Greatest Hits

39 votes
5 answers

How many squares actually ARE in this picture? Is this a trick question with no right answer?

This is one of those popular pictures on sites like Facebook. I always see a huge variation of answers such as $8, 9, 16, 17, 24, 28, 30, 40, 41, 52,$ etc., yet I've never seen a definitive answer on ...
user1092719's user avatar
1625 votes
88 answers

Visually stunning math concepts which are easy to explain

Since I'm not that good at (as I like to call it) 'die-hard-mathematics', I've always liked concepts like the golden ratio or the dragon curve, which are easy to understand and explain but are ...
1 vote
8 answers

Simplification of: AB + A'C + BC in boolean algebra

I am trying to understand the simplification of the boolean expression: AB + A'C + BC I know it simplifies to A'C + BC And I understand why, but I cannot figure out how to perform the ...
M. S.'s user avatar
  • 11
132 votes
8 answers

What is the difference between independent and mutually exclusive events?

Two events are mutually exclusive if they can't both happen. Independent events are events where knowledge of the probability of one doesn't change the probability of the other. Are these ...
Adnan Ali's user avatar
  • 1,493
24 votes
4 answers

Meaning of coprime = (relatively) prime = mutually prime

I've never encountered this question in any of my math classes and it just shows up randomly in my comsci class with no further info about it. I've wiki'ed it, but can't even understand that. Could ...
Jonathan's user avatar
  • 359
587 votes
14 answers

Integral $\int_{-1}^1\frac1x\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2\,x^2+2\,x+1}{2\,x^2-2\,x+1}\right) \mathrm dx$

I need help with this integral: $$I=\int_{-1}^1\frac1x\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2\,x^2+2\,x+1}{2\,x^2-2\,x+1}\right)\ \mathrm dx.$$ The integrand graph looks like this: $\hspace{1in}$ The ...
Laila Podlesny's user avatar
64 votes
12 answers

Easy way of memorizing values of sine, cosine, and tangent

My math professor recently told us that she wanted us to be able to answer $\sin\left(\frac{\pi }{2}\right)$ in our head on the snap. I know I can simply memorize the table for the test by this Friday,...
James Smith's user avatar
  • 1,729
7 votes
10 answers

A loss and gain problem

This is a very simple but confusing puzzle. A customer buys goods worth $200$ rupees from a shop. The shopkeeper selling these goods makes zero profit from this purchase. The lady gives him a $1000$ ...
constantlearner's user avatar
64 votes
2 answers

What is the integral of 1/x?

What is the integral of $\frac{1}{x}$? Do you get $\ln(x)$ or $\ln|x|$? In general, does integrating $f'(x)/f(x)$ give $\ln(f(x))$ or $\ln|f(x)|$? Also, what is the derivative of $|f(x)|$? Is it $f'(x)...
hollow7's user avatar
  • 2,465
266 votes
10 answers

Derivative of sigmoid function $\sigma (x) = \frac{1}{1+e^{-x}}$

In my AI textbook there is this paragraph, without any explanation. The sigmoid function is defined as follows $$\sigma (x) = \frac{1}{1+e^{-x}}.$$ This function is easy to differentiate ...
Bryan Glazer's user avatar
  • 3,044
18 votes
3 answers

Show that $(a+b+c)^3 = a^3 + b^3 + c^3+ (a+b+c)(ab+ac+bc)$

As stated in the title, I'm supposed to show that $(a+b+c)^3 = a^3 + b^3 + c^3 + (a+b+c)(ab+ac+bc)$. My reasoning: $$(a + b + c)^3 = [(a + b) + c]^3 = (a + b)^3 + 3(a + b)^2c + 3(a + b)c^2 + c^3$$ ...
Sawyier's user avatar
  • 409
14 votes
2 answers

How to find mean and median from histogram

Solution for finding mean : The problem faced when i saw a video to evaluate the mean According to the video, the lecturer said that, we can take ...
user avatar
42 votes
5 answers

How to determine if a function is one-to-one?

I am looking for the "best" way to determine whether a function is one-to-one, either algebraically or with calculus. I know a common, yet arguably unreliable method for determining this answer would ...
Oliver Spryn's user avatar
  • 1,523
66 votes
4 answers

taylor series of $\ln(1+x)$?

Compute the taylor series of $\ln(1+x)$ I've first computed derivatives (up to the 4th) of ln(1+x) $f^{'}(x)$ = $\frac{1}{1+x}$ $f^{''}(x) = \frac{-1}{(1+x)^2}$ $f^{'''}(x) = \frac{2}{(1+x)^3}$ $f^{'...
Minu's user avatar
  • 981
23 votes
18 answers

Piecewise functions: Got an example of a real world piecewise function?

Looking for something beyond a contrived textbook problem concerning jelly beans or equations that do not represent anything concrete. Not just a piecewise function for its own sake. Anyone?
8 votes
2 answers

Calculus question taking derivative to find horizontal tangent line

How would I solve this problem? Find the point where the tangent line is horizontal in the following function: $$f(x)=(x-2)(x^2-x-11)$$ I computed the derivative: $\quad f'(x)=(x-2)(2x-1)+(1)(x^2-x-...
Fernando Martinez's user avatar
124 votes
11 answers

What does it mean to have a determinant equal to zero?

After looking in my book for a couple of hours, I'm still confused about what it means for a $(n\times n)$-matrix $A$ to have a determinant equal to zero, $\det(A)=0$. I hope someone can explain this ...
user2171775's user avatar
  • 1,375
47 votes
7 answers

How to calculate the intersection of two planes?

How to calculate the intersection of two planes ? These are the planes and the result is gonna be a line in $\Bbb R^3$: $x + 2y + z - 1 = 0$ $2x + 3y - 2z + 2 = 0$
user1111261's user avatar
  • 1,149
318 votes
9 answers

Is a matrix multiplied with its transpose something special?

In my math lectures, we talked about the Gram-Determinant where a matrix times its transpose are multiplied together. Is $A A^\mathrm T$ something special for any matrix $A$?
Martin Ueding's user avatar
65 votes
3 answers

What does "∈" mean?

I have started seeing the "∈" symbol in math. What exactly does it mean? I have tried googling it but google takes the symbol out of the search.
Locke's user avatar
  • 815
49 votes
4 answers

Find the sum of all the multiples of 3 or 5 below 1000

How to solve this problem, I can not figure it out: If we list all the natural numbers below 10 that are multiples of 3 or 5, we get 3, 5, 6 and 9. The sum of these multiples is 23. Find the sum of ...
Sarfraz's user avatar
  • 635
65 votes
4 answers

How to tell if a set of vectors spans a space?

I want to know if the set $\{(1, 1, 1), (3, 2, 1), (1, 1, 0), (1, 0, 0)\}$ spans $\mathbb{R}^3$. I know that if it spans $\mathbb{R}^3$, then for any $x, y, z, \in \mathbb{R}$, there exist $c_1, c_2, ...
Javier's user avatar
  • 7,328
4 votes
3 answers

What is the perimeter of a sector?

I don't understand this. So we have: \begin{align} r &= 12 \color{gray}{\text{ (radius of circle)}} \\ d &= 24 \text{ (r}\times2) \color{gray}{\text{ (diameter of circle)}} \\ c &= 24\pi \...
Jawad's user avatar
  • 139
112 votes
5 answers

Difference between "≈", "≃", and "≅"

In mathematical notation, what are the usage differences between the various approximately-equal signs "≈", "≃", and "≅"? The Unicode standard lists all of them inside the Mathematical Operators ...
GOTO 0's user avatar
  • 1,812
7 votes
7 answers

Probability of getting exactly 2 heads in 3 coins tossed with order not important?

I have been thinking of this problem for the post 3-4 hours, I have come up with this problem it is not a home work exercise Let's say I have 3 coins and I toss them, Here order is not important ...
Max's user avatar
  • 495
53 votes
4 answers

Determine if vectors are linearly independent

Determine if the following set of vectors is linearly independent: $$\left[\begin{array}{r}2\\2\\0\end{array}\right],\left[\begin{array}{r}1\\-1\\1\end{array}\right],\left[\begin{array}{r}4\\2\\-2\...
Mirrana's user avatar
  • 9,087
3 votes
3 answers

Plotting $(x^2 + y^2 - 1)^3 - x^2 y^3 = 0$

I have no idea how this equation: \begin{equation} (x^2 + y^2 - 1)^3 - x^2 y^3 = 0 \end{equation} Produces this picture: Can someone provide a general explanation of plotting this function?
m0nhawk's user avatar
  • 1,779
45 votes
4 answers

Find the coordinates of a point on a circle

I have a circle like so Given a rotation θ and a radius r, how do I find the coordinate (x,y)? Keep in mind, this rotation could be anywhere between 0 and 360 degrees. For example, I have a radius r ...
CoderTheTyler's user avatar
61 votes
11 answers

How to calculate the area of a 3D triangle?

I have coordinates of 3d triangle and I need to calculate its area. I know how to do it in 2D, but don't know how to calculate area in 3d. I have developed data as follows. ...
iamgopal's user avatar
  • 643
202 votes
14 answers

How many sides does a circle have?

My son is in 2nd grade. His math teacher gave the class a quiz, and one question was this: If a triangle has 3 sides, and a rectangle has 4 sides, how many sides does a circle have? My first ...
Fixee's user avatar
  • 11.6k
31 votes
4 answers

Prove that if events A and B are independent, then the complement events of A and B are also independent.

I know that: $$\begin{gathered}P\left(A\cap B\right)=P\left(A\right)P\left(B\right)\\ P\left(A^{C}\right)=1-P\left(A\right)\\ P\left(B^{C}\right)=1-P\left(B\right) \end{gathered} $$ My proof so far: $...
Preston Charmelle's user avatar
78 votes
11 answers

Proof of triangle inequality

I understand intuitively that this is true, but I'm embarrassed to say I'm having a hard time constructing a rigorous proof that $|a+b| \leq |a|+|b|$. Any help would be appreciated :)
ivan's user avatar
  • 3,247
81 votes
10 answers

Finding out the area of a triangle if the coordinates of the three vertices are given

What is the simplest way to find out the area of a triangle if the coordinates of the three vertices are given in $x$-$y$ plane? One approach is to find the length of each side from the coordinates ...
TSP1993's user avatar
  • 969
446 votes
14 answers

Fourier transform for dummies

What is the Fourier transform? What does it do? Why is it useful (in math, in engineering, physics, etc)? This question is based on the question of Kevin Lin, which didn't quite fit in Mathoverflow. ...
84 votes
19 answers

How to find perpendicular vector to another vector?

How do I find a vector perpendicular to a vector like this: $$3\mathbf{i}+4\mathbf{j}-2\mathbf{k}?$$ Could anyone explain this to me, please? I have a solution to this when I have $3\mathbf{i}+4\...
niko's user avatar
  • 959
3 votes
2 answers

how to convert 1e+11 into number?

What will be 1e+11 in number? I know e2 means * 10^2 but i am confused with this above question. What will its value be? I know how to use exponential function ...
ashutosh's user avatar
  • 141
17 votes
3 answers

What does the small number on top of the square root symbol mean?

I just came across this annotation in my school's maths compendium: The compendium is very brief and doesn't explain what this means.
Hubro's user avatar
  • 317
46 votes
11 answers

Finding a unit vector perpendicular to another vector

For example we have the vector $8i + 4j - 6k$, how can we find a unit vector perpendicular to this vector?
adil's user avatar
  • 507
14 votes
3 answers

"$x$ times as many as" versus "$x$ times more than"?

I keep coming across such questions during my GRE preparation: A is how many times of B versus A is how many times greater than B A is what percentage of B versus A is what percentage greater than B ...
Max's user avatar
  • 495
33 votes
10 answers

Proving that the number of vertices of odd degree in any graph G is even

I'm having a bit of a trouble with the below question Given $G$ is an undirected graph, the degree of a vertex $v$, denoted by $\mathrm{deg}(v)$, in graph $G$ is the number of neighbors of $v$. ...
sandman001's user avatar
41 votes
9 answers

Probability of 3 Heads in 10 Coin Flips

What’s the probability of getting 3 heads and 7 tails if one flips a fair coin 10 times. I just can’t figure out how to model this correctly.
Newbie1923's user avatar
145 votes
12 answers

What does the dot product of two vectors represent?

I know how to calculate the dot product of two vectors alright. However, it is not clear to me what, exactly, does the dot product represent. The product of two numbers, $2$ and $3$, we say that it ...
Saturn's user avatar
  • 7,201
54 votes
13 answers

Real world application of Fourier series

What are some real world applications of Fourier series? Particularly the complex Fourier integrals?
Namit Sinha's user avatar
70 votes
7 answers

What’s the difference between analytical and numerical approaches to problems?

I don't have much (good) math education beyond some basic university-level calculus. What do "analytical" and "numerical" mean? How are they different?
jbrennan's user avatar
  • 819
211 votes
7 answers

Transpose of inverse vs inverse of transpose

Given a square matrix, is the transpose of the inverse equal to the inverse of the transpose? $$ (A^{-1})^T = (A^T)^{-1} $$
Void Star's user avatar
  • 2,555
362 votes
11 answers

What is the importance of eigenvalues/eigenvectors?

What is the importance of eigenvalues/eigenvectors?
Ryan's user avatar
  • 5,519
16 votes
2 answers

Sxx in linear regression

What is the meaning of the symbols $S_{xx}$ and $S_{xy}$ in simple linear regression? I know the formula but what is the meaning of those symbols?
Dkova's user avatar
  • 359
11 votes
4 answers

In which order do I graph transformations of functions?

In which order do I graph transformations of functions? The 6 function transformations are: Vertical Shifts Horizontal Shifts Reflection about the x-axis Reflection about the y-axis Vertical ...
BlueMagic1923's user avatar
141 votes
36 answers

Proof that $1+2+3+4+\cdots+n = \frac{n\times(n+1)}2$

Why is $1+2+3+4+\ldots+n = \dfrac{n\times(n+1)}2$ $\space$ ?
b1_'s user avatar
  • 1,585
170 votes
7 answers

What is the difference between Fourier series and Fourier transformation?

What's the difference between Fourier transformations and Fourier Series? Are they the same, where a transformation is just used when its applied (i.e. not used in pure mathematics)?
Dean's user avatar
  • 1,881