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1500 questions
121
votes
7 answers
What would base $1$ be?
Base $10$ uses these digits: $\{0,1,2,3,4,5,6,7,8,9\};\;$ base $2$ uses: $\{0,1\};\;$ but what would base $1$ be? Let's say we define Base $1$ to use: $\{0\}$. Because $10_2$ is equal to $010_2$, would all numbers be equal? The way I have thought…
Justin
- 2,719
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
- 1,275
121
votes
7 answers
Implies ($\Rightarrow$) vs. Entails ($\models$) vs. Provable ($\vdash$)
Consider A $\Rightarrow$ B, A $\models$ B, and A $\vdash$ B. What are some examples contrasting their proper use? For example, give A and B such that A $\models$ B is true but A $\Rightarrow$ B is false. I'd appreciate pointers to any tutorial-level…
user287424
- 1,329
121
votes
10 answers
Good 1st PDE book for self study
What is a good PDE book suitable for self study? I'm looking for a book that doesn't require much prerequisite knowledge beyond undergraduate-level analysis. My goal is to understand basic solutions techniques as well as some basic theory.
Mykie
- 7,285
121
votes
6 answers
How Do You Actually Do Your Mathematics?
Better yet, what I'm asking is how do you actually write your mathematics? I think I need to give brief background: Through most of my childhood, I'd considered myself pretty good at math, up through the high school level. I easily followed…
Uticensis
- 3,401
121
votes
6 answers
Why don't analysts do category theory?
I'm a mathematics student in abstract algebra and algebraic geometry. Most of my books cover a great deal of category theory and it is an essential tool in understanding these two subjects. Recently, I started taking some functional analysis courses…
gary
- 1,211
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 'general…
userX
- 2,049
120
votes
16 answers
Do mathematicians, in the end, always agree?
I've been trying to study some different sciences in my life, ranging from biology to mathematics, and if I try to explain to people why I like mathematics above the others, I think the most important reason for me is that mathematicians, in the…
Kasper
- 14,040
- 18
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- 129
120
votes
2 answers
Determinant of a non-square matrix
I wrote an answer to this question based on determinants, but subsequently deleted it because the OP is interested in non-square matrices, which effectively blocks the use of determinants and thereby undermined the entire answer. However, it can be…
goblin GONE
- 69,755
120
votes
16 answers
Ways to evaluate $\int \sec \theta \, \mathrm d \theta$
The standard approach for showing $\int \sec \theta \, \mathrm d \theta = \ln|\sec \theta + \tan \theta| + C$ is to multiply by $\dfrac{\sec \theta + \tan \theta}{\sec \theta + \tan \theta}$ and then do a substitution with $u = \sec \theta + \tan…
Mike Spivey
- 57,148
120
votes
15 answers
The math behind Warren Buffett's famous rule – never lose money
This is a question about a mathematical concept, but I think I will be able to ask the question better with a little bit of background first. Warren Buffett famously provided 2 rules to investing: Rule No. 1: Never lose money. Rule No. 2: Never…
120
votes
1 answer
Finding primes so that $x^p+y^p=z^p$ is unsolvable in the $p$-adic units
On my number theory exam yesterday, we had the following interesting problem related to Fermat's last theorem: Suppose $p>2$ is a prime. Show that $x^p+y^p=z^p$ has a solution in $\mathbb{Z}_p^{\times}$ if and only if there exists an integer $a$…
ArtW
- 3,591
120
votes
26 answers
If squaring a number means multiplying that number with itself then shouldn't taking square root of a number mean to divide a number by itself?
If squaring a number means multiplying that number with itself then shouldn't taking square root of a number mean to divide a number by itself? For example the square of $2$ is $2^2=2 \cdot 2=4 $ . But square root of $2$ is not $\frac{2}{2}=1$ .
bluebellae
- 1,643
120
votes
18 answers
Fastest way to meet, without communication, on a sphere?
I was puzzled by a question my colleague asked me, and now seeking your help. Suppose you and your friend* end up on a big sphere. There are no visual cues on where on the sphere you both are, and the sphere is way bigger than you two. There are no…
Rob Audenaerde
- 1,245
119
votes
11 answers
Am I just not smart enough?
When I was doing math, let us say for example, introductory number theory, it seems to take me a lot of time to fully understand a theorem. By understanding, I mean, both intuitively and also rigorously (know how to prove or derive). However, I…
Kun
- 2,566