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1500 questions
150
votes
8 answers
Can you explain the "Axiom of choice" in simple terms?
As I'm sure many of you do, I read the XKCD webcomic regularly. The most recent one involves a joke about the Axiom of Choice, which I didn't get.
I went to Wikipedia to see what the Axiom of Choice is, but as often happens with things like this,…
Tyler
- 3,067
150
votes
6 answers
Do Arithmetic Mean and Geometric Mean of Prime Numbers converge?
I was looking at a list of primes. I noticed that $ \frac{AM (p_1, p_2, \ldots, p_n)}{p_n}$ seemed to converge.
This led me to try $ \frac{GM (p_1, p_2, \ldots, p_n)}{p_n}$ which also seemed to converge.
I did a quick Excel graph and regression and…
Soham
- 1,553
150
votes
7 answers
Apparently sometimes $1/2 < 1/4$?
My son brought this home today from his 3rd-grade class. It is from an official Montgomery County, Maryland mathematics assessment test:
True or false? $1/2$ is always greater than $1/4$.
Official answer: false
Where has he gone wrong?
Addendum,…
SDiv
- 2,520
150
votes
11 answers
Are we allowed to compare infinities?
I'm in middle school and had a question (my dad is helping me with formatting).
We're learning about infinity in math class and there are a lot of problems like how it's not a number and how if you add one to infinity it doesn't change value.
But…
Alice
- 1,347
149
votes
41 answers
Why is negative times negative = positive?
Someone recently asked me why a negative $\times$ a negative is positive, and why a negative $\times$ a positive is negative, etc.
I went ahead and gave them a proof by contradiction like this:
Assume $(-x) \cdot (-y) = -xy$
Then divide both sides…
Sev
- 2,093
149
votes
6 answers
What's 4 times more likely than 80%?
There's an 80% probability of a certain outcome, we get some new information that means that outcome is 4 times more likely to occur.
What's the new probability as a percentage and how do you work it out?
As I remember it the question was posed like…
Jim
- 1,241
149
votes
7 answers
$\pi$ in arbitrary metric spaces
Whoever finds a norm for which $\pi=42$ is crowned nerd of the day!
Can the principle of $\pi$ in euclidean space be generalized to 2-dimensional metric/normed spaces in a reasonable way?
For Example, let $(X,||.||)$ be a 2-dimensional normed vector…
CBenni
- 1,940
148
votes
4 answers
Partial derivative in gradient descent for two variables
I've started taking an online machine learning class, and the first learning algorithm that we are going to be using is a form of linear regression using gradient descent. I don't have much of a background in high level math, but here is what I…
voithos
- 1,763
148
votes
15 answers
Are "if" and "iff" interchangeable in definitions?
In some books the word "if" is used in definitions and it is not clear if they actually mean "iff" (i.e "if and only if").
I'd like to know if in mathematical literature in general "if" in definitions means "iff".
For example I am reading "Essential…
fiftyeight
- 2,687
148
votes
12 answers
Why do we use the word "scalar" and not "number" in Linear Algebra?
During a year and half of studying Linear Algebra in academy, I have never questioned why we use the word "scalar" and not "number". When I started the course our professor said we would use "scalar" but he never said why.
So, why do we use the word…
LiziPizi
- 2,855
147
votes
11 answers
Why is the volume of a cone one third of the volume of a cylinder?
The volume of a cone with height $h$ and radius $r$ is $\frac{1}{3} \pi r^2 h$, which is exactly one third the volume of the smallest cylinder that it fits inside.
This can be proved easily by considering a cone as a solid of revolution, but I would…
bryn
- 9,746
146
votes
5 answers
Can someone explain the math behind tessellation?
Tessellation is fascinating to me, and I've always been amazed by the drawings of M.C.Escher, particularly interesting to me, is how he would've gone about calculating tessellating shapes.
In my spare time, I'm playing a lot with a series of…
ocodo
- 1,765
146
votes
4 answers
What is category theory useful for?
Okay, so I understand what calculus, linear algebra, combinatorics and even topology try to answer (update: this is not the case in hindsight), but why invent category theory?
In Wikipedia it says it is to formalize. As far as I can tell, it sort of…
Asinomás
- 105,651
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 is $2$ added to itself $3$ times or something like…
Saturn
- 7,191
145
votes
16 answers
What should be the intuition when working with compactness?
I have a question that may be regarded by many as duplicate since there's a similar one at MathOverflow.
In $\mathbb{R}^n$ the compact sets are those that are closed and bounded, however the guy who answered this question and had his answer…
Gold
- 26,547