For questions about the writing of proofs. But instead of focusing the mathematics behind the proof, these questions ask about the details and implementation of the proof.

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143
votes
13answers
6k views

Can every proof by contradiction also be shown without contradiction?

Are there some proofs that can only be shown by contradiction or can everything that can be shown by contradiction also be shown without contradiction? What are the advantage/disadvantages of proving ...
90
votes
9answers
4k views

Why do people use “it is easy to prove”?

Math is not generally what I am doing, but I have to read some literature and articles in dynamic systems and complexity theory. What I noticed is that authors tend to use (quite frequently) the ...
68
votes
14answers
4k views

Formal proof for $(-1) \times (-1) = 1$

Is there a formal proof for $(-1) \times (-1) = 1$? It's a fundamental formula not only in arithmetic but also in the whole of math. Is there a proof for it or is it just assumed?
65
votes
14answers
7k views

Why are mathematical proofs that rely on computers controversial?

There are many theorems in mathematics that have been proved with the assistance of computers, take the famous four color theorem for example. Such proofs are often controversial among some ...
54
votes
4answers
6k views

Can an irrational number raised to an irrational power be rational?

Can an irrational number raised to an irrational power be rational? If it can be rational, how can one prove it?
44
votes
3answers
6k views

Proof by contradiction vs Prove the contrapositive

What is the difference between a "proof by contradiction" and "proving the contrapositive"? Intuitive, it feels like doing the exact same thing. And when I compare an exercise, one person proves by ...
44
votes
2answers
1k views

On a long proof

On wikipedia there is a claim that the Abel–Ruffini theorem was nearly proved by Paolo Ruffini, and that his proof spanned over $500$ pages, is this really true? I don't really know much abstract ...
43
votes
13answers
3k views

What is a proof?

I am just a high school student, and I haven't seen much in mathematics (calculus and abstract algebra). Mathematics is a system of axioms which you choose yourself for a set of undefined entities, ...
41
votes
7answers
2k views

Could I be using proof by contradiction too much?

Lately, I've developed a habit of proving almost everything by contradiction. Even for theorems for which direct proofs are the clear choice, I'd just start by writing "Assume not" then prove it ...
38
votes
7answers
2k views

When working proof exercises from a textbook with no solutions manual, how do you know when your proof is sound/acceptable?

When working proof exercises from a textbook with no solutions manual, how do you know when your proof is sound/acceptable? Often times I "feel" as if I can write a proof to an exercise but most ...
36
votes
7answers
3k views

How does a non-mathematician go about publishing a proof in a way that ensures it to be up to the mathematical community's standards?

I'm a computer science student who is a maths hobbyist. I'm convinced that I've proven a major conjecture. The problem lies in that I've never published anything before and am not a mathematician by ...
32
votes
5answers
5k views

Are all prime numbers finite?

If we answer false, then there must be an infinite prime number. But infinity is not a number and we have a contradiction. If we answer true, then there must be a greatest prime number. But Euclid ...
31
votes
6answers
1k views

Why do we write proofs “forward?”

I am aware that this might turn into a discussion, but I have a feeling this might have an answer (maybe something historical?) instead. I'm hoping that those with speculations keep it in the ...
30
votes
5answers
3k views

Is it okay to reverse engineer proofs in homework questions?

In a linear algebra text book, one homework question I received was: Prove that $\mathbf{a \cdot b} = \frac{1}{4}(\|\mathbf{a + b}\|^2 - \|\mathbf{a - b}\|^2)$. Where $\mathbf{a}$ and ...
26
votes
2answers
680 views

English words in written mathematics

I recently marked over $100$ assignments for a multivariable calculus course. One question which a lot of people did poorly was proving a given set was open. Aside from issues relating to rigour and ...
23
votes
4answers
911 views

Are there any common practices in mathematics to guard against mistakes?

It occurred to me that math is somewhat like programming (or vice-versa, if you prefer) because, in both, it is easy to make mistakes or overlook them, and the smallest error or misguided assumption ...
22
votes
5answers
863 views

Level of Rigor in Mathematical Physics

I am a physics/math undergrad and I have recently become familiar with some more rigorous formalisms of mechanics, such as Lagrangian mechanics and Noether's Theorem. However, I've noticed that the ...
21
votes
10answers
1k views

Prove if $56x = 65y$ then $x + y$ is divisible by $11$

If $x$ and $y$ are natural numbers, and $56x = 65y$, prove that $x + y$ is divisible by $11$. I tried taking the $\gcd(56x,65y)$ using the Euclidean algorithm, but I got nowhere with it and do not ...
21
votes
5answers
3k views

Is a brute force method considered a proof?

Say we have some finite set, and some theory about a set, say "All elements of the finite set $X$ satisfy condition $Y$". If we let a computer check every single member of $X$ and conclude that the ...
21
votes
8answers
2k views

Why don't Venn diagrams count as formal proofs?

Just curious. If the purpose of a proof is to inform and persuade, why don't Venn diagrams count? Is it just convention or is there a more, umm, formal reason haha. Thanks!
21
votes
1answer
804 views

How to prove convergence of polynomials in $e$ (Euler's number)

These polynomials in $e$ converge to 2$$f(i)=e^i - i \sum_{k=1}^{i-1}\frac{(i-k)^{k-1}{e^{i-k}}{(-1)^{k+1}}}{k!}, \text{ where } i>1$$ This function goes to 2. I've calculated this with sage math ...
20
votes
7answers
2k views

LaTeX/TeX Vs. Mathematica for Typesetting

I know Mathematica like the back of my hand, but I do not know a speck of $\LaTeX$ or $\TeX$. With regards to mathematical typesetting, is there something significant I can do in $\LaTeX$/$\TeX$ that ...
19
votes
1answer
2k views

How would one be able to prove mathematically that $1+1 = 2$?

Is it possible to prove that $1+1 = 2$? Or rather, how would one prove this algebraically or mathematically?
19
votes
5answers
5k views

Use of “without loss of generality”

Why do we use "without loss of generality" when writing proofs? Is it necessary or convention? What "synonym" can be used? Thanks.
18
votes
6answers
2k views

What are some common proof strategies in mathematics?

I want to start out by saying that I am new at proof based mathematics. I am used to seeing patterns and using them to solve similar problems. However, I have found this is not a very good way to ...
18
votes
6answers
803 views

Difficulty in Mathematical Writing

Lots of people (including myself) face lot of problems in tackling Mathematical Problems, which appear as if we can solve it, but then writing out a solution becomes difficult. Let us consider some ...
17
votes
10answers
3k views

Proving $\sqrt 3$ is irrational.

There is a very simple proof by means of divisivility that $\sqrt 2$ is irrational. I have to prove that $\sqrt 3$ is irrational too, as a homework. I have done it as follows, ad absurdum: Suppose ...
17
votes
3answers
710 views

My first induction proof (very simple number theory). Please mark/grade.

What do you think about my first induction proof? Please mark/grade. Theorem The sum of the cubes of three consecutive natural numbers is a multiple of 9. Proof First, introducing a predicate ...
16
votes
7answers
3k views

how to be good at proving?

I'm starting my Discrete Math class, and I was taught proving techniques such as proof by contradiction, contrapositive proof, proof by construction, direct proof, equivalence proof etc. I know how ...
15
votes
10answers
2k views

Critiques on proof showing $\sqrt{12}$ is irrational.

My only exposure to proofs was in a math logic class I took in University. I was wondering if my attempt at proving that $\sqrt{12}$ is irrational is OK. $$\Big(\frac{m}{n}\Big)^2 = 12$$ ...
15
votes
3answers
622 views

prove $\sqrt{a_n b_n}$ and $\frac{1}{2}(a_n+b_n)$ have same limit

i am given this problem: let $a\ge0$,$b\ge0$, and the sequences $a_n$ and $b_n$ are defined in this way: $a_0:=a$, $b_0:=b$ and $a_{n+1}:= \sqrt{a_nb_n}$ and $b_{n+1}:=\frac{1}{2}(a_n+b_n)$ for all ...
15
votes
3answers
620 views

Starting sentences with mathematical symbols.

I apologise if this is a duplicate in any way or is too opinion-based. To what extent is it best not to start a sentence with a mathematical symbol? I find that when trying to solve a problem or ...
15
votes
4answers
657 views

Advice for writing good mathematics?

It's been a (far-fetched, possibly) goal of mine to some day write a math Textbook. I've been thinking about writing this question for a while, but reading an exceedingly mediocre text on Mathematical ...
15
votes
4answers
591 views

Prove $\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab + bc + ca$

Let $a,b,c$ are non-negative numbers, such that $a+b+c = 3$. Prove that $\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab + bc + ca$ Here's my idea: $\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab + bc + ca$ ...
14
votes
7answers
885 views

$\infty - \infty = 0$ ?

I am given this sequence with square root. $a_n:=\sqrt{n+1000}-\sqrt{n}$. I have read that sequence converges to $0$, if $n \rightarrow \infty$. Then I said, well, it may be because $\sqrt{n}$ goes to ...
14
votes
6answers
695 views

Can an algorithm be part of a proof?

I am an undergraduate student. I have read several papers in graph theory and found something may be strange: an algorithm is part of a proof. In the paper, except the last two sentences, other ...
14
votes
3answers
253 views

Showing that $1^k+2^k + \dots + n^k$ is divisible by $n(n+1)\over 2$

For any odd positive integer $k\geq1$, the sum $1^k+2^k + \dots + n^k$ is divisible by $n(n+1)\over 2$. I used induction principle for the solution but cannot prove it. I took $P(k) = ...
13
votes
7answers
962 views

Prove that the additive inverse of an odd integer is an odd integer

This is a homework problem, but I don't want the answer, just a little guidance: Prove that the additive inverse of an odd integer is an odd integer. When approaching a problem like this, how ...
13
votes
7answers
1k views

How do we know whether certain mathematical theorems are circular?

There are countless mathematical theorems and lemmata, some of which, obviously, depend on others. My question is: how do we know that, say, Theorem $A_1$- which uses a result proved in Theorem $A_2$ ...
13
votes
4answers
710 views

Prove that $x = 2$ is the unique solution to $3^x + 4^x = 5^x$ where $x \in \mathbb{R}$ [duplicate]

Yesterday, my uncle asked me this question: Prove that $x = 2$ is the unique solution to $3^x + 4^x = 5^x$ where $x \in \mathbb{R}$. How can we do this? Note that this is not a diophantine ...
13
votes
3answers
656 views

Counterexamples to proofs of correct statements

This question is in part inspired by a quote I saw in an answer to another question: The problem with incorrect proofs to correct statements is that it is hard to come up with a counterexample. ...
13
votes
2answers
437 views

Proofs from the “Ugly Book”

There is a famous saying in mathematics from Paul Erdős: "You don't have to believe in God, but you should believe in The Book." "The Book" is an imaginary book in which God had written down the best ...
13
votes
1answer
186 views

Why are proofs written in first person plural? Were they ever written differently?

It's probably a silly question but it interests me when was the convention of writing proofs in first person plural introduced? Is there any historical examples of a different POV for proof writing?
13
votes
2answers
215 views

Question from Putnam '89: Primes of the form $101\ldots01$

I'm not a math major, but would like to compete in the Putnam. As suggested in other questions here, I'm working some old contest problems. I'd like some input on this attempted proof--general input ...
12
votes
8answers
8k views

How do I prove that a function is well defined?

How do you in general prove that a function is well-defined? $$f:X\to Y:x\mapsto f(x)$$ I learned that I need to prove that every point has exactly one image. Does that mean that I need to prove the ...
12
votes
4answers
330 views

Tips for writing proofs

When writing formal proofs in abstract and linear algebra, what kind of jargon is useful for conveying solutions effectively to others? In general, how should one go about structuring a formal proof ...
12
votes
1answer
371 views

Is there such a thing as a mathematical thesaurus?

I want this for two reasons: When writing proofs, I am constantly in need of synonyms of basic words like thus, there exists, for all, such as, contains, etc. A lot of mathematical concepts have ...
12
votes
2answers
297 views

Prove that the multiplicative groups $\mathbb{R} - \{0\}$ and $\mathbb{C} - \{0\}$ are not isomorphic.

Is my proof correct? I have made use of the fact isomorphism preserves order of elements, which I proved couple of exercises back. I am also interested in other ways of proving it. Is there a more ...
11
votes
5answers
574 views

Finite Abelian Group

Let $G$ be a finite abelian group, $G = \{e, a_{1}, a_{2}, ..., a_{n} \}$. Prove that $(a_{1}a_{2}\cdot \cdot \cdot a_{n})^{2}$ = $e$. I've been stuck on this problem for quite some time. Could ...
11
votes
6answers
2k views

Proof that there are infinitely many positive rational numbers smaller than any given positive rational number.

I'm trying to prove this statement:- "Let $x$ be a positive rational number. There are infinitely many positive rational numbers less than $x$." This is my attempt of proving it:- Assume that ...