Questions tagged [proof-writing]

For questions about the formulation of a proof. This tag should not be the only tag for a question and should not be used to ask for a proof of a statement.

314
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14answers
18k 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 ...
154
votes
9answers
12k 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 ...
121
votes
10answers
19k views

Is the blue area greater than the red area?

Problem: A vertex of one square is pegged to the centre of an identical square, and the overlapping area is blue. One of the squares is then rotated about the vertex and the resulting overlap is ...
100
votes
5answers
23k 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?
100
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12answers
11k 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 ...
96
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10answers
12k views

Does notation ever become “easier”?

I'm in my first semester of college going for a math major and it's pretty great. I'm doing well, however, there seems to be huge gap between how difficult /complex an idea is and how convoluted it is ...
84
votes
14answers
13k 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?
69
votes
12answers
6k views

Will assuming the existence of a solution ever lead to a contradiction?

I'm reading Manfredo Do Carmo's differential geometry book. In section 1-7, he discusses the "Isoperimetric Inequality" which is related to the question of what 2-dimensional shape maximizes the ...
69
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9answers
5k views

Is it bad form to write mysterious proofs without explaining what one intends to do? [closed]

Often when doing assignments, I find myself deliberately writing in a "mysterious" way. By this I mean that the reader usually will not understand what exactly is going on and what for, until the very ...
67
votes
4answers
4k views

A strange integral: $\int_{-\infty}^{+\infty} {dx \over 1 + \left(x + \tan x\right)^2} = \pi.$

While browsing on Integral and Series, I found a strange integral posted by @Sangchul Lee. His post doesn't have a response for more than a month, so I decide to post it here. I hope he doesn't mind ...
66
votes
15answers
33k views

Prove if $n^2$ is even, then $n$ is even.

I am just learning maths, and would like someone to verify my proof. Suppose $n$ is an integer, and that $n^2$ is even. If we add $n$ to $n^2$, we have $n^2 + n = n(n+1)$, and it follows that $n(n+1)...
65
votes
11answers
9k views

Is it technically incorrect to write proofs forward?

A question on an assignment was similar to prove: $$2a^2-7ab+2b^2 \geq -3ab.$$ and my proof was: $$2a^2-4ab+2b^2\geq0$$ $$a^2-2ab+b^2\geq0$$ $$(a-b)^2\geq0$$ which is true. However, my professor ...
57
votes
8answers
10k 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? [closed]

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 ...
57
votes
14answers
5k 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, ...
56
votes
7answers
4k 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 ...
53
votes
2answers
2k 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 ...
51
votes
9answers
14k views

Is an irrational number odd or even?

My sister just asked this question to me: "Is an irrational number odd or even?" I told her that decimals are not odd or even and that does imply that not recurring and non repeating decimals will ...
51
votes
13answers
12k views

How can you prove that the square root of two is irrational?

I have read a few proofs that $\sqrt{2}$ is irrational. I have never, however, been able to really grasp what they were talking about. Is there a simplified proof that $\sqrt{2}$ is irrational?
51
votes
7answers
30k 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 ...
51
votes
5answers
29k 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?
48
votes
8answers
6k views

Systems of linear equations: Why does no one plug back in?

When someone wants to solve a system of linear equations like $$\begin{cases} 2x+y=0 \\ 3x+y=4 \end{cases}\,,$$ they might use this logic: $$\begin{align} \begin{cases} 2x+y=0 \\ 3x+y=4 \end{cases}...
46
votes
3answers
9k views

Where is the flaw in this “proof” of the Collatz Conjecture?

Edit I've highlighted the area in the proof where the mistake was made, for the benefit of anyone stumbling upon this in the future. It's the same mistake, made in two places: This has proven the ...
43
votes
6answers
3k 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 comments....
42
votes
2answers
3k views

The 'Factorialth Root'

I was dealing with the following question, given by my friend: Let $\xi(x)=\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{\cdots}}}}$ Define the series $X$ as $\xi(1),\xi(2),\xi(3),\dots$ Find $n$ for ...
42
votes
7answers
3k 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 of ...
41
votes
7answers
61k 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 ...
40
votes
3answers
7k 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?
39
votes
7answers
6k views

How to attack “if true, prove it; if not true, give a counterexample” question?

I am taking a basic analysis course. This is a general question that I often encounter in weekly homework. How should we start to attack this type of question: if the statement is true, prove it; if ...
38
votes
4answers
3k views

Is this a new method for finding powers?

Playing with a pencil and paper notebook I noticed the following: $x=1$ $x^3=1$ $x=2$ $x^3=8$ $x=3$ $x^3=27$ $x=4$ $x^3=64$ $64-27 = 37$ $27-8 = 19$ $8-1 = 7$ $19-7=12$ $37-19=18$ $18-...
38
votes
3answers
57k views

Prove: If a sequence converges, then every subsequence converges to the same limit.

I need some help understanding this proof: Prove: If a sequence converges, then every subsequence converges to the same limit. Proof: Let $s_{n_k}$ denote a subsequence of $s_n$. Note that $n_k \...
37
votes
8answers
5k views

Should a mathematical proof be 'convincing'?

I just read a description of what is a mathematical proof in my mathematical logic textbook, and I'm a bit puzzled by it. It goes like this: A mathematical proof is a finite sequence of mathematical ...
37
votes
5answers
3k 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 ...
37
votes
4answers
5k views

What really is mathematical rigor? How can I be more rigorous?

I'm an undergraduate mathematics student who has received some constructive feedback from two instructors at the end of my exams. Namely, that I am a bit hand-wavey and not always very rigorous. While ...
37
votes
1answer
1k views

The art of proof summarizing. Are there known rules, or is it a purely common sense matter?

When a proof is long and difficult, it can be really nice vis-à-vis the reader to give a summary or an outline of the deduction before beginning hard work. Are there known rules to give a good proof ...
36
votes
2answers
13k views

How to efficiently use a calculator in a linear algebra exam, if allowed

We are allowed to use a calculator in our linear algebra exam. Luckily, my calculator can also do matrix calculations. Let's say there is a task like this: Calculate the rank of this matrix: $...
36
votes
3answers
3k views

How can I answer this Putnam question more rigorously?

Given real numbers $a_0, a_1, ..., a_n$ such that $\dfrac {a_0}{1} + \dfrac {a_1}{2} + \cdots + \dfrac {a_n}{n+1}=0,$ prove that $a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n=0$ has at least one real ...
35
votes
8answers
172k views

What is the integral of 0?

I am trying to convince my friend that the integral of $0$ is $C$, where $C$ is an arbitrary constant. He can't seem to grasp this concept. Can you guys help me out here? He keeps saying it is $0$.
35
votes
1answer
957 views

Can a number be palindrome in 4 consecutive number bases?

$4$ consecutive bases? Are there numbers that are a palindrome in $4$ consecutive number bases? Note that I'm not counting one digit palindromes, since one digit numbers $x$ are trivially ...
33
votes
4answers
5k views

Why do proof authors use natural language sentences to write proofs?

I haven't read very many proofs. The majority of the ones that I've read, I've read in my first-year proofs textbook. Nevertheless, its first chapter expatiates on the proper use of English in ...
33
votes
10answers
6k 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!
33
votes
5answers
4k 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 $\mathbf{b}$...
33
votes
3answers
1k 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 ...
31
votes
6answers
4k views

When has one sufficiently mastered an area of mathematics?

This is a rather soft question regarding the mastery of various mathematical subjects, such as undergraduate subjects. In particular, say, when has one mastered undergraduate analysis? Is it ...
30
votes
2answers
49k views

What is the proof that covariance matrices are always semi-definite?

Suppose that we have two different discreet signal vectors of $N^\text{th}$ dimension, namely $\mathbf{x}[i]$ and $\mathbf{y}[i]$, each one having a total of $M$ set of samples/vectors. $\mathbf{x}[m]...
28
votes
2answers
14k views

Proof: How many digits does a number have? $\lfloor \log_{10} n \rfloor +1$

I read recently that you can find the number of digits in a number through the formula $\lfloor \log_{10} n \rfloor +1$ What's the logic behind this rather what's the proof?
27
votes
3answers
3k views

In proofs, are “for each” and “for any” synonyms?

In proofs, are "for each" and "for any" synonyms? Or some context is usually required to determine this?
27
votes
7answers
7k views

Can you use both sides of an equation to prove equality?

For example: $\color{red}{\text{Show that}}$$$\color{red}{\frac{4\cos(2x)}{1+\cos(2x)}=4-2\sec^2(x)}$$ In high school my maths teacher told me To prove ...
27
votes
4answers
3k views

Can't find mistake in an easy proof.

Consider the following theorem. $\textbf{Theorem:}$ for any sets $A, B, C, D$, if $A \times B \subseteq C \times D$ then $A \subseteq C$ and $B \subseteq D$. Then the following proof is given. $\...
26
votes
15answers
3k views

Is there more to explain why a hypothesis doesn't hold, rather than that it arrives at a contradiction?

Yesterday, I had the pleasure of teaching some maths to a high-school student. She wondered why the following doesn't work: $\sqrt{a+b}=\sqrt{a}+\sqrt{b}$. I explained it as follows (slightly less ...
26
votes
12answers
4k views

How do I make a student understand contradiction?

We were trying to prove that if $3p^2=q^2$ for nonnegative integers $p$ and $q$, then $3$ divides both $p$ and $q$. I finished writing the solution (using Euclid's lemma) when a student asked me "...