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1500 questions
121
votes
21 answers
How do you explain to a 5th grader why division by zero is meaningless?
I wish to explain my younger brother: he is interested and curious, but he cannot grasp the concepts of limits and integration just yet. What is the best mathematical way to justify not allowing division by zero?
Shubh Khandelwal
- 1,265
121
votes
1 answer
Is there a characterization of groups with the property $\forall N\unlhd G,\:\exists H\leq G\text{ s.t. }H\cong G/N$?
A common mistake for beginning group theory students is the belief that a quotient of a group $G$ is necessarily isomorphic to a subgroup of $G$. Is there a characterization of the groups in which this property holds?
If this question is too broad,…
Alexander Gruber
- 26,963
121
votes
12 answers
Is there an "inverted" dot product?
The dot product of vectors $\mathbf{a}$ and $\mathbf{b}$ is defined as:
$$\mathbf{a} \cdot \mathbf{b} =\sum_{i=1}^{n}a_{i}b_{i}=a_{1}b_{1}+a_{2}b_{2}+\cdots +a_{n}b_{n}$$
What about the quantity?
$$\mathbf{a} \star \mathbf{b} = \prod_{i=1}^{n}…
doc
- 1,307
121
votes
7 answers
What are the issues in modern set theory?
This is spurred by the comments to my answer here. I'm unfamiliar with set theory beyond Cohen's proof of the independence of the continuum hypothesis from ZFC. In particular, I haven't witnessed any real interaction between set-theoretic issues…
Paul VanKoughnett
- 5,444
121
votes
10 answers
Motivation for the rigour of real analysis
I am about to finish my first year of studying mathematics at university and have completed the basic linear algebra/calculus sequence. I have started to look at some real analysis and have really enjoyed it so far.
One thing I feel I am lacking in…
1729
- 2,157
121
votes
26 answers
Examples of problems that are easier in the infinite case than in the finite case.
I am looking for examples of problems that are easier in the infinite case than in the finite case. I really can't think of any good ones for now, but I'll be sure to add some when I do.
Asinomás
- 105,651
121
votes
13 answers
Why can't calculus be done on the rational numbers?
I was once told that one must have a notion of the reals to take limits of functions. I don't see how this is true since it can be written for all functions from the rationals to the rationals, which I will denote $f$, that
$$\forall…
Praise Existence
- 1,445
121
votes
15 answers
What is the smallest unknown natural number?
There are several unknown numbers in mathematics, such as optimal constants in some inequalities.
Often it is enough to some estimates for these numbers from above and below, but finding the exact values is also interesting.
There are situations…
Joonas Ilmavirta
- 25,809
121
votes
17 answers
How can I understand and prove the "sum and difference formulas" in trigonometry?
The "sum and difference" formulas often come in handy, but it's not immediately obvious that they would be true.
\begin{align}
\sin(\alpha \pm \beta) &= \sin \alpha \cos \beta \pm \cos \alpha \sin \beta \\
\cos(\alpha \pm \beta) &= \cos \alpha \cos…
Tyler
- 3,067
121
votes
39 answers
What's your favorite proof accessible to a general audience?
What math statement with proof do you find most beautiful and elegant, where such is accessible to a general audience, meaning you could state, prove, and explain it to a general audience in roughly $5 \pm\epsilon$ minutes.
Let's define…
userX
- 2,029
121
votes
1 answer
Solving Special Function Equations Using Lie Symmetries
The Lie group and representation theory approach to special functions, and how they solve the ODEs arising in physics is absolutely amazing. I've given an example of its power below on Bessel's equation.
Kaufman's article describes algebraic methods…
bolbteppa
- 4,389
120
votes
4 answers
What is the difference and relationship between the binomial and Bernoulli distributions?
How should I understand the difference or relationship between binomial and Bernoulli distribution?
user122358
- 2,712
120
votes
8 answers
Lebesgue integral basics
I'm having trouble finding a good explanation of the Lebesgue integral. As per the definition, it is the expectation of a random variable. Then how does it model the area under the curve? Let's take for example a function $f(x) = x^2$. How do we…
user957
- 3,387
120
votes
8 answers
Are the "proofs by contradiction" weaker than other proofs?
I remember hearing several times the advice that, we should avoid using a proof by contradiction, if it is simple to convert to a direct proof or a proof by contrapositive. Could you explain the reason? Do logicians think that proofs by…
AgCl
- 6,292
120
votes
10 answers
The deep reason why $\int \frac{1}{x}\operatorname{d}x$ is a transcendental function ($\log$)
In general, the indefinite integral of $x^n$ has power $n+1$. This is the standard power rule. Why does it "break" for $n=-1$? In other words, the derivative rule $$\frac{d}{dx} x^{n} = nx^{n-1}$$ fails to hold for $n=0$.
Is there some deep…
Shuheng Zheng
- 949