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1500 questions
105
votes
13 answers
Don't see the point of the Fundamental Theorem of Calculus.
$$\frac{d}{dx}\int_a^xf(t)\,dt$$ I would love to to understand what exactly is the point of FTC. I'm not interested in mechanically churning out solutions to problems. It doesn't state anything that isn't already known. Prior to reading about FTC,…
JackOfAll
- 4,921
104
votes
1 answer
What's the largest possible volume of a taco, and how do I make one that big?
Let $f$ be a continuous, even function over some interval $I=[-a,a]$ such that the total arc length of $f$ over $I$ is at least $2$ , $f(0)=0$ , and $f$ is increasing on $(0,a)$ . [You might imagine something like $f(x)=x^2$ .] View the graph of…
2'5 9'2
- 57,884
104
votes
5 answers
True or false? $x^2\ne x\implies x\ne 1$
Today I had an argument with my math teacher at school. We were answering some simple True/False questions and one of the questions was the following: $$x^2\ne x\implies x\ne 1$$ I immediately answered true, but for some reason, everyone (including…
Chris
- 1,061
104
votes
15 answers
Different proofs of the Pythagorean theorem?
The Pythagorean Theorem is one of the most popular to prove by mathematicians, and there are many proofs available (including one from James Garfield ). What's are some of the most elegant proofs? My favorite is this graphical one: According to…
Shane
- 111
104
votes
8 answers
Does L'Hôpital's work the other way?
As referred in Wikipedia (see the specified criteria there), L'Hôpital's rule says, $$ \lim_{x\to c}\frac{f(x)}{g(x)}=\lim_{x\to c}\frac{f'(x)}{g'(x)} $$ As $$ \lim_{x\to c}\frac{f'(x)}{g'(x)}= \lim_{x\to c}\frac{\int f'(x)\ dx}{\int g'(x)\ dx} $$…
JMCF125
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104
votes
22 answers
What are Your Favourite Maths Puzzles?
We all love a good puzzle To a certain extent, any piece of mathematics is a puzzle in some sense: whether we are classifying the homological intersection forms of four manifolds or calculating the optimum dimensions of a cylinder, it is an element…
Tom Boardman
- 3,785
104
votes
7 answers
What kind of "symmetry" is the symmetric group about?
There are two concepts which are very similar literally in abstract algebra: symmetric group and symmetry group . By definition, the symmetric group on a set is the group consisting of all bijections of the set (all one-to-one and onto functions)…
user9464
104
votes
19 answers
Good Physical Demonstrations of Abstract Mathematics
I like to use physical demonstrations when teaching mathematics (putting physics in the service of mathematics, for once, instead of the other way around), and it'd be great to get some more ideas to use. I'm looking for nontrivial ideas in abstract…
Jamie Banks
- 13,460
104
votes
10 answers
What is $\sqrt{i}$?
If $i=\sqrt{-1}$, is $\large\sqrt{i}$ imaginary? Is it used or considered often in mathematics? How is it notated?
Gordon Gustafson
- 3,545
104
votes
11 answers
Is there any conjecture that has been proved to be solvable/provable but whose direct solution/proof is not yet known?
In mathematics, is there any conjecture about the existence of an object that was proven to exist but that has not been explicitly constructed to this day? Here object could be any mathematical object, such as a number, function, algorithm, or even…
104
votes
9 answers
Division by $0$ and its restrictions
Consider the following expression: $$\frac{1}{2} \div \frac{4}{x}$$ Over here, one would state the restriction as $x \neq 0 $, as that would result in division by $0$. But if we rearrange the expression, then: $$\begin{align} \frac12\div\frac4x &=…
Devansh Sharma
- 1,145
104
votes
4 answers
How is a group made up of simple groups?
I've read more than once the analogy between simple groups and prime numbers, stating that any group is built up from simple groups, like any number is built from prime numbers. I've recently started self-studying subgroup series, which is supposed…
Bruno Stonek
- 13,319
104
votes
4 answers
Probability density function vs. probability mass function
I've a confession to make. I've been using PDF's and PMF's without actually knowing what they are. My understanding is that density equals area under the curve, but if I look at it that way, then it doesn't make sense to refer to the "mass" of a…
0x0
- 2,711
104
votes
13 answers
Limit of sequence in which each term is defined by the average of preceding two terms
We have a sequence of numbers $x_n$ determined by the equality $$x_n = \frac{x_{n-1} + x_{n-2}}{2}$$ The first and zeroth term are $x_1$ and $x_0$.The following limit must be expressed in terms of $x_0$ and $x_1$ $$\lim_{n\rightarrow\infty} x_n $$…
Ananth Kamath
- 2,313
103
votes
11 answers
Mathematician vs. Computer: A Game
A mathematician and a computer are playing a game: First, the mathematician chooses an integer from the range $2,...,1000$. Then, the computer chooses an integer uniformly at random from the same range. If the numbers chosen share a prime factor,…
user139000