For posts seeking explanation or clarification of a specific step in a proof. "Please explain this proof" is off topic (too broad, missing context). Instead, the question must identify precisely which step in the proof requires explanation, and why so. This should not be the only tag for a question, and should not be used to circumvent site policies regarding duplicate questions.
Questions tagged [proof-explanation]
11824 questions
47
votes
5 answers
In a proof by contradiction, how do we know the assumption is the cause of the contradiction?
In a proof by contradiction, how do we know the assumption is the cause of the contradiction? And not just the result of some other property more fundamental to numbers?
In other words, how can we be sure we arrived at the contradiction because…
Stephen
- 3,682
12
votes
3 answers
How can we prove that in order to the product to be maximum, the addends must be equal?
Let me explain my deal. We can split any positive number into addends, and then we can take those addends and use them now as factors. The product will be maximum if all addends were equal.
E.g., the number 10 can be split apart as 9 + 1, or 8 +…
Alexander
- 325
7
votes
2 answers
Deriving proof that $x^2 = 2^x$ has three real solutions
I'm new to proofs and although I find this proof fascinating, I'm hoping someone could help me understand how the seasoned mathematician arrived at this result... I'm assuming it involves some abstraction of the Lambert W function.
This is a…
ClownInTheMoon
- 1,987
5
votes
1 answer
Dividing up a fortune
I am trying to solve this below problem:
At his death, a millionaire left his 10 children a million dollars in cash, all in $\$100$, $\$10$, $\$1$ bills,10-cent, and 1-cent coins. Show that there is a way for them to split the fortune into ten…
5
votes
1 answer
Why is $e^π$ transcendental?
the title and the tags might be correct for the question.
In the wikipedia page about gelfond's constant I saw that $e^{π}$ is transcendental and the proof was
$e^{π}=(e^{iπ})^{-i}=(-1)^{-i}$
Since $-1$ and $-i$ is algebraic and $-i$ is not…
Rounak Sarkar
- 2,463
5
votes
2 answers
Prove $x=\sqrt{3} - \sqrt{2}$ is irrational
The question is: prove $x=\sqrt{3} - \sqrt{2}$ is not rational.
I can "prove" the above (ie. I saw the answer in my book) but can't quite understand it.
$x = \sqrt{3} - \sqrt{2}$,
$x+\sqrt{2}=\sqrt{3}$,
$(x + \sqrt{2})^2 = 3$,
$x^2+2\sqrt{2}x+2 =…
5
votes
4 answers
Proof of Theorem: The principle of mathematical induction
I have a question from my textbook and wanted to make sure that I understood it. I have marked in a green box the question that I have.
ALEXANDER
- 2,099
4
votes
2 answers
Don't understand a line for the proof of $\displaystyle \sum_{1≤k≤n}\frac{\sin{kx}}{k}≥0$
In this math exchange post Inequality $\sum\limits_{1\le k\le n}\frac{\sin kx}{k}\ge 0$ (Fejer-Jackson), in the answer posted by River Li (the third one), they write "We have $\sin{mz}<0$ due to the smallestness of $m$". Since $z\in{(0,\pi)}$, I…
Pen and Paper
- 1,371
4
votes
2 answers
For all $n \in Z$, $n^2+2n+3$ is even if and only if $4 | (n^2−2n− 7)$.
I understand that this is a bi-conditional proof and have been able to successfully solve the backwards direction, but am having difficulty proving the forwards direction. This is my work I have done for my attempt at a direct proof:
For the first…
4
votes
2 answers
Is this proof of Euler's Formula circular?
I am looking at this proof (in the image) from a textbook for engineering mathematics:
I don't understand this proof, and my reasoning is as follows:
Known True Statements:
$z=r(cos\theta + isin\theta)$
$z_1z_2 = r_1r_2[cos(\theta_1 + \theta_2) +…
masiewpao
- 2,217
4
votes
2 answers
Can someone please explain why this proof using strong induction makes intuitive sense?
Theorem: Every integer $n>1$ is either prime or a product of primes.
Proof: By strong induction.
Let $n$ be an arbitrary natural number greater than 1.
Inductive hypothesis: Assume that for every integer $1
IgnorantCuriosity
- 1,423
4
votes
4 answers
Proof by contradiction concerning prime numbers. Where is the contradiction?
Statement to be proved:
Prove that if $n > 1$ is not divisible by any prime number $p$ where $p \le \sqrt{n}$ then $n$ is a prime number.
Suppose we assume that $n$ is composite. We then prove that $n$ is divisible by a prime number $\le…
sammy
- 523
3
votes
4 answers
How do I show that the sum of the reciprocals of perfect cubes is less than 3/2?
I have done research to find that this value is known as Apery's constant, but I haven't found anything about calculating it using purely math.
I have tried to find patterns between the numbers but have come up with nothing.
Please help and explain…
user1172200
3
votes
1 answer
Confusion on proof strategy: steps are reversible
There are two proofs I'm trying to compare. I asked about one of them here: Proof organization: proving the set of functions $f: \mathbb{R} \to \mathbb{R}$ is a direct sum of even and odd functions, but am posting a new question because that was a…
JohnT
- 1,368
3
votes
3 answers
Why are these triangles similar?
I'm lookoing at leonbloy's answer here: Intuitive understanding of the derivatives of $\sin x$ and $\cos x$
Can somebody explain to me why $\phi=\theta$? Why is it that the triangles in question are similar? Thank you
user637978