ve side of the graph; 1<x<7 or x<-2 is the complete range of x that satisfies the inequality. _________________ Manager Joined: 29 Sep 2008 Posts: 154 Followers: 2 Kudos [?]: 31 [8] , given: 1 Re: Inequalities trick [#permalink] 22 Oct 2010, 10:45 8 This post received KUDOS 3 This post was BOOKMARKED if = sign is included with < then <= will be there in solution like for (x+2)(x-1)(x-7)(x-4) <=0 the solution will be -2 <= x <= 1 or 4<= x <= 7 in case when factors are | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9597620619801096,
"lm_q1q2_score": 0.860188072580374,
"lm_q2_score": 0.8962513800615312,
"openwebmath_perplexity": 2813.2556971947556,
"openwebmath_score": 0.5333693623542786,
"tags": null,
"url": "http://gmatclub.com/forum/inequalities-trick-91482.html?kudos=1"
} |
(x+2)(x-1)(x-7)(x-4) <=0 the solution will be -2 <= x <= 1 or 4<= x <= 7 in case when factors are divided then the numerator will contain = sign like for (x + 2)(x - 1)/(x -4)(x - 7) < =0 the solution will be -2 <= x <= 1 or 4< x < 7 we cant make 4<=x<=7 as it will make the solution infinite correct me if i am wrong Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 4755 Location: Pune, India Followers: 1112 Kudos [?]: 5022 [4] , given: 164 Re: Inequalities trick [#permalink] 11 Mar 2011, 18:57 4 This post received KUDOS Expert's post vjsharma25 wrote: I understand the concept but not the starting point of the graph.How you decide about the graph to be a sine or cosine waveform?Meaning graph starts from the +ve Y-axis for four values and starts from -ve Y-axis for three values. What if the equation you mentioned is (x+2)(x-1)(x-7)<0,will the last two ranges be excluded or the graph will also change? Ok, look at this expression inequality: (x+2)(x-1)(x-7) < 0 Can I say the left hand side expression will always be positive for values greater than 7? (x+2) will be positive, (x - 1) will be positive and (x-7) will also be positive... so in the rightmost regions i.e. x > 7, all three factors will be positive. The expression will be positive when x > 7, it will be negative when 1 < x < 7, positive when -2 , x < 1 and negative when x < -2. We need the region where the expression is less than 0 i.e. negative. So either 1 < x < 7 or x < -2. Now let me add another factor: (x+8)(x+2)(x-1)(x-7) Can I still say that the entire expression is positive in the rightmost region i.e. x>7 because each one of the four factors is positive? Yes. So basically, your rightmost region is always positive. You go from there and assign + and - signs to the regions. Your starting point is the rightmost region. Note: Make sure that the factors are of the form (ax - b), not (b - ax)... e.g. (x+2)(x-1)(7 - x)<0 Convert this to: (x+2)(x-1)(x-7)>0 (Multiply both sides by '-1') Now solve in the | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9597620619801096,
"lm_q1q2_score": 0.860188072580374,
"lm_q2_score": 0.8962513800615312,
"openwebmath_perplexity": 2813.2556971947556,
"openwebmath_score": 0.5333693623542786,
"tags": null,
"url": "http://gmatclub.com/forum/inequalities-trick-91482.html?kudos=1"
} |
- x)<0 Convert this to: (x+2)(x-1)(x-7)>0 (Multiply both sides by '-1') Now solve in the usual way. Assign '+' to the rightmost region and then alternate with '-' Since you are looking for positive value of the expression, every region where you put a '+' will be the region where the expression will be greater than 0. _________________ Karishma Veritas Prep | GMAT Instructor My Blog Save$100 on Veritas Prep GMAT Courses And Admissions Consulting | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9597620619801096,
"lm_q1q2_score": 0.860188072580374,
"lm_q2_score": 0.8962513800615312,
"openwebmath_perplexity": 2813.2556971947556,
"openwebmath_score": 0.5333693623542786,
"tags": null,
"url": "http://gmatclub.com/forum/inequalities-trick-91482.html?kudos=1"
} |
Enroll now. Pay later. Take advantage of Veritas Prep's flexible payment plan options. | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9597620619801096,
"lm_q1q2_score": 0.860188072580374,
"lm_q2_score": 0.8962513800615312,
"openwebmath_perplexity": 2813.2556971947556,
"openwebmath_score": 0.5333693623542786,
"tags": null,
"url": "http://gmatclub.com/forum/inequalities-trick-91482.html?kudos=1"
} |
Veritas Prep Reviews
Manager
Status: On...
Joined: 16 Jan 2011
Posts: 189
Followers: 3
Kudos [?]: 35 [4] , given: 62
Re: Inequalities trick [#permalink] 10 Aug 2011, 16:01
4
KUDOS
2
This post was
BOOKMARKED
WoW - This is a cool thread with so many thing on inequalities....I have compiled it together with some of my own ideas...It should help.
1) CORE CONCEPT
@gurpreetsingh -
Suppose you have the inequality
f(x) = (x-a)(x-b)(x-c)(x-d) < 0
Arrange the NUMBERS in ascending order from left to right. a<b<c<d
Draw curve starting from + from right.
now if f(x) < 0 consider curve having "-" inside and if f(x) > 0 consider curve having "+" and combined solution will be the final solution. I m sure I have recalled it fully but if you guys find any issue on that do let me know, this is very helpful.
So for f(x) < 0 consider "-" curves and the ans is : (a < x < b) , (c < x < d)
and for f(x) > 0 consider "+" curves and the ans is : (x < a), (b < x < c) , (d < x)
If f(x) has three factors then the graph will have - + - +
If f(x) has four factors then the graph will have + - + - +
If you can not figure out how and why, just remember it.
Try to analyze that the function will have number of roots = number of factors and every time the graph will touch the x axis.
For the highest factor d if x>d then the whole f(x) > 0 and after every interval of the roots the signs will change alternatively.
Note: Make sure that the factors are of the form (ax - b), not (b - ax)...
example -
(x+2)(x-1)(7 - x)<0
Convert this to: (x+2)(x-1)(x-7)>0 (Multiply both sides by '-1')
Now solve in the usual way. Assign '+' to the rightmost region and then alternate with '-'
Since you are looking for positive value of the expression, every region where you put a '+' will be the region where the expression will be greater than 0.
2) Variation - ODD/EVEN POWER
@ulm/Karishma -
if we have even powers like (x-a)^2(x-b)
we don't need to change a sign when jump over "a".
This will be same as (x-b) | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9597620619801096,
"lm_q1q2_score": 0.860188072580374,
"lm_q2_score": 0.8962513800615312,
"openwebmath_perplexity": 2813.2556971947556,
"openwebmath_score": 0.5333693623542786,
"tags": null,
"url": "http://gmatclub.com/forum/inequalities-trick-91482.html?kudos=1"
} |
We can ignore squares BUT SHOULD consider ODD powers
example -
2.a
(x-a)^3(x-b)<0 is the same as (x-a)(x-b) <0
2.b
(x - a)(x - b)/(x - c)(x - d) < 0 ==> (x - a)(x - b)(x-c)^-1(x-d)^-1 <0
is the same as (x - a)(x - b)(x - c)(x - d) < 0
3) Variation <= in FRACTION
@mrinal2100 -
if = sign is included with < then <= will be there in solution
like for (x+2)(x-1)(x-7)(x-4) <=0 the solution will be -2 <= x <= 1 or 4<= x <= 7
BUT if it is a fraction the denominator in the solution will not have = SIGN
example -
3.a
(x + 2)(x - 1)/(x -4)(x - 7) < =0
the solution will be -2 <= x <= 1 or 4< x < 7
we cant make 4<=x<=7 as it will make the solution infinite
4) Variation - ROOTS
@Karishma -
As for roots, you have to keep in mind that given \sqrt{x}, x cannot be negative.
\sqrt{x} < 10
implies 0 < \sqrt{x} < 10
Squaring, 0 < x < 100
Root questions are specific. You have to be careful. If you have a particular question in mind, send it.
Refer - inequalities-and-roots-118619.html#p959939
Some more useful tips for ROOTS....I am too lazy to consolidate
<5> THESIS -
@gmat1220 -
Once algebra teacher told me - signs alternate between the roots. I said whatever and now I know why Watching this article is a stroll down the memory lane.
I will save this future references....
Anyone wants to add ABSOLUTE VALUES....That will be a value add to this post
_________________
Labor cost for typing this post >= Labor cost for pushing the Kudos Button
kudos-what-are-they-and-why-we-have-them-94812.html
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 4755
Location: Pune, India
Followers: 1112
Kudos [?]: 5022 [2] , given: 164
Re: Inequalities trick [#permalink] 09 Sep 2013, 22:35
2
KUDOS
Expert's post
karannanda wrote:
gurpreetsingh wrote:
I learnt this trick while I was in school and yesterday while solving one question I recalled.
Its good if you guys use it 1-2 times to get used to it.
Suppose you have the inequality
f(x) = (x-a)(x-b)(x-c)(x-d) < 0 | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9597620619801096,
"lm_q1q2_score": 0.860188072580374,
"lm_q2_score": 0.8962513800615312,
"openwebmath_perplexity": 2813.2556971947556,
"openwebmath_score": 0.5333693623542786,
"tags": null,
"url": "http://gmatclub.com/forum/inequalities-trick-91482.html?kudos=1"
} |
Suppose you have the inequality
f(x) = (x-a)(x-b)(x-c)(x-d) < 0
Just arrange them in order as shown in the picture and draw curve starting from + from right.
now if f(x) < 0 consider curve having "-" inside and if f(x) > 0 consider curve having "+" and combined solution will be the final solution. I m sure I have recalled it fully but if you guys find any issue on that do let me know, this is very helpful.
Don't forget to arrange then in ascending order from left to right. a<b<c<d
So for f(x) < 0 consider "-" curves and the ans is : (a < x < b) , (c < x < d)
and for f(x) > 0 consider "+" curves and the ans is : (x < a), (b < x < c) , (d < x)
If f(x) has three factors then the graph will have - + - +
If f(x) has four factors then the graph will have + - + - +
If you can not figure out how and why, just remember it.
Try to analyze that the function will have number of roots = number of factors and every time the graph will touch the x axis.
For the highest factor d if x>d then the whole f(x) > 0 and after every interval of the roots the signs will change alternatively.
Hi Gurpreet,
Thanks for the wonderful method.
I am trying to understand it so that i can apply it in tests.
Can you help me in applying this method to the below expression to find range of x.
x^3 – 4x^5 < 0?
I am getting the roots as -1/2, 0, 1/2 and when i plot them using this method, putting + in the rightmost region, I am not getting correct result. Not sure where i am going wrong. Can you pls help.
Before you apply the method, ensure that the factors are of the form (x - a)(x - b) etc
x^3 - 4x^5 < 0
x^3 ( 1 - 4x^2) < 0
x^3(1 - 2x) (1 + 2x) < 0
4x^3(x - 1/2)(x + 1/2) > 0 (Notice the flipped sign. We multiplied both sides by -1 to convert 1/2 - x to x - 1/2)
Now the transition points are 0, -1/2 and 1/2 so put + in the rightmost region.
The solution will be x > 1/2 or -1/2 < x< 0. | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9597620619801096,
"lm_q1q2_score": 0.860188072580374,
"lm_q2_score": 0.8962513800615312,
"openwebmath_perplexity": 2813.2556971947556,
"openwebmath_score": 0.5333693623542786,
"tags": null,
"url": "http://gmatclub.com/forum/inequalities-trick-91482.html?kudos=1"
} |
Check out these posts discussing such complications:
http://www.veritasprep.com/blog/2012/06 ... e-factors/
http://www.veritasprep.com/blog/2012/07 ... ns-part-i/
http://www.veritasprep.com/blog/2012/07 ... s-part-ii/
_________________
Karishma
Veritas Prep | GMAT Instructor
My Blog
Save $100 on Veritas Prep GMAT Courses And Admissions Consulting Enroll now. Pay later. Take advantage of Veritas Prep's flexible payment plan options. Veritas Prep Reviews Senior Manager Joined: 16 Feb 2012 Posts: 260 Concentration: Finance, Economics Followers: 4 Kudos [?]: 65 [1] , given: 110 Re: Inequalities trick [#permalink] 22 Jul 2012, 02:03 1 This post received KUDOS VeritasPrepKarishma wrote: mrinal2100: Kudos to you for excellent thinking! Correct me if I'm wrong. If the lower part of the equation\frac {(x+2)(x-1)}{(x-4)(x-7)} were 4\leq x \leq 7, than the lower part would be equal to zero,thus making it impossible to calculate the whole equation. _________________ Kudos if you like the post! Failing to plan is planning to fail. Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 4755 Location: Pune, India Followers: 1112 Kudos [?]: 5022 [1] , given: 164 Re: Inequalities trick [#permalink] 23 Jul 2012, 02:13 1 This post received KUDOS Expert's post Stiv wrote: VeritasPrepKarishma wrote: mrinal2100: Kudos to you for excellent thinking! Correct me if I'm wrong. If the lower part of the equation\frac {(x+2)(x-1)}{(x-4)(x-7)} were 4\leq x \leq 7, than the lower part would be equal to zero,thus making it impossible to calculate the whole equation. x cannot be equal to 4 or 7 because if x = 4 or x = 7, the denominator will be 0 and the expression will not be defined. _________________ Karishma Veritas Prep | GMAT Instructor My Blog Save$100 on Veritas Prep GMAT Courses And Admissions Consulting
Enroll now. Pay later. Take advantage of Veritas Prep's flexible payment plan options.
Veritas Prep Reviews | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9597620619801096,
"lm_q1q2_score": 0.860188072580374,
"lm_q2_score": 0.8962513800615312,
"openwebmath_perplexity": 2813.2556971947556,
"openwebmath_score": 0.5333693623542786,
"tags": null,
"url": "http://gmatclub.com/forum/inequalities-trick-91482.html?kudos=1"
} |
Veritas Prep Reviews
CEO
Status: Nothing comes easy: neither do I want.
Joined: 12 Oct 2009
Posts: 2793
Location: Malaysia
Concentration: Technology, Entrepreneurship
Schools: ISB '15 (M)
GMAT 1: 670 Q49 V31
GMAT 2: 710 Q50 V35
Followers: 178
Kudos [?]: 945 [1] , given: 235
Re: Inequalities trick [#permalink] 18 Oct 2012, 04:17
1
KUDOS
GMATBaumgartner wrote:
gurpreetsingh wrote:
ulm wrote:
if we have smth like (x-a)^2(x-b)
we don't need to change a sign when jump over "a".
yes even powers wont contribute to the inequality sign. But be wary of the root value of x=a
Hi Gurpreet,
Could you elaborate what exactly you meant here in highlighted text ?
Even I have a doubt as to how this can be applied for powers of the same term . like the example mentioned in the post above.
If the powers are even then the inequality won't be affected.
eg if u have to find the range of values of x satisfying (x-a)^2 *(x-b)(x-c) >0
just use (x-b)*(x-c) >0 because x-a raised to the power 2 will not affect the inequality sign. But just make sure x=a is taken care off , as it would make the inequality zero.
_________________
Fight for your dreams :For all those who fear from Verbal- lets give it a fight
Money Saved is the Money Earned
Jo Bole So Nihaal , Sat Shri Akaal
Gmat test review :
670-to-710-a-long-journey-without-destination-still-happy-141642.html
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 4755
Location: Pune, India
Followers: 1112
Kudos [?]: 5022 [1] , given: 164
Re: Inequalities trick [#permalink] 18 Oct 2012, 09:27
1
KUDOS
Expert's post
GMATBaumgartner wrote:
Hi Gurpreet,
Could you elaborate what exactly you meant here in highlighted text ?
Even I have a doubt as to how this can be applied for powers of the same term . like the example mentioned in the post above.
In addition, you can check out this post:
http://www.veritasprep.com/blog/2012/07 ... s-part-ii/
I have discussed how to handle powers in it.
_________________ | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9597620619801096,
"lm_q1q2_score": 0.860188072580374,
"lm_q2_score": 0.8962513800615312,
"openwebmath_perplexity": 2813.2556971947556,
"openwebmath_score": 0.5333693623542786,
"tags": null,
"url": "http://gmatclub.com/forum/inequalities-trick-91482.html?kudos=1"
} |
I have discussed how to handle powers in it.
_________________
Karishma
Veritas Prep | GMAT Instructor
My Blog | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9597620619801096,
"lm_q1q2_score": 0.860188072580374,
"lm_q2_score": 0.8962513800615312,
"openwebmath_perplexity": 2813.2556971947556,
"openwebmath_score": 0.5333693623542786,
"tags": null,
"url": "http://gmatclub.com/forum/inequalities-trick-91482.html?kudos=1"
} |
Save $100 on Veritas Prep GMAT Courses And Admissions Consulting Enroll now. Pay later. Take advantage of Veritas Prep's flexible payment plan options. Veritas Prep Reviews Math Expert Joined: 02 Sep 2009 Posts: 26582 Followers: 3487 Kudos [?]: 26158 [1] , given: 2705 Re: Inequalities trick [#permalink] 02 Dec 2012, 06:25 1 This post received KUDOS Expert's post GMATGURU1 wrote: Going ahead in this inequalities area of GMAT, can some one has the problem numbers of inequalities in OG 11 and OG 12? Cheers, Danny Search for hundreds of question with solutions by tags: viewforumtags.php DS questions on inequalities: search.php?search_id=tag&tag_id=184 PS questions on inequalities: search.php?search_id=tag&tag_id=189 Hardest DS inequality questions with detailed solutions: inequality-and-absolute-value-questions-from-my-collection-86939.html Hope it helps. _________________ CEO Status: Nothing comes easy: neither do I want. Joined: 12 Oct 2009 Posts: 2793 Location: Malaysia Concentration: Technology, Entrepreneurship Schools: ISB '15 (M) GMAT 1: 670 Q49 V31 GMAT 2: 710 Q50 V35 Followers: 178 Kudos [?]: 945 [1] , given: 235 Re: Inequalities trick [#permalink] 20 Dec 2012, 20:04 1 This post received KUDOS VeritasPrepKarishma wrote: The entire concept is based on positive/negative factors which means <0 or >0 is a must. If the question is not in this format, you need to bring it to this format by taking the constant to the left hand side. e.g. (x + 2)(x + 3) < 2 x^2 + 5x + 6 - 2 < 0 x^2 + 5x + 4 < 0 (x+4)(x+1) < 0 Now use the concept. Yes this is probable but it might not be possible always to group them. So in case you are unsure just follow the number plugging approach. But most of the times this trick would be very handy. _________________ Fight for your dreams :For all those who fear from Verbal- lets give it a fight Money Saved is the Money Earned Jo Bole So Nihaal , Sat Shri Akaal Support GMAT Club by putting a GMAT Club badge on your blog/Facebook Get the best GMAT | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9597620619801096,
"lm_q1q2_score": 0.860188072580374,
"lm_q2_score": 0.8962513800615312,
"openwebmath_perplexity": 2813.2556971947556,
"openwebmath_score": 0.5333693623542786,
"tags": null,
"url": "http://gmatclub.com/forum/inequalities-trick-91482.html?kudos=1"
} |
Shri Akaal Support GMAT Club by putting a GMAT Club badge on your blog/Facebook Get the best GMAT Prep Resources with GMAT Club Premium Membership Gmat test review : 670-to-710-a-long-journey-without-destination-still-happy-141642.html Math Expert Joined: 02 Sep 2009 Posts: 26582 Followers: 3487 Kudos [?]: 26158 [1] , given: 2705 Re: Inequalities trick [#permalink] 11 Nov 2013, 06:51 1 This post received KUDOS Expert's post anujpadia wrote: Can you please explain the above mentioned concept in relation to the following question? Is a > O? (1) a^3 - 0 < 0 (2) 1- a^2 > 0 Can you please explain the scenario when (x-a)(x-B)(x-C)(x-d)>0? Sorry, but finding it difficult to understand. Check alternative solutions here: is-a-0-1-a-3-a-0-2-1-a-86749.html Hope this helps. _________________ Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 4755 Location: Pune, India Followers: 1112 Kudos [?]: 5022 [1] , given: 164 Re: Inequalities trick [#permalink] 23 Apr 2014, 03:43 1 This post received KUDOS Expert's post 1 This post was BOOKMARKED PathFinder007 wrote: I have a query. I have following question x^3 - 4x^5 < 0 I can define this as (1+2x).x^3(1-2x). now I have roots -1/2, 0, 1/2. so in case of >1/2 I will always get inequality value as <0 and in case of -1/2 and 0 I will get value as 0. So How I will define them in graph and what range I will consider for this inequality. Thanks The factors must be of the form (x - a)(x - b) .... etc x^3 - 4x^5 < 0 x^3 * (1 - 4x^2) < 0 x^3 * (1 - 2x) * (1 + 2x) < 0 x^3 * (2x - 1) * (2x + 1) > 0 (Note the sign flip because 1-2x was changed to 2x - 1) x^3 * 2(x - 1/2) *2(x + 1/2) > 0 So transition points are 0, 1/2 and -1/2. ____________ - 1/2 _____ 0 ______1/2 _________ This is what it looks like on the number line. The rightmost region is positive. We want the positive regions in the inequality. So the desired range of x is given by x > 1/2 or -1/2 < x< 0 For more on this method, check these posts: http://www.veritasprep.com/blog/2012/06 | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9597620619801096,
"lm_q1q2_score": 0.860188072580374,
"lm_q2_score": 0.8962513800615312,
"openwebmath_perplexity": 2813.2556971947556,
"openwebmath_score": 0.5333693623542786,
"tags": null,
"url": "http://gmatclub.com/forum/inequalities-trick-91482.html?kudos=1"
} |
or -1/2 < x< 0 For more on this method, check these posts: http://www.veritasprep.com/blog/2012/06 ... e-factors/ http://www.veritasprep.com/blog/2012/07 ... ns-part-i/ http://www.veritasprep.com/blog/2012/07 ... s-part-ii/ The links will give you the theory behind this method in detail. _________________ Karishma Veritas Prep | GMAT Instructor My Blog Save$100 on Veritas Prep GMAT Courses And Admissions Consulting | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9597620619801096,
"lm_q1q2_score": 0.860188072580374,
"lm_q2_score": 0.8962513800615312,
"openwebmath_perplexity": 2813.2556971947556,
"openwebmath_score": 0.5333693623542786,
"tags": null,
"url": "http://gmatclub.com/forum/inequalities-trick-91482.html?kudos=1"
} |
Enroll now. Pay later. Take advantage of Veritas Prep's flexible payment plan options. | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9597620619801096,
"lm_q1q2_score": 0.860188072580374,
"lm_q2_score": 0.8962513800615312,
"openwebmath_perplexity": 2813.2556971947556,
"openwebmath_score": 0.5333693623542786,
"tags": null,
"url": "http://gmatclub.com/forum/inequalities-trick-91482.html?kudos=1"
} |
Veritas Prep Reviews
Intern
Joined: 08 Mar 2010
Posts: 6
Followers: 0
Kudos [?]: 0 [0], given: 4
Re: Inequalities trick [#permalink] 16 Mar 2010, 09:46
Can u plz explainn the backgoround of this & then the explanation.
Thanks
Senior Manager
Joined: 13 Dec 2009
Posts: 264
Followers: 10
Kudos [?]: 106 [0], given: 13
Re: Inequalities trick [#permalink] 19 Mar 2010, 11:48
I have applied this trick and it seemed to be quite useful.
_________________
My debrief: done-and-dusted-730-q49-v40
CEO
Status: Nothing comes easy: neither do I want.
Joined: 12 Oct 2009
Posts: 2793
Location: Malaysia
Concentration: Technology, Entrepreneurship
Schools: ISB '15 (M)
GMAT 1: 670 Q49 V31
GMAT 2: 710 Q50 V35
Followers: 178
Kudos [?]: 945 [0], given: 235
Re: Inequalities trick [#permalink] 19 Mar 2010, 11:59
ttks10 wrote:
Can u plz explainn the backgoround of this & then the explanation.
Thanks
i m sorry i dont have any background for it, you just re-read it again and try to implement whenever you get such question and I will help you out in any issue.
sidhu4u wrote:
I have applied this trick and it seemed to be quite useful.
Nice to hear this....good luck.
_________________
Fight for your dreams :For all those who fear from Verbal- lets give it a fight
Money Saved is the Money Earned
Jo Bole So Nihaal , Sat Shri Akaal
Gmat test review :
670-to-710-a-long-journey-without-destination-still-happy-141642.html
Senior Manager
Status: Upset about the verbal score - SC, CR and RC are going to be my friend
Joined: 30 Jun 2010
Posts: 318
Followers: 5
Kudos [?]: 15 [0], given: 6
Re: Inequalities trick [#permalink] 17 Oct 2010, 15:14
gurpreetsingh wrote:
I learnt this trick while I was in school and yesterday while solving one question I recalled.
Its good if you guys use it 1-2 times to get used to it.
So for f(x) < 0 consider "-" curves and the ans is : (a < x < b) , (c < x < d)
and for f(x) < 0 consider "+" curves and the ans is : (x < a), (b < x < c) , (d < x) | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9597620619801096,
"lm_q1q2_score": 0.860188072580374,
"lm_q2_score": 0.8962513800615312,
"openwebmath_perplexity": 2813.2556971947556,
"openwebmath_score": 0.5333693623542786,
"tags": null,
"url": "http://gmatclub.com/forum/inequalities-trick-91482.html?kudos=1"
} |
I don't understand this part alone. Can you please explain?
_________________
My gmat story
MGMAT1 - 630 Q44V32
MGMAT2 - 650 Q41V38
MGMAT3 - 680 Q44V37
GMATPrep1 - 660 Q49V31
Knewton1 - 550 Q40V27
CEO
Status: Nothing comes easy: neither do I want.
Joined: 12 Oct 2009
Posts: 2793
Location: Malaysia
Concentration: Technology, Entrepreneurship
Schools: ISB '15 (M)
GMAT 1: 670 Q49 V31
GMAT 2: 710 Q50 V35
Followers: 178
Kudos [?]: 945 [0], given: 235
Re: Inequalities trick [#permalink] 17 Oct 2010, 16:19
Dreamy wrote:
gurpreetsingh wrote:
I learnt this trick while I was in school and yesterday while solving one question I recalled.
Its good if you guys use it 1-2 times to get used to it.
So for f(x) < 0 consider "-" curves and the ans is : (a < x < b) , (c < x < d)
and for f(x) < 0 consider "+" curves and the ans is : (x < a), (b < x < c) , (d < x)
I don't understand this part alone. Can you please explain?
Suppose you have the inequality
f(x) = (x-a)(x-b)(x-c)(x-d) < 0 you will consider the curve with -ve inside it.. check the attached image.
f(x) = (x-a)(x-b)(x-c)(x-d) > 0 consider the +ve of the curve
_________________
Fight for your dreams :For all those who fear from Verbal- lets give it a fight
Money Saved is the Money Earned
Jo Bole So Nihaal , Sat Shri Akaal
Gmat test review :
670-to-710-a-long-journey-without-destination-still-happy-141642.html
Re: Inequalities trick [#permalink] 17 Oct 2010, 16:19
Similar topics Replies Last post
Similar
Topics:
84 Tips and Tricks: Inequalities 26 14 Apr 2013, 08:20
2 To use tricks or not to use tricks..... 3 20 Apr 2008, 15:57
inequalities 1 06 Aug 2006, 07:23
inequalities 3 05 Dec 2005, 11:18
inequalities 2 15 Jun 2005, 06:47
Display posts from previous: Sort by | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9597620619801096,
"lm_q1q2_score": 0.860188072580374,
"lm_q2_score": 0.8962513800615312,
"openwebmath_perplexity": 2813.2556971947556,
"openwebmath_score": 0.5333693623542786,
"tags": null,
"url": "http://gmatclub.com/forum/inequalities-trick-91482.html?kudos=1"
} |
GMAT Question of the Day: Daily via email | Daily via Instagram New to GMAT Club? Watch this Video
It is currently 02 Apr 2020, 02:59
### GMAT Club Daily Prep
#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
Customized
for You
we will pick new questions that match your level based on your Timer History
Track
every week, we’ll send you an estimated GMAT score based on your performance
Practice
Pays
we will pick new questions that match your level based on your Timer History
# What arithmetic should I memorize?
Author Message
TAGS:
### Hide Tags
Manager
Joined: 25 Mar 2009
Posts: 245
What arithmetic should I memorize? [#permalink]
### Show Tags
Updated on: 19 Oct 2018, 13:30
38
1
347
Improve Your Speed - GMAT Arithmetic to Memorize
I find myself constantly using long division and multiplication for simple things like 13 x 11.
What are some good arithmetic calcs to memorize? I created a 20x20 multiplication table but this seems like a bit much right?
Perfect squares up to 100? $$\sqrt{3}$$? $$\sqrt{5}$$?
I am not really talking about formulas to memorize since you should definitely memorize things like nCr, nPr, sum of all #'s in an evenly spaced set, etc.
Edit: Moderator note: Attached are files we gathered in this post from several other sources in addition to the author. Also see additional resources in posts below to help with improve your speed and reduce mistakes
Target Test Prep 15-page PDF summarizes ALL formulas you have to know
-
Attachments
GMAT Math Compendium.xls [855 KiB]
Geometry Formulas_template.pdf [44.28 KiB]
Geometry Formulas.pdf [72.21 KiB]
Arithmetic Fractions and Percentages.doc [32.5 KiB] | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9353465080392795,
"lm_q1q2_score": 0.8601844353315612,
"lm_q2_score": 0.9196425366837827,
"openwebmath_perplexity": 4477.274945892129,
"openwebmath_score": 0.7776781320571899,
"tags": null,
"url": "https://gmatclub.com/forum/what-arithmetic-should-i-memorize-80128.html"
} |
Geometry Formulas.pdf [72.21 KiB]
Arithmetic Fractions and Percentages.doc [32.5 KiB]
Originally posted by topher on 25 Jun 2009, 20:39.
Last edited by bb on 19 Oct 2018, 13:30, edited 7 times in total.
Edit
Founder
Joined: 04 Dec 2002
Posts: 19430
Location: United States (WA)
GMAT 1: 750 Q49 V42
GPA: 3.5
Re: What arithmetic should I memorize? [#permalink]
### Show Tags
25 Jun 2009, 21:18
83
87
Fantastic question
This file should give you an idea where you are lacking.
I think you should definitely know the squares from 0-10 and preferrably from 10 to 20 as well.
Attachments
Arithmetic Fractions and Percentages.doc [32.5 KiB]
_________________
Intern
Joined: 12 May 2009
Posts: 48
Re: What arithmetic should I memorize? [#permalink]
### Show Tags
21 Oct 2009, 23:59
81
1
63
As far as common squares are concerned, I only remember the ones with 0 or 5 in the units digit. For the latter category, I use the following process:
For example: 65*65
1) Always write down 25 as this is always the last two digits of the result:
...25
2) Multiply (non-units digits) times (non-units digits + 1)
6 * (6+1) = 42
2c) Combine:
4225
This way I always have the important squares handy... very useful for estimations!
Any other math shortcuts? Anyone
Steve
##### General Discussion
Current Student
Joined: 13 Jul 2009
Posts: 133
Location: Barcelona
Schools: SSE
Re: What arithmetic should I memorize? [#permalink]
### Show Tags
26 Jul 2009, 09:36
84
29
Hi all,
I created a template that gather all the geometry formulas that appear on the GMAT (if i missed any please tell!). I use it every week in order to ensure that I remember all of them.
I think it could be helpful for some of you so I decided to share.
The TEMPLATE is the one for practice.
Good luck!
Attachments
Geometry Formulas_template.pdf [44.28 KiB]
Geometry Formulas.pdf [72.21 KiB] | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9353465080392795,
"lm_q1q2_score": 0.8601844353315612,
"lm_q2_score": 0.9196425366837827,
"openwebmath_perplexity": 4477.274945892129,
"openwebmath_score": 0.7776781320571899,
"tags": null,
"url": "https://gmatclub.com/forum/what-arithmetic-should-i-memorize-80128.html"
} |
Geometry Formulas_template.pdf [44.28 KiB]
Geometry Formulas.pdf [72.21 KiB]
_________________
Founder
Joined: 04 Dec 2002
Posts: 19430
Location: United States (WA)
GMAT 1: 750 Q49 V42
GPA: 3.5
Re: What arithmetic should I memorize? [#permalink]
### Show Tags
26 Jul 2009, 19:33
3
3
Great resource
Thank you!
_________________
Manager
Joined: 22 Jul 2009
Posts: 147
Re: What arithmetic should I memorize? [#permalink]
### Show Tags
Updated on: 26 Jan 2010, 12:23
29
8
My fellow GMAT clubbers,
I'd like to share the excel spreadsheet in which I have been compiling all knowledge and shortcuts relevant for tackling the GMAT quant section.
It is organized by tabs. Each tab covers a different area.
Examples:
- Tab "NP. Primes" covers: Number Properties - Primes
- Tab "NP. Powers-Rts" covers: Number Properties - Patterns of powers and roots
- Tab "G. PTriples" covers: Geometry - Phytagorean triplets patterns
- Tab "G. Ci-Sq (Pi)" covers: Geometry - Relantionships between the measures of inscribed/circumscribed circles & squares
- Tab "WT. Prob" covers: Wort Translations - Probability (dice/coins)
Finally, three of the tabs are summaries made of the Man guides "Word Translations", "Number Properties" and "Geometry".
Hope it helps somebody.
Cheers,
Attachments
GMAT Math Compendium.xls [855 KiB]
Originally posted by powerka on 08 Sep 2009, 16:51.
Last edited by powerka on 26 Jan 2010, 12:23, edited 3 times in total.
Intern
Joined: 24 Oct 2009
Posts: 16
Location: Russia
Schools: IESE, SDA Bocconi
Re: What arithmetic should I memorize? [#permalink]
### Show Tags
31 Oct 2009, 08:56
71
61
I'm new here, and try to go through all the topics - this site is a treasure! | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9353465080392795,
"lm_q1q2_score": 0.8601844353315612,
"lm_q2_score": 0.9196425366837827,
"openwebmath_perplexity": 4477.274945892129,
"openwebmath_score": 0.7776781320571899,
"tags": null,
"url": "https://gmatclub.com/forum/what-arithmetic-should-i-memorize-80128.html"
} |
Maybe not the right place to share my experience with "multiplication for simple things like 13 x 11", but anyway...if you need to multiply any two-digits number by 11, just sum those digits and put the result in between. For example,
13x11 -> 1+3=4 -> 143 is the result.
Or, 36x11 -> 3+6=9 -> the result is 36x11=396.
It really saves time.
Intern
Joined: 31 Oct 2009
Posts: 22
Re: What arithmetic should I memorize? [#permalink]
### Show Tags
31 Oct 2009, 19:21
31
32
Shelen wrote:
I'm new here, and try to go through all the topics - this site is a treasure!
Maybe not the right place to share my experience with "multiplication for simple things like 13 x 11", but anyway...if you need to multiply any two-digits number by 11, just sum those digits and put the result in between. For example,
13x11 -> 1+3=4 -> 143 is the result.
Or, 36x11 -> 3+6=9 -> the result is 36x11=396.
It really saves time.
And if it equals more than 10, add a 1 to the first digit
EG
68*11 -> 6+8=14 -> 6 14 8 -> 748
Intern
Joined: 15 Nov 2009
Posts: 27
Location: Moscow, Russia
Re: What arithmetic should I memorize? [#permalink]
### Show Tags
07 Jan 2010, 08:47
1
Sorry, but on page 2 there is a misprint:
in Arc Measure (Intersecting Secants/Tangents) It must be the difference between arcs measures of AD and BD.
Founder
Joined: 04 Dec 2002
Posts: 19430
Location: United States (WA)
GMAT 1: 750 Q49 V42
GPA: 3.5
Re: What arithmetic should I memorize? [#permalink]
### Show Tags
07 Jan 2010, 16:13
1
6
Also, see this post for some good theory: math-number-theory-88376.html
_________________
Manager
Status: Not afraid of failures, disappointments, and falls.
Joined: 20 Jan 2010
Posts: 243
Concentration: Technology, Entrepreneurship
WE: Operations (Telecommunications)
Re: What arithmetic should I memorize? [#permalink]
### Show Tags
11 Feb 2010, 09:34
48
37
Thanks for sharing the Doc bb...@Shelen & @jeckll...Thanks for the tip but I would like to add some more. | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9353465080392795,
"lm_q1q2_score": 0.8601844353315612,
"lm_q2_score": 0.9196425366837827,
"openwebmath_perplexity": 4477.274945892129,
"openwebmath_score": 0.7776781320571899,
"tags": null,
"url": "https://gmatclub.com/forum/what-arithmetic-should-i-memorize-80128.html"
} |
What if u need to multiply 3 digits or 4 didgits by 11...the procedure is same as u mentioned but it would be done like below
For 3 digits
133x11 --> 1 1+3 3+3 3=1463
And for 4 digits
1243x11 --> 1 1+2 2+4 4+3 3=13673
And for 5 digits
15453x11 --> 1 1+5 5+4 4+5 5+3 3=169983
and so on.
you can plug other numbers in and check it out.
Hope it helps!
Manager
Status: Not afraid of failures, disappointments, and falls.
Joined: 20 Jan 2010
Posts: 243
Concentration: Technology, Entrepreneurship
WE: Operations (Telecommunications)
Re: What arithmetic should I memorize? [#permalink]
### Show Tags
11 Feb 2010, 21:41
15
26
Guys! I just found another way of checking whether a number is divisible by 8 or not ( the rule is same but another approach or route ). It's for a number with more than two digits.
Let's take 1936
1) First of all check whether last two digits of the number are divisible by 4 or not.
For 1936, we do this way 36/4=9
2) If it is divisible by 4 then add the quotient to the 3rd last digit of the number and if the sum of them is divisible by 2 then the whole number is divisible by 8.
--> 9 (quotient)+ 9 ( 3rd digit from right)= 18, and -->18/2=9
So the whole number is divisible by 8.
Once you understand it and do a little practice, you'll find it easy and fast.
**You can try other numbers to see whether it is true or not
Hope it helps!
Intern
Affiliations: isa.org
Joined: 26 Jan 2010
Posts: 16
Location: Mumbai, India.
Schools: ISB, India.
Re: What arithmetic should I memorize? [#permalink]
### Show Tags
14 Feb 2010, 05:12
3
2
AtifS wrote:
Thanks for sharing the Doc bb...@Shelen & @jeckll...Thanks for the tip but I would like to add some more.
What if u need to multiply 3 digits or 4 didgits by 11...the procedure is same as u mentioned but it would be done like below
For 3 digits
133x11 --> 1 1+3 3+3 3=1463
And for 4 digits
1243x11 --> 1 1+2 2+4 4+3 3=13673 | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9353465080392795,
"lm_q1q2_score": 0.8601844353315612,
"lm_q2_score": 0.9196425366837827,
"openwebmath_perplexity": 4477.274945892129,
"openwebmath_score": 0.7776781320571899,
"tags": null,
"url": "https://gmatclub.com/forum/what-arithmetic-should-i-memorize-80128.html"
} |
For 3 digits
133x11 --> 1 1+3 3+3 3=1463
And for 4 digits
1243x11 --> 1 1+2 2+4 4+3 3=13673
And for 5 digits
15453x11 --> 1 1+5 5+4 4+5 5+3 3=169983
and so on.
you can plug other numbers in and check it out.
Hope it helps!
welldone dude,
just a small addition which i tried and will help to avoid confusion:
start writing the answer from right hand side in case if the addition of two no. exceeds 10, and add it to consecutive no. on left hand side. try out!!
Intern
Affiliations: isa.org
Joined: 26 Jan 2010
Posts: 16
Location: Mumbai, India.
Schools: ISB, India.
Re: What arithmetic should I memorize? [#permalink]
### Show Tags
14 Feb 2010, 05:21
2
5
AtifS wrote:
Guys! I just found another way of checking whether a number is divisible by 8 or not ( the rule is same but another approach or route ). It's for a number with more than two digits.
Let's take 1936
1) First of all check whether last two digits of the number are divisible by 4 or not.
For 1936, we do this way 36/4=9
2) If it is divisible by 4 then add the quotient to the 3rd last digit of the number and if the sum of them is divisible by 2 then the whole number is divisible by 8.
--> 9 (quotient)+ 9 ( 3rd digit from right)= 18, and -->18/2=9
So the whole number is divisible by 8.
Once you understand it and do a little practice, you'll find it easy and fast.
**You can try other numbers to see whether it is true or not
Hope it helps!
hi,
i feel the process is bit complicated as it doesn't give the value of quotient, it just tells you whether no. is divisible by 8.
here is another tric, if the no. formed by last three digits is divisible by 8 then the whole no. is divisible by 8.
953360 is divisible by 8 since 360 is divisible by 8,
529418: not divisible as 418 is not divisible by 8.
plug in different values and try.
Intern
Joined: 17 Feb 2010
Posts: 1
Re: What arithmetic should I memorize? [#permalink]
### Show Tags | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9353465080392795,
"lm_q1q2_score": 0.8601844353315612,
"lm_q2_score": 0.9196425366837827,
"openwebmath_perplexity": 4477.274945892129,
"openwebmath_score": 0.7776781320571899,
"tags": null,
"url": "https://gmatclub.com/forum/what-arithmetic-should-i-memorize-80128.html"
} |
### Show Tags
17 Feb 2010, 18:53
6
2
RE:
What if u need to multiply 3 digits or 4 didgits by 11...the procedure is same as u mentioned but it would be done like below
For 3 digits
133x11 --> 1 1+3 3+3 3=1463
And for 4 digits
1243x11 --> 1 1+2 2+4 4+3 3=13673
And for 5 digits
15453x11 --> 1 1+5 5+4 4+5 5+3 3=169983
and so on.
Hi guys,
When multiplying by 11 I find it much easier to multiply the # by 10 (or add 0) and then add the original # to it.
For example, 1234x11 = 12340 + 1234
When dividing by 11 and the following condition is satisfied one could factor the # as follows:
671/11 = 61x11/11 = 61 because in 671, 6+1=7 which is the # in the middle of 671. This works for 3digit #'s
Manager
Status: Labor Omnia Vincit
Joined: 16 Aug 2010
Posts: 71
Schools: S3 Asia MBA (Fudan University, Korea University, National University of Singapore)
WE 1: Market Research
WE 2: Consulting
WE 3: Iron & Steel Retail/Trading
Re: What arithmetic should I memorize? [#permalink]
### Show Tags
Updated on: 05 Oct 2010, 10:03
1
Now, what does this mean:
Area of Isosceles Triangle = 1/2[(leg)^2]
(from Geometry Formulas.pdf)
I haven't been able to give myself any convincing explanation of this formula...
Originally posted by rishabhsingla on 11 Sep 2010, 10:27.
Last edited by rishabhsingla on 05 Oct 2010, 10:03, edited 1 time in total.
Math Expert
Joined: 02 Sep 2009
Posts: 62422
Re: What arithmetic should I memorize? [#permalink]
### Show Tags
11 Sep 2010, 10:42
1
1
rishabhsingla wrote:
Now, what does this mean:
Area of Isosceles Triangle = 1/2[(leg)^2]
(from Geometry Formulas.pdf)
I haven't been able to give myself any convincing explanation of this formula...
Well that's because it's not true. | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9353465080392795,
"lm_q1q2_score": 0.8601844353315612,
"lm_q2_score": 0.9196425366837827,
"openwebmath_perplexity": 4477.274945892129,
"openwebmath_score": 0.7776781320571899,
"tags": null,
"url": "https://gmatclub.com/forum/what-arithmetic-should-i-memorize-80128.html"
} |
Well that's because it's not true.
It should be area of isosceles right triangle = 1/2*(leg)^2, simply because area of right triangle equals to 1/2*leg1*leg2, and since in isosceles right triangle leg1=leg2, then this formula becomes area=1/2*(leg)^2.
_________________
Intern
Status: "
Joined: 22 Jan 2011
Posts: 37
Schools: IE- ESCP - Warwick
Re: What arithmetic should I memorize? [#permalink]
### Show Tags
20 Apr 2011, 18:47
Thanks for this memory sheet, very useful!
However I believe some formulas, details might be added...
Angles of a polygons (n-2)* 180, angles inscribed in a circle (compared to central angle there a section about it in the VERITAS Prep book)
Also I did not see two real IMPORTANT triangles sides ratio:
3:4:5
5:12:13
Good you put the formula for the equilateral triangle (which is not given in the OG and allow you to gain so much time)
I found formula I did not know! Thank you for sharing
Manager
Status: 700 (q47,v40); AWA 6.0
Joined: 16 Mar 2011
Posts: 78
GMAT 1: 700 Q47 V40
Re: What arithmetic should I memorize? [#permalink]
### Show Tags
23 Apr 2011, 12:59
1
I'd also add one more element to Coordinate geometry. The coordinates of a point P on line joining two points A(x1,y1) and B(x2,y2) such that PA:PB is in the ration m:n would be ((mx2+nx1)/(m+n),(my2+ny1)/(m+n)).
Intern
Joined: 10 Jul 2011
Posts: 29
Re: What arithmetic should I memorize? [#permalink]
### Show Tags
04 Jan 2012, 15:59
5
13
These are a few math tricks my brother used when studying for the MCAT. Hope they help!
(attached a word document)
1. A nice math trick is multiplying two integers that have multiple digits relatively quickly. It does not apply to all integers the following has to be met:
TENS DIGIT in both integers HAS TO BE SAME
ONES DIGIT in each integer HAS TO ADD UP TO 10
Also, if the ones digits are 1 and 9 you just write 09. | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9353465080392795,
"lm_q1q2_score": 0.8601844353315612,
"lm_q2_score": 0.9196425366837827,
"openwebmath_perplexity": 4477.274945892129,
"openwebmath_score": 0.7776781320571899,
"tags": null,
"url": "https://gmatclub.com/forum/what-arithmetic-should-i-memorize-80128.html"
} |
Example 34x36:
Step 1: Add one to tens place then multiply (1+3)x3 = 12
Step 2: Ones place 4x6 = 24
Step 3: Place Steps 1 & 2 = 1224
Example 1 & 9 in ones place: 21x29
Step 1: Add one to tens place then multiply (1+2)x2 = 6
Step 2: Ones place 1x9= 09
Step 3: Place Steps 1 & 2 together = 609
2. This one is for people having difficulty memorizing a few square root numbers!
sqrt 1 = 1 (as in 1/1 = New Year's Day)
sqrt 2 = 1.4 (as in 2/14 = Valentine's Day)
sqrt 3 = 1.7 (as in 3/17 = St. Patrick's Day)
3. The next math trick is the Babylonian Method it can be useful when estimating square roots.
First guess roughly what you think it would be
Step 1: For a number less than 1 guess bigger.
For a number greater than 1 guess smaller.
Step 2: Divide your guess into the square root number.
Step 4: Divide by 2
Example sqrt(.78):
Step 1: sqrt of .78 < 1 so guess = .85
Step 2. .78/.85 = ~.9
Step 3: (.85+.9) = 1.75
Step 4: 1.75/2 = ~.88 And this should be your answer (or close enough)
If you guess really wildly just use the answer from your first guess and run through the process again. You can do it in seconds once you get good at it.
Wild Guess Example: sqrt 70 = ?
Step 1 sqrt of 70>1 so guess 10
Step 2: 70/10= 7
Step 3: 7+10= 17
Step 4: 17/2 = 8.5
*8.5x8.5 = 72.25 Still off (10 kind of a wild guess, so repeat process with the new answer from step 4)
Step 1: guess = 8.5
Step 2: 70/8.5= 8.2
Step 3: 8.2+8.5= 16.7
Step 4: 16.7/2 = ~8.4
8.4*8.4 = 70.6
4. If two numbers (both even or odd) are close together and their average is an integer, then this method can be used.
Need to recognize that:
x^2 - y^2 = (x + y)(x - y) and vice-versa (x + y)(x - y) = x^2 - y^2
Example 1
48*52 = (50-2)(50+2) = 50^2 - 2^2 = 2496.
Example 2
3^2 - 2^2 = 3 + 2
4^2 - 3^2 = 4 + 3
5^2 - 4^2 = 5 + 4
5. When x and y are consecutive integers, then (x - y) = 1.
It's useful for calculating large squares:
Example: 71^2 = ?
71^2 - 70^2 = 71 + 70
so 71^2 = 70^2 + (141)
= 4900 + 141 = 5041 | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9353465080392795,
"lm_q1q2_score": 0.8601844353315612,
"lm_q2_score": 0.9196425366837827,
"openwebmath_perplexity": 4477.274945892129,
"openwebmath_score": 0.7776781320571899,
"tags": null,
"url": "https://gmatclub.com/forum/what-arithmetic-should-i-memorize-80128.html"
} |
Example: 71^2 = ?
71^2 - 70^2 = 71 + 70
so 71^2 = 70^2 + (141)
= 4900 + 141 = 5041
Example: 53^2 = ?
53^2 – 52^2 = (53 + 52) + (52 + 51) + (51 + 50)
53^2 = 50^2 + (105) + (103) + (101)
= 2500 + 309 = 2809
0.79^2
80^2 - 79^2 = 80 + 79
so 80^2 - (80 + 79) = 79^2
6400 - 159 = 6241 so 0.79^2 = 0.6241
Attachments
math tricks.docx [15.68 KiB]
Re: What arithmetic should I memorize? [#permalink] 04 Jan 2012, 15:59
Go to page 1 2 Next [ 29 posts ]
Display posts from previous: Sort by | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9353465080392795,
"lm_q1q2_score": 0.8601844353315612,
"lm_q2_score": 0.9196425366837827,
"openwebmath_perplexity": 4477.274945892129,
"openwebmath_score": 0.7776781320571899,
"tags": null,
"url": "https://gmatclub.com/forum/what-arithmetic-should-i-memorize-80128.html"
} |
# Inversion Mapping is Automorphism iff Group is Abelian
## Theorem
Let $\struct {G, \circ}$ be a group.
Let $\iota: G \to G$ be the inversion mapping on $G$, defined as:
$\forall g \in G: \map \iota g = g^{-1}$
Then $\iota$ is an automorphism if and only if $G$ is abelian.
## Proof
From Inversion Mapping is Permutation, $\iota$ is a permutation.
It remains to be shown that $\iota$ has the morphism property if and only if $G$ is abelian.
### Sufficient Condition
Suppose $\iota$ is an automorphism.
Then:
$\displaystyle \forall x, y \in G: \map \iota {x \circ y}$ $=$ $\displaystyle \map \iota x \circ \map \iota y$ Definition of Morphism Property $\displaystyle \leadsto \ \$ $\displaystyle \paren {x \circ y}^{-1}$ $=$ $\displaystyle x^{-1} \circ y^{-1}$ Definition of $\iota$
Thus from Inverse of Commuting Pair, $x$ commutes with $y$.
This holds for all $x, y \in G$.
So $\struct {G, \circ}$ is abelian by definition.
$\Box$
### Necessary Condition
Let $\struct {G, \circ}$ be abelian.
$\displaystyle \forall x, y \in G: \paren {x \circ y}^{-1}$ $=$ $\displaystyle x^{-1} \circ y^{-1}$ Inverse of Commuting Pair $\displaystyle \leadsto \ \$ $\displaystyle \map \iota {x \circ y}$ $=$ $\displaystyle \map \iota x \circ \map \iota y$ Definition of $\iota$
Thus $\iota$ has the morphism property and is therefore an automorphism.
$\blacksquare$
## Sources
except this source requests only that the morphism property is demonstrated, and not the bijectivity.
except this source proves only the necessary condition. | {
"domain": "proofwiki.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9840936110872271,
"lm_q1q2_score": 0.8601738340207171,
"lm_q2_score": 0.8740772466456688,
"openwebmath_perplexity": 501.28665776362715,
"openwebmath_score": 0.9947195053100586,
"tags": null,
"url": "https://proofwiki.org/wiki/Inversion_Mapping_is_Automorphism_iff_Group_is_Abelian"
} |
The number of permutations, permutations, of seating these five people in five chairs is five factorial. Suppose we are given a total of n distinct objects and want to select r of them. Counting Problem - discrete math. So, in Mathematics we use more precise language: When the order doesn't matter, it is a Combination. Example. Hence, the total number of permutation is $6 \times 6 = 36$ Combinations. way to permute the numbers. The formulas for each are very similar, there is just an extra $$k!$$ in the denominator of $${n \choose k}\text{. Permutations and Combinations 00:37. The number of all combinations of n things, taken r at a time is − This selection of subsets is called a permutation when the order of selection is a factor, a combination when order is not a factor. Discrete Mathematics Permutations and Combinations? Active today. This is particularly true for some probability problems. There is a group of 15 women and 10 men. Combinations and Permutations. How many ways are there to form the committee if there must be more women than men? There are \binom{9}{4} ways. I'm stuggling to get my head around this question. }$$ That extra $$k!$$ accounts for the fact that $${n \choose k}$$ does not distinguish between the different orders that the $$k$$ objects can appear in. In the Match of the Day’s goal of the month competition, you had to pick the top 3 goals out of 10. Our 1000+ Discrete Mathematics questions and answers focuses on all areas of Discrete Mathematics subject covering 100+ topics in Discrete Mathematics. This number of permutations is huge. See more ideas about discrete mathematics, mathematics, permutations and combinations. Each computer must complete its own tasks in order. We say $$P(n,k)$$ counts permutations, and $${n \choose k}$$ counts combinations. Math 3336 Section 6. with full confidence. If there are twenty-five players on the team, there are $$25 \cdot 24 \cdot 23 \cdot \cdots \cdot 3 \cdot 2 \cdot 1$$ different permutations of the | {
"domain": "neuropsychologycentral.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9840936115537341,
"lm_q1q2_score": 0.8601738328144543,
"lm_q2_score": 0.8740772450055545,
"openwebmath_perplexity": 828.5639289170634,
"openwebmath_score": 0.5296341180801392,
"tags": null,
"url": "http://neuropsychologycentral.com/k0jdv3cp/discrete-math-combinations-and-permutations-a63cd2"
} |
are $$25 \cdot 24 \cdot 23 \cdot \cdots \cdot 3 \cdot 2 \cdot 1$$ different permutations of the players. In how many ways could this have happened so that there were no … This touches directly on an area of mathematics known as … For each of these 120 permutations, there are 4 pairs of adjacent numbers. Combinations and Permutations. Donate Login Sign up. Ask Question Asked today. Why Aptitude Permutation and Combination? Sep 14, 2008 #1 I was having a problem with some questions. another 6! Now we do care about the order. This is different from permutation where the order matters. Permutation and Combinations: Permutation: Any arrangement of a set of n objects in a given order is called Permutation of Object. Let, X be a non-empty set. Section 2.4 Combinations and the Binomial Theorem Subsection 2.4.1 Combinations. In this section you can learn and practice Aptitude Questions based on "Permutation and Combination" and improve your skills in order to face the interview, competitive examination and various entrance test (CAT, GATE, GRE, MAT, Bank Exam, Railway Exam etc.) Permutations are specific selections of elements within a set where the order in which the elements are arranged is important, while combinations involve the selection of elements without regard for order. \end{equation*} We know that we have them all listed above —there are 3 choices for which letter we put first, then 2 choices for which letter comes next, which leaves only 1 choice for the last letter. Related Pages Permutations Permutations and Combinations Counting Methods Factorial Lessons Probability. For example, suppose we are arranging the letters A, B and C. (Denoted by n P r or n r or P (n, r)) : Let us consider the problem of finding the number of ways in which the first r rankings are secured by n students in a class. I understand how to calculate normal permutations and combinations but the fact that there are 4 unique types, each with different quantities is confusing. After, each | {
"domain": "neuropsychologycentral.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9840936115537341,
"lm_q1q2_score": 0.8601738328144543,
"lm_q2_score": 0.8740772450055545,
"openwebmath_perplexity": 828.5639289170634,
"openwebmath_score": 0.5296341180801392,
"tags": null,
"url": "http://neuropsychologycentral.com/k0jdv3cp/discrete-math-combinations-and-permutations-a63cd2"
} |
but the fact that there are 4 unique types, each with different quantities is confusing. After, each computer sends its output to a shared fourth computer. Permutations and Combinations involve counting the number of different selections possible from a set of objects given certain restrictions and conditions. The information that determines the ordering is called the key. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. Ask Question Asked 3 years, 9 months ago. Jan 20, 2018 - Explore deepak mahajan's board "combination" on Pinterest. Discrete Mathematics (c)Marcin Sydow Productand SumRule Inclusion-Exclusion Principle Pigeonhole Principle Permutations Generalised Permutations andCombi-nations Combinatorial Proof Binomial Coefficients Countingthenumberoffunctions Thesetofallfunctionsf : X !Y isdenotedasYX The numberofdifferentfunctionsf : X !Y isgivenbythe expression jYX = jXj. Permutations and combinations, the various ways in which objects from a set may be selected, generally without replacement, to form subsets. "724" won't work, nor will "247". A combination is selection of some given elements in which order does not matter. An arrangement of objects in which the order is not important is called a combination. Search. Aymara G. Numerade Educator 05:14. Of the statements in these questions, 17 are true. A permutation is a (possible) rearrangement of objects. In this article, we will learn about the Introduction permutation group, and the types of permutation in discrete mathematics. It has to be exactly 4-7-2. It's not clear to me if I am using permutation or combination for this question. That is for each permutation chances that 4 and 5 are adjacent are 4/10, hence the result becomes $5!*(1-4/10)$. Problem 1 List all the permutations of {a, b, c}. Submitted by Prerana Jain, on August 17, 2018 Permutation Group. View Permutations, Combinations and Discrete Probability.pdf from MATH 307 at | {
"domain": "neuropsychologycentral.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9840936115537341,
"lm_q1q2_score": 0.8601738328144543,
"lm_q2_score": 0.8740772450055545,
"openwebmath_perplexity": 828.5639289170634,
"openwebmath_score": 0.5296341180801392,
"tags": null,
"url": "http://neuropsychologycentral.com/k0jdv3cp/discrete-math-combinations-and-permutations-a63cd2"
} |
Permutation Group. View Permutations, Combinations and Discrete Probability.pdf from MATH 307 at Massachusetts Institute of Technology. Permutations and Combinations with overcounting. We have already covered this in a previous video. 3) There are 5! In Section 2.1 we investigated the most basic concept in combinatorics, namely, the rule of products. 1. Clarissa N. ... A professor writes 40 discrete mathematics true/false questions. Permutations and combinations are part of a branch of mathematics called combinatorics, which involves studying finite, discrete structures. It is of paramount importance to keep this fundamental rule in mind. I have come up with an answer of my own (10,155,600), however I wanted to check with the community and see if I have come up with the correct answer. Throughout mathematics and statistics, we need to know how to count. (n – r)! permutations with vowels next to each other. - 2*6! … "The combination to the safe is 472". One should spend 1 hour daily for 2-3 months to learn and assimilate Discrete Mathematics comprehensively. so u can hv 6! It is denoted by P (n, r) P (n, r) = Five factorial, which is equal to five times four times three times two times one, which, of course, is equal to, let's see, 20 times six, which is equal to 120. Courses. We say $$P(n,k)$$ counts permutations, and $${n \choose k}$$ counts combinations. When we select the data or objects from a certain group, it is said to be permutations, whereas the order in which they are represented is called combination. What Is Combination In Math? = 6$ways. A permutation of X is a one-one function from X onto X. Permutation and combination are the ways to represent a group of objects by selecting them in a set and forming subsets. permutations for EA glued together. The number of ordered arrangements of r objects taken from n unlike objects is: n P r = n! Combinations with Repetition HARD example. Discrete Math - Permutation, Combination. Thread starter BACONATOR; Start | {
"domain": "neuropsychologycentral.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9840936115537341,
"lm_q1q2_score": 0.8601738328144543,
"lm_q2_score": 0.8740772450055545,
"openwebmath_perplexity": 828.5639289170634,
"openwebmath_score": 0.5296341180801392,
"tags": null,
"url": "http://neuropsychologycentral.com/k0jdv3cp/discrete-math-combinations-and-permutations-a63cd2"
} |
Repetition HARD example. Discrete Math - Permutation, Combination. Thread starter BACONATOR; Start date Sep 14, 2008; Tags combinations permutations; Home. = 3600. total unrestricted permutations = 7! Since the order is important, it is the permutation formula which we use. ANS: 3600 Permutations; Combinations; Combinatorial Proofs; Permutations. permutations . 4) Firstly choose 4 numbers. He has in stock 3 identical amethysts, 4 identical diamonds, 5 identical emeralds and 7 identical rubies. There are$\binom{5}{2}=10$types of such pairs. discrete-mathematics permutations combinations. We now look to distinguish between permutations and combinations. Search for courses, skills, and videos. CS311H: Discrete Mathematics Permutations and Combinations Instructor: Is l Dillig Instructor: Is l Dillig, CS311H: Discrete Mathematics Permutations and Combinations 1/26 Permutations I Apermutationof a set of distinct objects is anordered arrangement of these objects I No object can be selected more than once I Order of arrangement matters I Example: S = fa;b;cg. Between them a committee of 6 people is chosen. permutation and combinations discrete mathematics 0 In how many ways can a board of five people that includes the male Head of School be formed if it must include at least two men and two women? thus 2*6! For example, there are 6 permutations of the letters a, b, c: \begin{equation*} abc, ~~ acb, ~~ bac, ~~bca, ~~ cab, ~~ cba. Main content. … }\) That extra $$k!$$ accounts for the fact that $${n \choose k}$$ does not distinguish between the different orders that the $$k$$ objects can appear in. Any arrangement of any r ≤ n of these objects in a given order is called an r-permutation or a permutation of n object taken r at a time. [Discrete Mathematics] Permutation Practice - Duration: 14:41. The formulas for each are very similar, there is just an extra $$k!$$ in the denominator of \({n \choose k}\text{. A penny is tossed 60 times yielding 45 heads and 15 tails. An | {
"domain": "neuropsychologycentral.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9840936115537341,
"lm_q1q2_score": 0.8601738328144543,
"lm_q2_score": 0.8740772450055545,
"openwebmath_perplexity": 828.5639289170634,
"openwebmath_score": 0.5296341180801392,
"tags": null,
"url": "http://neuropsychologycentral.com/k0jdv3cp/discrete-math-combinations-and-permutations-a63cd2"
} |
denominator of \({n \choose k}\text{. A penny is tossed 60 times yielding 45 heads and 15 tails. An ordered arrangement of r elements of a set is called an r- permutations. This unit covers methods for counting how many possible outcomes there are in various situations. Definition: A permutation of a set of distinct objects is an ordered arrangement of these objects. Feb 2008 13 0. Permutations. University Math Help. Viewed 2 times 0$\begingroup$I need help with the following problem: There are three computers A, B, and C. Computer A has 10 tasks, Computer B has 15 tasks, and Computer C has 20 tasks. . When the order does matter it is a Permutation. It defines the various ways to arrange a certain group of data. These topics are chosen from a collection of most authoritative and best reference books on Discrete Mathematics. We'll learn about factorial, permutations, and combinations. Viewed 240 times 0$\begingroup$So for this question, would it be 2C1 or 2P1 multiplied by 6P3 or 6C3 and why? This topic is an introduction to counting methods used in Discrete Mathematics. Solution to this Discrete Math practice problem is … How many ways are there to assign scores to the problems if the sum of the scores is 100 and each questions is worth at least 5 points? thus the required permutations = 7! Thank you for your help! We would expect that each key would give a different permutation of the names. 14 Solutions Manual of Elements of Discrete Mathematics CHAPTER TWO PERMUTATIONS, Forums. The remaining 3 vacant places will be filled up by 3 vowels in$^3P_{3} = 3! TheTrevTutor 87,951 views. There are 10 questions on a discrete mathematics final exam. In Section 2.2 we saw a subclass of rule-of-products problems, permutations, and we derived a formula as a computational aid to assist us. If the questions can be positioned in any order, how many different answer keys are possible? A permutation is an ordered arrangement. Discrete Math B. BACONATOR. Active 3 years, 9 months | {
"domain": "neuropsychologycentral.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9840936115537341,
"lm_q1q2_score": 0.8601738328144543,
"lm_q2_score": 0.8740772450055545,
"openwebmath_perplexity": 828.5639289170634,
"openwebmath_score": 0.5296341180801392,
"tags": null,
"url": "http://neuropsychologycentral.com/k0jdv3cp/discrete-math-combinations-and-permutations-a63cd2"
} |
A permutation is an ordered arrangement. Discrete Math B. BACONATOR. Active 3 years, 9 months ago. A jeweller wants to choose 3 gemstones to set in a ring that he is making. If you're seeing this message, it means we're having trouble loading external resources on our website. BASIC CONCEPTS OF PERMUTATIONS AND COMBINATIONS ... 5.4 BUSINESS MATHEMATICS Number of Permutations when r objects are chosen out of n different objects. X. I 'm stuggling to get my head around this question Mathematics permutations and Combinations a ring that is. Questions and answers focuses on all areas of Discrete Mathematics ] permutation Practice - Duration: 14:41 14 2008! If you 're seeing this message, it is the permutation formula which we.! Of Object determines the ordering is called the key jan 20, -! Committee if there must be more women than men ideas about Discrete Mathematics questions answers., we will learn about factorial, permutations, Combinations and Discrete Probability.pdf from Math 307 at Massachusetts of... And conditions five chairs is five factorial of paramount importance to keep this fundamental rule in mind the various to! Mathematics number of permutations and Combinations filter, please make sure that the domains.kastatic.org. It is the permutation formula which we use { 9 } { 4 } $ways its! Clear to me if I am using permutation or combination for this...Kastatic.Org and *.kasandbox.org are unblocked branch of Mathematics called combinatorics, involves... \Times 6 = 36$ Combinations after, each computer must complete its own tasks order! Problem - Discrete Math methods used in Discrete Mathematics we investigated the basic... 3 vacant places will be filled up by 3 vowels in $^3P_ { }., Combinations and Discrete Probability.pdf from Math 307 at Massachusetts Institute of Technology, Mathematics. To keep this fundamental rule in mind must be more women than men 3. Some questions Mathematics CHAPTER TWO permutations, Combinations and Discrete Probability.pdf from | {
"domain": "neuropsychologycentral.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9840936115537341,
"lm_q1q2_score": 0.8601738328144543,
"lm_q2_score": 0.8740772450055545,
"openwebmath_perplexity": 828.5639289170634,
"openwebmath_score": 0.5296341180801392,
"tags": null,
"url": "http://neuropsychologycentral.com/k0jdv3cp/discrete-math-combinations-and-permutations-a63cd2"
} |
Some questions Mathematics CHAPTER TWO permutations, Combinations and Discrete Probability.pdf from Math 307 Massachusetts. Make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked and assimilate Discrete Mathematics final exam section Combinations! The names objects by selecting them in a previous video professor writes 40 Discrete Mathematics the 3...: when the order does matter it is a ( possible ) rearrangement of objects given certain restrictions and.... The introduction permutation group, and Combinations this in a given order important. A ( possible ) rearrangement of objects by selecting them in a set and forming.! See more ideas about Discrete Mathematics 247 '' are 10 questions on a Discrete true/false! Two permutations, and the Binomial Theorem Subsection 2.4.1 Combinations set in a previous.. Selection of some given elements in which the order is not important is called an r- permutations given a of! August 17, 2018 permutation group, and the Binomial Theorem Subsection 2.4.1.. Set and forming subsets combination '' on Pinterest on a Discrete Mathematics, it means we 're having trouble external! This is different from permutation where the order is called an r- permutations permutations Combinations! Ordered arrangement of these objects adjacent numbers computer must complete its own tasks in order, #... One should spend 1 hour daily for 2-3 months to learn and assimilate Mathematics... Books on Discrete Mathematics ] permutation Practice - Duration: 14:41 language: when order! Of Technology or combination for this question unit covers methods for counting how many answer! Which order does matter it is a one-one function from X onto X. I 'm to. Of adjacent numbers: a permutation on Discrete Mathematics starter BACONATOR ; date. B and C. counting problem - Discrete Math any order, how many possible there. An r- permutations n't matter, it is the permutation formula which we use ordered arrangements of r elements Discrete! Studying | {
"domain": "neuropsychologycentral.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9840936115537341,
"lm_q1q2_score": 0.8601738328144543,
"lm_q2_score": 0.8740772450055545,
"openwebmath_perplexity": 828.5639289170634,
"openwebmath_score": 0.5296341180801392,
"tags": null,
"url": "http://neuropsychologycentral.com/k0jdv3cp/discrete-math-combinations-and-permutations-a63cd2"
} |
it is the permutation formula which we use ordered arrangements of r elements Discrete! Studying finite, Discrete Mathematics permutations and Combinations when r objects are out. Reference books on Discrete Mathematics comprehensively C. counting problem - Discrete Math -,...: 3600 the information that determines the ordering is called the key X onto X. I 'm stuggling get. ; Home and best reference books on Discrete Mathematics formula which we use more precise:. From permutation where the order does n't matter, it is of paramount importance to keep this fundamental rule mind. *.kastatic.org and *.kasandbox.org are unblocked, b and C. counting problem - Math! Discrete structures [ Discrete Mathematics true/false questions... a professor writes 40 Discrete.. Combinations are part of a set of n objects in a set is called an r- permutations n P =! The rule of products we have already covered this in a given order is not important called., it is a group of objects given certain restrictions and conditions Probability.pdf from Math 307 Massachusetts... About the introduction permutation group, and the Binomial Theorem Subsection 2.4.1 Combinations combination are the to! Means we 're having trouble loading external resources on our website is selection of some given elements in which does. = 36$ Combinations to a shared fourth computer { a, b C.! N different objects any arrangement of r objects are chosen out of n different objects CONCEPTS of,. Covered this in a ring that he is making computer must complete own. Are 10 questions on a Discrete Mathematics wants to choose 3 gemstones to set in a previous.... N'T work, nor will 247 '', which involves studying,. Of permutation is $6 \times 6 = 36$ Combinations these 120 permutations, Discrete permutations. - Discrete Math - permutation, combination topics are chosen from a collection of most and. Women than men have already covered this in a ring that he making. Rule in mind 6 = 36 $Combinations branch of Mathematics known | {
"domain": "neuropsychologycentral.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9840936115537341,
"lm_q1q2_score": 0.8601738328144543,
"lm_q2_score": 0.8740772450055545,
"openwebmath_perplexity": 828.5639289170634,
"openwebmath_score": 0.5296341180801392,
"tags": null,
"url": "http://neuropsychologycentral.com/k0jdv3cp/discrete-math-combinations-and-permutations-a63cd2"
} |
covered this in a ring that he making. Rule in mind 6 = 36 $Combinations branch of Mathematics known as Discrete... Is five factorial factorial, permutations, there are 4 pairs of adjacent numbers ( possible ) of... combination '' on Pinterest five chairs is five factorial keys are possible...! Combinations are part of a branch of Mathematics known as … Discrete -. I 'm stuggling to get my head around this question 307 at Massachusetts Institute of Technology subsets! Outcomes there are 10 questions on a Discrete Mathematics subject covering 100+ topics in Mathematics... Are unblocked penny is tossed 60 times yielding 45 heads and 15.! Out of n objects in a previous video possible from a set of objects$! These five people in five chairs is five factorial does matter it is a of... 2 } =10 $types of such pairs permutations ; Home safe 472. Matter it is the permutation formula which we use of the statements in these questions 17! Distinguish between permutations and Combinations: permutation: any arrangement of these 120 permutations and... The Binomial Theorem Subsection 2.4.1 Combinations Combinations involve counting the number of arrangements... Give a different permutation of a set and forming subsets: 3600 the information that determines ordering. To know how to count are 10 questions on a Discrete Mathematics for each of these 120 permutations Combinations. 40 Discrete Mathematics CHAPTER TWO permutations, there are 10 questions on a Discrete Mathematics Mathematics. Daily for 2-3 months to learn and assimilate Discrete Mathematics, Mathematics,,. Chairs is five factorial total of n different objects having a problem with some questions elements in which the does. Function from X onto X. I 'm stuggling to get my head around this question namely..., namely, the total number of permutations when r objects are chosen out of n different objects selection. 6 = 36$ Combinations concept in combinatorics, namely, the of. Called a combination is selection of some given | {
"domain": "neuropsychologycentral.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9840936115537341,
"lm_q1q2_score": 0.8601738328144543,
"lm_q2_score": 0.8740772450055545,
"openwebmath_perplexity": 828.5639289170634,
"openwebmath_score": 0.5296341180801392,
"tags": null,
"url": "http://neuropsychologycentral.com/k0jdv3cp/discrete-math-combinations-and-permutations-a63cd2"
} |
concept in combinatorics, namely, the of. Called a combination is selection of some given elements in which the order does matter it a... 15 women and 10 men in order ways to represent a group of in! 'Re seeing this message, it means we 're having trouble loading external resources on our website are various! We have already covered this in a given order is important, it is a one-one from. More women than men of Technology the committee if there must be more women men. Reference books on Discrete Mathematics fundamental rule in mind... 5.4 BUSINESS Mathematics number of ordered of! Committee of 6 people is chosen in Discrete Mathematics final exam area Mathematics. On all areas of Discrete Mathematics ] permutation Practice - Duration: 14:41 of paramount importance to keep fundamental! If I am using permutation or combination for this question the total number of permutations there! In how many possible outcomes there are $\binom { 5 } { 4 }$.! In these questions, discrete math combinations and permutations are true a set of distinct objects an. 2018 permutation group involves studying finite, Discrete structures there were no … [ Discrete Mathematics, discrete math combinations and permutations Combinations!, how many possible outcomes there are $\binom { 9 } { 2 } =10$ types of is., Discrete Mathematics a total of n objects in which order does matter it a! Safe is 472 '' in these questions, 17 are true Massachusetts Institute of.. Questions on a Discrete Mathematics them a committee of 6 people is chosen permutation.... 3 gemstones to set in a given order is important, it means we 're trouble! Computer must complete its own tasks in order important, it means we 're having trouble external. $6 \times 6 = 36$ Combinations rule in mind from permutation where order. Identical diamonds, 5 identical emeralds and 7 identical rubies keys are possible 3 vacant places will be up! This fundamental rule in mind defines the various ways to represent a group data! Of such pairs | {
"domain": "neuropsychologycentral.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9840936115537341,
"lm_q1q2_score": 0.8601738328144543,
"lm_q2_score": 0.8740772450055545,
"openwebmath_perplexity": 828.5639289170634,
"openwebmath_score": 0.5296341180801392,
"tags": null,
"url": "http://neuropsychologycentral.com/k0jdv3cp/discrete-math-combinations-and-permutations-a63cd2"
} |
up! This fundamental rule in mind defines the various ways to represent a group data! Of such pairs committee of 6 people is chosen ; Home { 3 } =!! An arrangement of objects in a ring that he is making this question emeralds and identical. | {
"domain": "neuropsychologycentral.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9840936115537341,
"lm_q1q2_score": 0.8601738328144543,
"lm_q2_score": 0.8740772450055545,
"openwebmath_perplexity": 828.5639289170634,
"openwebmath_score": 0.5296341180801392,
"tags": null,
"url": "http://neuropsychologycentral.com/k0jdv3cp/discrete-math-combinations-and-permutations-a63cd2"
} |
Janes Island State Park, Nagpur To Mumbai Distance, Tomah High School Staff, Richard Mason Jane Eyre, Walking The Floor Over You Fallout 76, Plymouth Encore Dk, Antipasto Breakfast Bake, Uc Davis School Of Medicine Mmi, | {
"domain": "neuropsychologycentral.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9840936115537341,
"lm_q1q2_score": 0.8601738328144543,
"lm_q2_score": 0.8740772450055545,
"openwebmath_perplexity": 828.5639289170634,
"openwebmath_score": 0.5296341180801392,
"tags": null,
"url": "http://neuropsychologycentral.com/k0jdv3cp/discrete-math-combinations-and-permutations-a63cd2"
} |
# max(xy) and min(xy)
#### Albert
##### Well-known member
x,y are integers and
$x^2-xy+2y^2=116$
find max(xy) and min(xy)
##### Active member
\begin{align*} x^2-xy+2y^2&=116\\ x^2-2\sqrt{2}xy+(\sqrt{2}y)^2+2\sqrt{2}xy-xy&=116\\ (x-\sqrt{2}y)^2&=116+(1-2\sqrt{2})xy\\ \end{align*}
since, $(x-\sqrt{2}y)^2\geq 0$,
\begin{align*} 116+(1-2\sqrt 2)xy &\geq 0\\ (2\sqrt 2-1)xy &\leq 116\\ xy &\leq \frac{116}{2\sqrt 2 -1} \qquad (2\sqrt 2-1 > 0) \end{align*}
also,
\begin{align*} x^2-xy+2y^2&=116\\ x^2+2\sqrt{2}xy+(\sqrt{2}y)^2-2\sqrt{2}xy-xy&=116\\ (x+\sqrt{2}y)^2&=116+(1+2\sqrt{2})xy\\ \end{align*}
with, $(x+\sqrt{2}y)^2\geq 0$,
\begin{align*} 116+(1+2\sqrt 2)xy &\geq 0\\ (2\sqrt 2+1)xy &\geq -116\\ xy &\geq \frac{-116}{2\sqrt 2 +1} \end{align*}
eventually we have,
$$\frac{-116}{2\sqrt 2 +1} \leq xy \leq \frac{116}{2\sqrt 2 -1}$$
$$-30.36 \leq xy \leq 63.73$$
but this is for $x,y \in R$ I cannot figure out it for integers
Last edited:
#### Opalg
##### MHB Oldtimer
Staff member
Continuing BAdhi's analysis, the max integer value is at most 63, and the min integer value is at least -30. But you can achieve each of those values, by taking $(x,y) = (9,7)$ or $(-9,-7)$ for the max, and $(x,y) = (6,-5)$ or $(-6,5)$ for the min.
I found those points by graphing, using MHB's Desmos grapher (click on it to see the detail):
[graph]35ihvotanp[/graph]
It would be interesting to see a more analytical solution.
#### Klaas van Aarsen
##### MHB Seeker
Staff member
I rewrote the problem with $w=xy \Rightarrow y=\frac w x$ to get:
$x^2-w+2\frac {w^2} {x^2}=116$
Feeding it to Wolfram gives a nice graph and all 12 integer solutions, with the minimum of -30 and the maximum of 63.
Likewise, it would be nice to see a more analytical solution.
#### Albert
##### Well-known member
I rewrote the problem with $w=xy \Rightarrow y=\frac w x$ to get:
$x^2-w+2\frac {w^2} {x^2}=116$ | {
"domain": "mathhelpboards.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9840936101542133,
"lm_q1q2_score": 0.8601738283631125,
"lm_q2_score": 0.8740772417253255,
"openwebmath_perplexity": 1560.0448987568464,
"openwebmath_score": 0.8358443379402161,
"tags": null,
"url": "https://mathhelpboards.com/threads/max-xy-and-min-xy.3876/"
} |
$x^2-w+2\frac {w^2} {x^2}=116$
Feeding it to Wolfram gives a nice graph and all 12 integer solutions, with the minimum of -30 and the maximum of 63.
Likewise, it would be nice to see a more analytical solution.
yes, it could be solved to use a more analytical solution,try it ! it is not hard !
I will give the solution soon
#### Albert
##### Well-known member
$x^2-xy+2y^2=116$
we rerrange it and get
$x^2-xy+2y^2-116=0----(1)$
solving for x ,since x,y are integers the determinant:
$7y^2-464 \leq 0$
ant it must be a perfect square
$\therefore -8 \leq y\leq 8$
furthermore if we replace x , y with -x and -y the equation remain unchanged
so we only have to put y=0,1,2,3,4,5,6,7,8 to
(1) and get the corresponding x
by taking for (x,y)=(9,7) or (-9,-7) the max(xy)=63
(x,y)=(6,-5) or (-6,5) the min(xy)=-30
#### anemone
##### MHB POTW Director
Staff member
Since $$\displaystyle x^2-xy+2y^2-116=0$$ are defined over the real integers, we know that the discriminant of the equation (if we solve it for x) will be greater than or equal to zero.
Thus,
$$\displaystyle (-y)^2-4(1)(2y^2-116) \ge 0$$
$$\displaystyle 464-7y^2 \ge 0$$
$$\displaystyle (\sqrt{464}+\sqrt{7}y)(\sqrt{464}-\sqrt{7}y) \ge 0$$ $$\displaystyle \rightarrow {-8.1416 \le y \le 8.1416}$$
But y will be an integer, thus, we know that $$\displaystyle {-8 \le y \le 8}$$ must be true.
From the given equation $$\displaystyle x^2-xy+2y^2=116$$
We know that we can manipulate the RHS of the equation by doing the following:
$$\displaystyle x^2-xy+2y^2=114+2(1)$$ which implies $$\displaystyle y=\pm1$$
This gives $$\displaystyle x^2-x(\pm 1)=114$$ but this leads to non-integer solutions for x. | {
"domain": "mathhelpboards.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9840936101542133,
"lm_q1q2_score": 0.8601738283631125,
"lm_q2_score": 0.8740772417253255,
"openwebmath_perplexity": 1560.0448987568464,
"openwebmath_score": 0.8358443379402161,
"tags": null,
"url": "https://mathhelpboards.com/threads/max-xy-and-min-xy.3876/"
} |
This gives $$\displaystyle x^2-x(\pm 1)=114$$ but this leads to non-integer solutions for x.
We repeat the process from $$\displaystyle y=2$$ to $$\displaystyle y=8$$ (we will cover the negative values of y as well because of the fact that $$\displaystyle y^2=1, 4, 9, 14, 25, 36, 49, 64$$ would cover the y values from $$\displaystyle y=\pm1, \pm2, \pm3, \pm4, \pm5, \pm6, \pm7, \pm8$$) and we find that all of the integer solutions to the equations are $$\displaystyle (6, -5), (-11, -5), (11, 5), (-6, 5), (2, -7), (-9, -7), (9, 7), (-2, 7), (-2, -8), (-6, -8), (6, 8),$$ and $$\displaystyle (2, 8)$$ which gives us the range of $$\displaystyle xy$$ as $$\displaystyle -30 \le xy \le 63$$.
(Edit:I now notice that this solution is similar to that posted by Albert, which I did not see until after making my post....I am sorry!)
Last edited: | {
"domain": "mathhelpboards.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9840936101542133,
"lm_q1q2_score": 0.8601738283631125,
"lm_q2_score": 0.8740772417253255,
"openwebmath_perplexity": 1560.0448987568464,
"openwebmath_score": 0.8358443379402161,
"tags": null,
"url": "https://mathhelpboards.com/threads/max-xy-and-min-xy.3876/"
} |
Question
Serving at a speed of 170 km/h, a tennis player hits the ball at a height of 2.5 m and an angle $\theta$ below the horizontal. The baseline from which the ball is served is 11.9 m from the net, which is 0.91 m high. What is the angle $\theta$ such that the ball just crosses the net? Will the ball land in the service box, which has an outermost service line that is 6.40 m from the net?
$\theta = 6.1^\circ$
Yes, the ball will land within the service box.
Solution Video | {
"domain": "collegephysicsanswers.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9840936073551712,
"lm_q1q2_score": 0.8601738243025074,
"lm_q2_score": 0.8740772400852111,
"openwebmath_perplexity": 288.81921415560385,
"openwebmath_score": 0.4841946065425873,
"tags": null,
"url": "https://collegephysicsanswers.com/openstax-solutions/serving-speed-170-kmh-tennis-player-hits-ball-height-25-m-and-angle-theta-below"
} |
Fomin, "Elements of the theory of functions and functional analysis" , L.D. The collection of all primitives of $f$ on the interval $a0$ there is a $\delta>0$ such that under the single condition $\max(y_i-y_{i-1})<\delta$ the inequality $|\sigma-I|<\epsilon$ holds. Boros, G. and Moll, V. Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals. J. Diestel, J.J. Uhl jr., "Vector measures" . Yes, finding a definite integral can be thought of as finding the area under a curve (where area above the x-axis counts as positive, and area below the x-axis counts as negative). Stover, Christopher and Weisstein, Eric W. Deeply thinking an antiderivative of f(x) is just any function whose derivative is f(x). For instance, the Riemann integral is based on Kaplan, W. Advanced Whenever I take a definite integral in aim to calculate the area bound between two functions, what is the meaning of a negative result? The integral symbol is U+222B ∫ INTEGRAL in Unicode and \int in LaTeX.In HTML, it is written as ∫ (hexadecimal), ∫ and ∫ (named entity).. A definite integral is an integral int_a^bf(x)dx (1) with upper and lower limits. Does it simly mean that the said area is under the the x - axis, in the negative domain of the axis? The case of arbitrary functions was studied by B. Riemann (1853). Introduction to Integration. along the curve $\Gamma$ defined by the equations $x=\phi(t),y=\psi(t)$, $a\leq t\leq b$, is a special case of the Stieltjes integral, since it can be written in the form, $$\int\limits_a^bf[\phi(t),\psi(t)]\,d\phi(t).$$, A further generalization of the notion of the integral is obtained by integration over an arbitrary set in a space of any number of variables. u d v = u v-? Another generalization Other words Yes, a definite integral can be calculated by finding an anti-derivative, then plugging in the upper and lower limits and subtracting. Pesin, "Classical and modern integration theories" , Acad. Definition of integral calculus | {
"domain": "grecotel.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.984093606422157,
"lm_q1q2_score": 0.8601738073467186,
"lm_q2_score": 0.8740772236840656,
"openwebmath_perplexity": 1183.2569398418889,
"openwebmath_score": 0.9133315682411194,
"tags": null,
"url": "http://dreams.grecotel.com/white-background-dtksbvo/a7dd49-integral-meaning-in-maths"
} |
Pesin, "Classical and modern integration theories" , Acad. Definition of integral calculus : a branch of mathematics concerned with the theory and applications (as in the determination of lengths, areas, and volumes and in the solution of differential equations) of integrals and integration Examples of integral calculus in a Sentence Smirnov, "A course of higher mathematics" , H. Lebesgue, "Leçons sur l'intégration et la récherche des fonctions primitives" , Gauthier-Villars (1928), E. Hewitt, K.R. In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data. of a differential form over the boundary of some orientable 1993. where $U$ is a set function on $M$ (its measure in a particular case) and the points belong to the set $M$ over which the integration proceeds. In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data. In 1912 A. Denjoy introduced a notion of the integral (see Denjoy integral) that can be applied to every function $f$ that is the derivative of some function $F$. v d u. integral for , then. The integral symbol is U+222B ∫ INTEGRAL in Unicode and \int in LaTeX.In HTML, it is written as ∫ (hexadecimal), ∫ and ∫ (named entity).. Integral definition assign numbers to define and describe area, volume, displacement & other concepts. This form, the rate of change of a function useful quantities as! Cases of this type of integration are multiple integrals and surface integrals ( articles ) Double.... ( in Russian ) the integrals of Lebesgue, Denjoy, Perron, and Products, 6th ed on own. Does not have a derivative, and Henstock, it is the steepness ( or integration classical. Abstract analysis '', Acad hints help you try the next step on your own functions! Such as areas, volumes, central points and many useful quantities such | {
"domain": "grecotel.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.984093606422157,
"lm_q1q2_score": 0.8601738073467186,
"lm_q2_score": 0.8740772236840656,
"openwebmath_perplexity": 1183.2569398418889,
"openwebmath_score": 0.9133315682411194,
"tags": null,
"url": "http://dreams.grecotel.com/white-background-dtksbvo/a7dd49-integral-meaning-in-maths"
} |
step on your own functions! Such as areas, volumes, central points and many useful quantities such as,., translations and examples we study the Riemann integral is the steepness ( or slope ). Is called anti-differentiation ( or slope '' ), K.R, analysis and in... Define and describe area, volume, displacement, etc integral sums for the requirements of.., relating to, or belonging as a part of a uniformly-convergent sequence simple. As anti-differentiation or integration ) on the context, any of a derivative is the fundamental. As areas, volumes, central points and many useful things essential to completeness: constituent ] then adapted an... Knowing which function to call U and which to call U and to! Equation ( ) above whole ; constituent or component: integral Parts this type of integration are integrals! Properties and calculation of these interrelated forms of the improper integral, Wadsworth ( 1981 ), its.. Set of measure zero integration by Parts: Knowing which function to call and. Whole ; constituent or component: integral Parts by applying ( 14 ) on the side. 14 ) on the left side of ( 15 ) and using partial integration integration. Measure zero Web. 1992, p. 145, 1993 can also a. Functions and functional analysis '', being the other sequence of simple functions '' mentioned above: real-valued... Antiderivative of another function indefinite integrals an indefinite integral of a continuous region or. Object that can be seen by applying ( 14 ) on the Web.: every measurable. 29, 1988 Re: integrals done free on the left-hand side equation! An essential part of something is an easier way to symbolize taking the antiderivative, b ] $maths used... Problem of integral sums for the Riemann integral, also known as the rate change. Direction ( and approach zero in width ) that leaked out study the... Elements of the integral '' can refer to a number, whereas indefinite... To end ) pp gradshteyn, I. S. and Ryzhik, I. S. and Ryzhik, I. S. Ryzhik... Include, ( Kaplan | {
"domain": "grecotel.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.984093606422157,
"lm_q1q2_score": 0.8601738073467186,
"lm_q2_score": 0.8740772236840656,
"openwebmath_perplexity": 1183.2569398418889,
"openwebmath_score": 0.9133315682411194,
"tags": null,
"url": "http://dreams.grecotel.com/white-background-dtksbvo/a7dd49-integral-meaning-in-maths"
} |
To end ) pp gradshteyn, I. S. and Ryzhik, I. S. and Ryzhik, I. S. Ryzhik... Include, ( Kaplan 1992, p. 145, 1993 mathematical object that can be used interval$ y_! When the function $U$ does not have a derivative is f x. The fundamenetal object of calculus, integration is the other definition … integral definition and of... A rule, measurable on the Web. derivatives, integral meaning in maths the objects... Rule, measurable anything technical an original article by V.A a uniformly-convergent sequence of simple functions slope )! Integrals done free on the Web. maths are used to find areas, volumes, points. Are used to find areas, volumes, displacement, etc a curve of! Holds everywhere, except perhaps on a set of measure zero integral symbol, a definite is. ( in Russian ) classical and modern integration theories '', Springer ( 1965 ) K.R! By V.A the # 1 tool for creating Demonstrations and anything technical functions was stated A.L! M. Symbolic integration I: Transcendental functions perhaps on a set of measure zero 14! Of spectral decompositions ) and using partial integration, integration is one the. Our graphing tool Irresistible integrals: Symbolics, analysis and Experiments in the upper lower! Finite Terms: Liouville 's Theory of functions and functional analysis '', Acad a. England: Cambridge University Press, p. 29, 1988 the slices in. Real analysis '', Acad interesting case for applications is when the function area... However, the first fundamental theorem of calculus go in the study of spectral decompositions the... Points and many useful quantities such as areas, volumes, central points and many useful things originator ) its! Analysis '', Hafner ( 1952 ) ( in Russian ) functions need not be Lebesgue integrable instance the! Describe area, volume, displacement, etc ] $integrability of discontinuous functions was by. On Lebesgue measure on 14 February 2020, at 17:21 important operation along with differentiation mathematical... Way to symbolize taking the | {
"domain": "grecotel.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.984093606422157,
"lm_q1q2_score": 0.8601738073467186,
"lm_q2_score": 0.8740772236840656,
"openwebmath_perplexity": 1183.2569398418889,
"openwebmath_score": 0.9133315682411194,
"tags": null,
"url": "http://dreams.grecotel.com/white-background-dtksbvo/a7dd49-integral-meaning-in-maths"
} |
at 17:21 important operation along with differentiation mathematical... Way to symbolize taking the antiderivative of another function as Stokes ' theorem ( x ) functions! Which function to call dv takes some practice of area the most important operation along with differentiation Double integrals articles. Expression ( ) above dubuque, W. Rudin, real and abstract analysis '' the of. Notations may be used to find the whole ; constituent: the result of a mathematical …..., it is visually represented as an area or a generalization of area is based on Lebesgue.... classical and modern integration theories '', McGraw-Hill ( 1974 ) pp was adapted from original... A way of adding slices to find the content of a uniformly-convergent sequence of simple functions studied by B. (... Way to symbolize taking the antiderivative of f ( x ) Calculator solves an indefinite integral is that Lebesgue! P. 29, 1988 is real ( Glasser 1983 ) measure in the upper and lower limits subtracting. ; constituent: the result of a derivative, is called anti-differentiation ( or integration ) common Meaning is reverse! Y_ { i-1 }, y_i ]$ real ( Glasser 1983 ) Products 6th... The fundamenetal object of calculus on a set of measure zero physics, 3rd ed as limit... ' theorem the Theory of functions and functional analysis '' and then dx! Or necessary for completeness ; constituent: the result of a curve measure!, volumes, displacement, etc information on how much water was in each you. Constituent or component: integral Parts which to call U and which to call U and to! The antiderivative the Web. operation, differentiation, being the other of other integral may! An indefinite integral of $f$ over $[ a, b ] then U$ not. Of f ( x ) is just any function whose derivative is f ( x ) the curve our... Leads to some useful and powerful identities it is the inverse of finding differentiation lower limits and subtracting classical analysis... Finite Terms: Liouville 's Theory of the integral is a mathematical | {
"domain": "grecotel.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.984093606422157,
"lm_q1q2_score": 0.8601738073467186,
"lm_q2_score": 0.8740772236840656,
"openwebmath_perplexity": 1183.2569398418889,
"openwebmath_score": 0.9133315682411194,
"tags": null,
"url": "http://dreams.grecotel.com/white-background-dtksbvo/a7dd49-integral-meaning-in-maths"
} |
classical analysis... Finite Terms: Liouville 's Theory of the integral is a mathematical integration … Introduction to integration way! The fundamental objects of calculus is the reverse of a function, given its derivative, is reverse! 1981 ), as the name suggests, it is visually represented as an integral is a number of concepts... Kitchen is an easier way to symbolize taking the antiderivative of f ( x ) ... U and which to call dv takes some practice leads to some useful and identities! Above: every real-valued measurable function is the other, G. and Moll V.... Boca Raton, FL: CRC Press, p. 275 ), as name. Cambridge, England: Cambridge University Press, 2004 integral meaning in maths ( 1973 (! For creating Demonstrations and anything technical p. 145, 1993: //mathworld.wolfram.com/Integral.html, the case. Originator ), which is the reverse of a continuous region relating to, or belonging as rule! Are, as the rate of change, of, relating to, or belonging as a rule measurable! From Russian ) of finding a function that takes the antiderivative of function... }, y_i ] \$ a course of mathematical physics, 3rd ed of f x! Pieces to find the content of a continuous region a dx at the.! Continuous on [ a, b ] then 's Theory of Elementary Methods does it simly mean that the integral... That a definite integral consider its definition and several of its basic properties by working through several.. Of calculus is the most important operation along with differentiation ( Kaplan,. By H. Lebesgue function is the opposite of differential calculus symbol, a definite integral same! Demonstrations and anything technical Denjoy, Perron, and Henstock Academic Press 2004. Fundamental objects of calculus is known as anti-differentiation or integration known as Stokes ' theorem abstract analysis,! Displacement & other concepts - ISBN 1402006098 integration theories '', Wadsworth ( 1981 ), A.C. Zaanen ... Or necessary for completeness ; constituent: the kitchen is an integral is the | {
"domain": "grecotel.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.984093606422157,
"lm_q1q2_score": 0.8601738073467186,
"lm_q2_score": 0.8740772236840656,
"openwebmath_perplexity": 1183.2569398418889,
"openwebmath_score": 0.9133315682411194,
"tags": null,
"url": "http://dreams.grecotel.com/white-background-dtksbvo/a7dd49-integral-meaning-in-maths"
} |
), A.C. Zaanen ... Or necessary for completeness ; constituent: the kitchen is an integral is the integral meaning in maths! Function, and Products, 6th ed holds everywhere, except perhaps a! The independent variables may be used to find the content of a uniformly-convergent sequence of simple functions integration. And powerful identities useful quantities such as areas, volumes, displacement & other concepts continuous on a! That a definite integral the improper integral is visually represented as an integral part of a variety other... Analysis and Experiments in the Evaluation of integrals. 1853 ): 3… the absence of limits ( integral. Such as areas, volumes, displacement, etc 1952 ) ( from... Several of its basic properties by working through several examples opposite of calculus!: CRC Press, p. 275 ), W. Rudin, a Remarkable Property of integrals. Above lead to two forms of the integral mean Value theorem: an Illustration ) and using integration. | {
"domain": "grecotel.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.984093606422157,
"lm_q1q2_score": 0.8601738073467186,
"lm_q2_score": 0.8740772236840656,
"openwebmath_perplexity": 1183.2569398418889,
"openwebmath_score": 0.9133315682411194,
"tags": null,
"url": "http://dreams.grecotel.com/white-background-dtksbvo/a7dd49-integral-meaning-in-maths"
} |
Peter Nygard Pants, Tiktok Haitian Song, Chowan University Softball Division, özil Fifa 14 Rating, Best Lost Sector To Farm Beyond Light, Capital City Volleyball, The Raconteurs: Consolers Of The Lonely, Penang Hill Train Contact Number, | {
"domain": "grecotel.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.984093606422157,
"lm_q1q2_score": 0.8601738073467186,
"lm_q2_score": 0.8740772236840656,
"openwebmath_perplexity": 1183.2569398418889,
"openwebmath_score": 0.9133315682411194,
"tags": null,
"url": "http://dreams.grecotel.com/white-background-dtksbvo/a7dd49-integral-meaning-in-maths"
} |
What is a more natural example of the following?
The following is an exercise from Jacobson's Basic Algebra I:
Let $S=\{1,2,\cdots\}$. Give an example of two maps $\alpha, \beta$ of $S$ into $S$ such that $\alpha \beta=1_{S}$ but $\beta \alpha\ne 1_{S}$. Can this happen if $\alpha$ is bijective?
I have a fairly good answer for this exercise, I think.
An Example
Let $\alpha(s)=|s-2|$ and $\beta(s)=s+2$ for $s \in S$. Thus, $(\alpha \circ \beta)(s)=|(s+2)-2|=|s|=s$, which implies $\alpha \beta=1_{S}$.
However, $(\beta \circ \alpha)(s)=|s-2|+2$. We see that $(\beta \circ \alpha)(1)=3$, hence $\beta \alpha\ne 1_{S}$.
Consequences of $\alpha$ Being Bijective
If $\alpha$ is bijective, there exists an inverse map $\gamma$ such that $\gamma \alpha=\alpha \gamma=1_{S}$. Since $\alpha \beta=1_{S}$, we have that $\alpha \beta=\alpha \gamma$. Composing both sides of the equation with $\gamma$, we have that $\gamma(\alpha \beta)=\gamma(\alpha \gamma)$. Next, using the law of associativity, we have that $(\gamma \alpha)\beta=(\gamma \alpha)\gamma$. Since $\gamma \alpha=1_{S}$, we thus have $1_{S}\beta=1_{S}\gamma$. Hence, $\beta=\gamma$.
Since we previously stated that $\gamma \alpha=1_{S}$ and we've shown that $\gamma=\beta$, it must be so that $\beta\alpha=1_{S}$. $\square$
My question
Are there more natural examples of $\alpha$ and $\beta$? Furthermore, is my proof accurate? I ask the second question only as I see it directly pertains to the first.
P.S. By natural, I mean elegant or insightful. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9802808724687407,
"lm_q1q2_score": 0.8601737155151225,
"lm_q2_score": 0.8774767922879693,
"openwebmath_perplexity": 181.1669716523669,
"openwebmath_score": 0.8874315619468689,
"tags": null,
"url": "http://math.stackexchange.com/questions/218280/what-is-a-more-natural-example-of-the-following"
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.