For questions on determining whether a number is rational, and related problems. If applicable, use this tag instead of (rational-numbers) and (irrational-numbers). Consider adding a tag (radicals) or (logarithms), depending on what the question is about.
Questions tagged [rationality-testing]
371 questions
6
votes
3 answers
Is this proof correct (Rationality of a number)?
Is $\sqrt[3] {3}+\sqrt[3]{9} $ a rational number? My answer is no, and there is my proof. I would like to know if this is correct:
Suppose this is rational. So there are positive integers $m,n$ such that…
Omer
- 2,490
5
votes
2 answers
Spivak's Calculus: chapter 2, problem 18(c)
In Spivak's calculus book, I cannot understand the solution proposed for question (c) of problem 18 in chapter 2:
Prove that $\sqrt{2}+\sqrt[3]{2}$ is irrational. Hint: Start by
working out the first 6 powers of this number.
Working out the powers…
Alexandros
- 224
5
votes
0 answers
Are there any proofs for (ir)rationality of the numbers $\sin(e)$, $\cos(e)$?
Are there any proofs for the (ir)rationality of the numbers $\sin(e)$, $\cos(e)$, $\tan(e)$, and $\cot(e)$?
Thanking in advance for any references.
M.R. Yegan
- 672
4
votes
1 answer
Prove $2^{1/p} - 2^{1/q}$ is irrational
Prove $2^{1/p} - 2^{1/q}$ is irrational for $p, q, \in \mathbb{N}, p \neq q$.
Not sure how to begin, as the usual trick of taking $p^{th}$ powers doesn't seem to work very well.
vukov
- 1,555
2
votes
1 answer
Is it rational or irrational?
I am a mathematical putz - please be kind.
From what I know, a rational number is a non-imaginary number that can be written as $\frac{p}{q}$. A repeating number can also be a rational number (i.e. $\frac{1}{3} = 0.3333...$). Ok, so far, so good. …
Jaco Van Niekerk
- 135
2
votes
5 answers
Irrationality of $\sqrt{24}$
Please, help me show that $\sqrt{24}$ is irrational by contradiction. I tried to prove using the fact that $24=2×2×2×3$ and that $\sqrt2$ and $\sqrt3$ are irrational, but the product of irrationals isn't always irrational.
Aaron
- 31
2
votes
1 answer
Is $\sqrt{2}^{\sqrt{2}}$ rational?
In §2.2 of her essay on mathematical morality, Eugenia Cheng includes the following example:
Why is it possible for an irrational to the power of an irrational to be rational?
Here is a nice little proof that it is possible:
Consider…
E.P.
- 2,481
- 17
- 32
1
vote
1 answer
If $\log_3(A)$ is rational, can $\log_3(A+1)$ be rational? (Where $A$ is a positive real number.)
Question:
If $\log_3(A)$ is rational, can $\log_3(A+1)$ be rational? (Where $A$ is a positive real number.)
My Working:
Starting by assuming
$$\log_3(A) = x \ {\rm{and}} \ \log_3(A+1) = y$$
it gives $$3^x = A \ {\rm {and}} \ 3^y = A + 1$$
Which…
user983206
0
votes
0 answers
Show that equation has no rational solutions
How do I prove that the equation $a^2-2b^2-3c^2+6d^2=0$ has no non trivial rational solutions?
What techniques are there to solve general problems like this?
Inspector gadget
- 45
- 1
- 7
0
votes
1 answer
If $a=b+c$ , $a^2$ is an integer, can $b^2$ or $c^2$ or both be irrational?
Say, $a=b+c$,
$a$ may be rational or irrational.
However, the constraint on $a$ is that $a^2$ is an integer.
b>0 , c>0 which means a>0.
Wanted to confirm that either $b^2$ or $c^2$ or both can't be irrational.
I reasoned like this :
$a^2…
0
votes
4 answers
For any positive integers $a$ and $b$, what relation must hold between them to ensure that $\dfrac{a}{b}$ is an irrational number?
When thinking about $\pi$, I came across this question myself, and don't know how to start approaching this problem.
I first thought that $a$ and $b$ must be prime numbers, but a simple calculation ($\dfrac{7}{3}$) tells me that this is not the…
Garmekain
- 125