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1500 questions
197
votes
12 answers
What is $dx$ in integration?
When I was at school and learning integration in maths class at A Level my teacher wrote things like this on the board.
$$\int f(x)\, dx$$
When he came to explain the meaning of the $dx$, he told us "think of it as a full stop". For whatever reason…
Sachin Kainth
- 6,643
196
votes
0 answers
Sorting of prime gaps
Let $g_i$ be the $i^{th}$ prime gap $p_{i+1}-p_i.$
If we rearrange the sequence $ (g_{n,i})_{i=1}^n$ so that for any finite $n$, if the gaps are arranged from smallest to largest, we have a new sequence $(\hat{g}_{n,i})_{i=1}^n.$
For example, for $n…
daniel
- 10,501
196
votes
22 answers
List of Interesting Math Blogs
I have the one or other interesting Math blog in my feedreader that I follow. It would be interesting to compile a list of Math blogs that are interesting to read, and do not require research-level math skills.
I'll start with my entries:
Division…
195
votes
16 answers
What's the intuition behind Pythagoras' theorem?
Today we learned about Pythagoras' theorem. Sadly, I can't understand the logic behind it.
$A^{2} + B^{2} = C^{2}$
$C^{2} = (5 \text{ cm})^2 + (7 \text{ cm})^2$
$C^{2} = 25 \text{ cm}^2 + 49 \text{ cm}^2$
$C^{2} = 74 \text{ cm}^2$
${x} =…
user123399
- 289
195
votes
14 answers
Intuition behind Matrix Multiplication
If I multiply two numbers, say $3$ and $5$, I know it means add $3$ to itself $5$ times or add $5$ to itself $3$ times.
But If I multiply two matrices, what does it mean ? I mean I can't think it in terms of repetitive addition.
What is the…
Happy Mittal
- 3,307
195
votes
9 answers
How to define a bijection between $(0,1)$ and $(0,1]$?
How to define a bijection between $(0,1)$ and $(0,1]$?
Or any other open and closed intervals?
If the intervals are both open like $(-1,2)\text{ and }(-5,4)$ I do a cheap trick (don't know if that's how you're supposed to do it):
I make a…
user1411893
- 2,173
195
votes
7 answers
What is the difference between "singular value" and "eigenvalue"?
I am trying to prove some statements about singular value decomposition, but I am not sure what the difference between singular value and eigenvalue is.
Is "singular value" just another name for eigenvalue?
Ramon
- 1,959
194
votes
25 answers
Can a coin with an unknown bias be treated as fair?
This morning, I wanted to flip a coin to make a decision but only had an SD card:
Given that I don't know the bias of this SD card, would flipping it be considered a "fair toss"?
I thought if I'm just as likely to assign an outcome to one side as…
Andrew Cheong
- 2,525
192
votes
31 answers
Proving the identity $\sum_{k=1}^n {k^3} = \big(\sum_{k=1}^n k\big)^2$ without induction
I recently proved that
$$\sum_{k=1}^n k^3 = \left(\sum_{k=1}^n k \right)^2$$
using mathematical induction. I'm interested if there's an intuitive explanation, or even a combinatorial interpretation of this property. I would also like to see any…
Fernando Martin
- 6,027
192
votes
2 answers
What are examples of functions with "very" discontinuous derivative?
Could someone give an example of a ‘very’ discontinuous derivative? I myself can only come up with examples where the derivative is discontinuous at only one point. I am assuming the function is real-valued and defined on a bounded interval.
user58273
- 2,037
192
votes
7 answers
Is the product of two Gaussian random variables also a Gaussian?
Say I have $X \sim \mathcal N(a, b)$ and $Y\sim \mathcal N(c, d)$. Is $XY$ also normally distributed?
Is the answer any different if we know that $X$ and $Y$ are independent?
jamaicanworm
- 4,694
- 11
- 36
- 53
191
votes
26 answers
Looking for an intuitive explanation why the row rank is equal to the column rank for a matrix
I am looking for an intuitive explanation as to why/how row rank of a matrix = column rank. I've read the proof on Wikipedia and I understand the proof, but I don't "get it". Can someone help me out with this ?
I find it hard to wrap my head around…
hari_sree
- 2,171
190
votes
29 answers
Good book for self study of a First Course in Real Analysis
Does anyone have a recommendation for a book to use for the self study of real analysis? Several years ago when I completed about half a semester of Real Analysis I, the instructor used "Introduction to Analysis" by Gaughan.
While it's a good book,…
CritChamp
- 229
188
votes
10 answers
Is $0$ a natural number?
Is there a consensus in the mathematical community, or some accepted authority, to determine whether zero should be classified as a natural number?
It seems as though formerly $0$ was considered in the set of natural numbers, but now it seems more…
bryn
- 10,045
188
votes
9 answers
There are apparently $3072$ ways to draw this flower. But why?
This picture was in my friend's math book:
Below the picture it says:
There are $3072$ ways to draw this flower, starting from the center of
the petals, without lifting the pen.
I know it's based on combinatorics, but I don't know how to…
user265554
- 2,933