Mathematics

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1500 questions
156
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
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156
votes
8 answers

Intuition behind Conditional Expectation

I'm struggling with the concept of conditional expectation. First of all, if you have a link to any explanation that goes beyond showing that it is a generalization of elementary intuitive concepts, please let me know. Let me get more specific. Let…
Stefan
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156
votes
1 answer

Is the following matrix invertible?

$$\begin{bmatrix} 1235 &2344 &1234 &1990\\ 2124 & 4123& 1990& 3026 \\ 1230 &1234 &9095 &1230\\ 1262 &2312& 2324 &3907 \end{bmatrix}$$ Clearly, its determinant is not zero and, hence, the matrix is invertible. Is there a more elegant way to do this?…
Yongkai
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155
votes
4 answers

Does $R[x] \cong S[x]$ imply $R \cong S$?

This is a very simple question but I believe it's nontrivial. I would like to know if the following is true: If $R$ and $S$ are rings and $R[x]$ and $S[x]$ are isomorphic as rings, then $R$ and $S$ are isomorphic. Thanks! If there isn't a proof (or…
Richard G
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155
votes
7 answers

Studying Euclidean geometry using hyperbolic criteria

You've spent your whole life in the hyperbolic plane. It's second nature to you that the area of a triangle depends only on its angles, and it seems absurd to suggest that it could ever be otherwise. But recently a good friend named Euclid has…
Zach Conn
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154
votes
10 answers

Are mathematical articles on Wikipedia reliable?

I know that Wikipedia gets a bad rap, and it seems like some teachers of mine have nothing better to do in class than harp on about the Great Academic Pastime of calling Wikipedia untrustworthy, but let's face it - Wikipedia is probably the single…
user140943
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154
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
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154
votes
8 answers

Do most mathematicians know most topics in mathematics?

How many topics outside of his or her specialization is an average mathematician familiar with? For example, does an average group theorist know enough of partial differential equations to pass a test in a graduate-level PDE course? Also, what are…
Sid Caroline
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154
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14 answers

Why does an argument similiar to 0.999...=1 show 999...=-1?

I accept that two numbers can have the same supremum depending on how you generate a decimal representation. So $2.4999\ldots = 2.5$ etc. Can anyone point me to resources that would explain what the below argument that shows $999\ldots = -1$ is…
CommonToad
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153
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 accepted…
Gold
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153
votes
7 answers

lim sup and lim inf of sequence of sets.

I was wondering if someone would be so kind to provide a very simple explanation of $\limsup$ and $\liminf$ of a sequence of sets. For a sequence of subsets $A_n$ of a set $X$ we have $$\limsup A_n= \bigcap_{N=1}^\infty \left( \bigcup_{n\ge N} A_n…
Comic Book Guy
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152
votes
3 answers

How to find the Galois group of a polynomial?

I've been learning about Galois theory recently on my own, and I've been trying to solve tests from my university. Even though I understand all the theorems, I seem to be having some trouble with the technical stuff. A specific example would be how…
IBS
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152
votes
5 answers

On Ph.D. Qualifying Exams

Where can I find Ph.D. qualifying exams questions? Is there any website that keeps a collection of such problems? I need it for doing some revision of the basic topics. I know of a book but that doesn't have the full collection.
Koushik
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152
votes
8 answers

When is matrix multiplication commutative?

I know that matrix multiplication in general is not commutative. So, in general: $A, B \in \mathbb{R}^{n \times n}: A \cdot B \neq B \cdot A$ But for some matrices, this equations holds, e.g. A = Identity or A = Null-matrix $\forall B \in…
Martin Thoma
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151
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
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