Questions tagged [rationality-testing]
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.
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Compact Proof that the constant $\sum_{k=1}^\infty \frac{(-1)^{k-1}}{k.k!}$ is irrational
In an answer some time ago I referenced a short proof that the constant $e$ is irrational by A. R. G. MacDivitt and Yukio Yanagisawa (The Mathematical Gazette , Volume 71 , Issue 457 , October 1987 , ...
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3
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If $0<x<1$ and $x$ is rational, does $2^x$ have to be irrational? [closed]
If $0<x<1$ and $x$ is rational, does $2^x$ have to be irrational? Why? Also, if $2^x$ is rational, and $0<x<1$ and $x$ is rational, does $x$ have to be irrational? (i.e. contrapositive of ...
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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?
4
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2
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$ֿ\sqrt{2}$ is irrational. Proof by contradiction or Proof of Negation?
I am just learning about proofs in an introductory course. I came across an example of "proof by contradiction" (see attachment) about $ֿ\sqrt{2}$ being irrational. Some online sources have ...
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Prove $\sqrt{2}+\sqrt{3}+\sqrt{5}+...+\sqrt{p_{n}}$ is irrational, where $p_{n}$ is the nth prime.
My motivation is making general proof , instead of trying to prove special cases.
To which branch of mathematics does my question belong?
I am highly interested in irrational numbers.
Is it good idea ,...
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1
answer
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Prove that number $\sqrt{2}$ is an irrational number using this theorem "if $a^2$ is even, $a$ must be even"
I have to prove the following:
Prove $\sqrt{2}$ is an irrational number using this theorem "if $a^2$ is even, $a$ must be even"
I made a proof by contradiction for the statement above, but ...
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0
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Directions for a proof of the irrationality of $\pi$ from Archimedes' approximation
I'm a complete amateur with respect to mathematics, but I looked up a few proofs of the irrationality of $\pi$ and was unsatisfied by the lack of proofs that would be elementary enough to be able to ...
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0
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52
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Does the maximum entropy Nash equilibrium with integer payoffs have rational probabilities?
I have a symmetric two-player zero-sum game, represented as an $n \times n$ skew-symmetric payoff matrix $M$. The components of $M$ are all integers. Are the probabilities in the maximum entropy Nash ...
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Regarding the rationality of $T_k = x^k + \frac{1}{x^k}$, given $T_m$ and $T_n$ are rational for some $m,n \in \mathbb{N}$.
We define $T_k = x^k + \frac{1}{x^k}$,for some non zero real number $x$. Then one can look at the algebra generated by $T_k$. Its simple to see that they satisfy
$$T_k T_l = T_{k+l} + T_{|k-l|} $$
...
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3
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What are some techniques to prove that a set of numbers is rationally independent?
I am solving a math problem for fun and it amounts to proving that a specified (finite) set of numbers, each defined by an infinite series involving polynomials and the factorials, is a rationally ...
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Prove that when the denominator of a rational number is of the form $2^n * 5^m$ it is a terminating decimal
What is the proof for when the denominator of a rational number is of the form $2^n * 5^m$ it is a terminating decimal?
For example:
$7/8$, where $8$ is of the form $2^3 * 5^0$
Therefore, $7/8$ is a ...
1
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1
answer
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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 $...
user983206
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If $a=b+c$ , $a^2$ is an integer, can $b^2$ or $c^2$ or both be irrational? [closed]
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 ...
0
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1
answer
183
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Is $\frac{2 \cdot 4\cdot 8 \cdot 16 \cdots}{1\cdot 3\cdot 7\cdot15 \cdots}$ irrational?
The number is the infinite product:$$\frac{2 \cdot 4\cdot 8 \cdot 16 \cdots}{1\cdot 3\cdot 7\cdot15 \cdots},$$ or about $3.4628.$
I have only proven this number is finite. Can you either prove it is ...
3
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1
answer
192
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Proof that there is no rational number $r$ satisfying $2^r=3$ [duplicate]
Hello guys it would be a massive help for me if you could take a look at my proof and point out any errors. I want to know whether its coherent and whether I need more or less detail.
Suppose $2^r = 3$...
2
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5
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Prove or disprove a claim regarding irrational numbers
I am trying to prove the following claim:
Let $ 0\leq n \in \Bbb Z$ and suppose that there exists a $k \in \Bbb Z$ such that $n=4k+3$.
Prove or disprove: $\sqrt n \notin \Bbb Q$ .
The problem I am ...
2
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1
answer
245
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Question in details of the proof of irrationality of $\zeta (3)$
I am reading through Wadim Zudilin's lecture notes and at the beginning of page 76 (i.e. in last part of Lemma 5.9.) things are getting really confusing :
How $g(y) = g(0) − 2^{3/2}y^2 + O(y^3)$ is ...
user200918
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2
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Is it true or false for all instances of $m>0$, $\sqrt{m^2+1}$ is irrational?
Is it true or false for all instances of $m>0$, $\sqrt{m^2+1}$ is irrational?
My textbook gives an answer
This is true for integers $m$:
$$m>0\implies m^2<m^2+1<m^2+2m+1=(m+1)^2$$
Since $...
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Rationalizing an expression with different nth-roots in the denominator
I was trying to help a friend of mine rationalize the following expression:
$$\frac{b}{\sqrt[3]{a^4}-\sqrt[3]{c^2}} $$
Simply multiplying by the conjugate won't work, even with multiple iterations.
Is ...
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5
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Let $x^3+\frac{1}{x^3}$ and $x^4+\frac{1}{x^4}$ are rational numbers. Show that $x+\frac{1}{x}$ is rational.
$x^3+\dfrac{1}{x^3}$ and $x^4+\dfrac{1}{x^4}$ are rational numbers. Prove that $x+\dfrac{1}{x}$ is rational number.
My solution:
$x^3+\dfrac{1}{x^3}=\left(x+\dfrac{1}{x}\right)\left(x^2+\dfrac{1}{x^...
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1
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Is it possible for the rationality of a computable number to be unknowable?
Main question: Suppose we have some real number $x$ that is proven to be not only be computable, but has had such an algorithm explicitly found. Could it be the case that it may be impossible to prove ...
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Novel (?) proof of the irrationality of $\sqrt3$ [duplicate]
A student of mine offered the following proof of the irrationality of $\sqrt{3}$: Suppose $(a/b)^2 = 3$ with $a,b$ having no common factor. Since $a^2=3b^2$, an easy parity argument (using the fact ...
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Can an infinite product of rationals be rational if the partial products become increasingly irreducible?
I've been curious about methods of proving the irrationality of some infinite products and had this idea.
Suppose
$$\prod_{n=1}^ka_n=\frac{b_k}{c_k}$$
where $a_n\in\mathbb{Q}$ and $\gcd(b_k,c_k)=1$. ...
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0
answers
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If $\alpha$ is an irrational number, then the minimal polynomial of $\alpha$ over $\Bbb{Q}$ does not have a rational root?
I'm reading Ross' "Elementary Analysis". He shows the rational zeros theorem and a proof of it. And then there are some examples such as the following ones:
The basic idea is that testing ...
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Is $\frac{\text{area of circle with diameter $d$}}{\text{area of square with side $d$}}$ irrational?
I understand that the area of a circle (based on diameter) with respect to the area of a square (same side length as circle diameter) is approximately 0.7854 the area of the square. My question:
Is
$$...
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Proving that $\sqrt[3] 7 -\sqrt 2$ is irrational.
I understand proving that $\sqrt{7}-\sqrt {2}$ is irrational, but how does the answer change if its cube root of $7$ instead of square root?
the way I solve $\sqrt{7}-\sqrt {2}$ is by assuming its ...
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Prove that $ 4 + 3 \sqrt 2 $ is irrational
I am trying to prove by contradiction that $ 4 + 3 \sqrt 2 $ is irrational.
(I may use the fact that $ \sqrt 2 $ is irrational, and must use the definition of rational numbers.)
Proofs by ...
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4
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If $x + y \in \mathbb Q$ and $x − 3 y \in \mathbb Q$, prove $x$ and $y$ are rational
Consider $𝑥, 𝑦 ∈ ℝ$, such that $𝑥 + 𝑦 ∈ ℚ$ and $𝑥 − 3𝑦 ∈ ℚ$. Then $𝑥, 𝑦 ∈ ℚ$.
Hey all. I'm recently working through a course-book that's involved with the math course I'm taking next semester ...
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Can I use Euler's formula to prove a series converges to a rational value?
Given the following infinite series with rational $x < 1$ && integer $n \geqslant 2$:
$$1-\frac{x}{n}+\frac{(1+n)x^2}{2n^2}-\frac{(1+n)(1+2n)x^3}{6n^3}+\frac{(1+n)(1+2n)(1+3n)x^4}{24n^4}-\...
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Does my proof of the irrationality of $\ln(2)$ hold up?
At first, I thought that proving the irrationality of $\ln(2)$ was so easy that it was trivial. However, someone in the comments of the $115$ vote answer at Can an irrational number raised to an ...
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How to prove that $\frac{\ln 5}{\ln 2}$ is irrational
I have to show that $\frac{\ln 5}{\ln 2}$ is irrational
I have tried the following:
Assume it's rational so
$\frac{\ln 5}{\ln 2} = \frac{p}{q}$
which becomes
$\log _2\left(5\right)=\frac{p}{q}$
...
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2
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If b is rational and c is real, can $x^2 + bx + c$ have one rational root and one irrational root? Use only definition of rational numbers to prove.
If $b$ is rational and $c$ is real, can $x^2 + bx + c$ have one rational root and one irrational root? Use only the definition of rational numbers to prove.
I think that it can't because since $b$ is ...
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Why was Apery's constant believed to be rational?
Apéry's theorem states that Apéry's constant $\zeta(3) := \sum_{n=0}^\infty n^{-3}$ is irrational. The Wikipedia article claims that this result was "wholly unexpected" to the point that the ...
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Is $\sqrt[3]{7+5 \sqrt{2}}+\sqrt[3]{20-14 \sqrt{2}}$ a rational number?
Is there a way to show that
$$\alpha=\sqrt[3]{7+5 \sqrt{2}}+\sqrt[3]{20-14 \sqrt{2}}$$
is a rational number?
I found $\alpha=3$ from doing simplifications. But, I would like to known a different ...
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How do I show that $\log_2(k)$ is an irrational number if $k$ is an odd integer greater than $1$?
Show that if $k$ is an odd integer grater than $1$, then $\log_2(k)$ is an irrational number.
So my approach for this problem was to use proof by contradiction so I wrote "Assume by ...
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Two specific irrational numbers summing to rational number that is not zero?
There already exist pages that show how two irrational numbers can sum to a rational number. However, is there an actual example of this? What are some of irrational numbers $x$ and $y$ such that $x + ...
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About union of two algebraically independent sets
My question might be very simple for those that have a deep understanding for algebraically independent sets.
Definition 1.1. Let $F$ be an extension field of $K$ and $S$ a subset of $F$. $S$ is ...
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How to make sure that solutions of an equation are rational multiple of $\pi$ or not?
I have this two-variable equation
$$\frac{\left(3 y^2+1\right)^2 \left(2 \cosh \frac{26 \pi y}{15} \cosh x y-\cosh (2 \pi -x) y\right)}{9 \left(y^2-1\right)^2 \cosh (x+2 \pi ) y+8 \left(3 y^2-1\...
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If $(a-b^2)b>0$, then $\sqrt[3]{a+\frac{9b^3+a}{3b}\sqrt{\frac{a-b^3}{3b}}}+\sqrt[3]{a-\frac{9b^3+a}{3b}\sqrt{\frac{a-b^3}{3b}}}$ is rational
From Hardy´s "A course of pure mathematics" 10th edition, problem 31 miscellaneous problems of chapter I.
If $(a-b^2)b>0$, then
$$
\sqrt[3]{a+\frac{9b^3+a}{3b}\sqrt{\frac{a-b^3}{3b}}}+\...
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When is $\log_{b}(a)$ irrational, for natural numbers $a,b$?
I've seen this result usually given as "if $a$ and $b$ are coprime, then $\log_{b}(a)$ is irrational." Wolfram|Alpha even says it is "unknown" whether or not $\log_{12}(6)$ is ...
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Miscellaneous Problem Chapter I ex.22 on G.H.Hardy's book "A course of pure mathematics"
If all the values of x and y given by
$$
ax^{2}+2hxy+by^{2}=1, a'x^{2}+2h'xy+b'y^2=1
$$
(where $a, h, b, a', h', b'$ are rational) are rational, then
$$
(h-h')^2 - (a-a')(b-b'), (ab'-a'b)^2+4(ah'-a'h)(...
- 79
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2
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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 ...
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0
answers
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Why is the irrationality of $\gamma$ so ellusive? (Euler–Mascheroni constant)
Perhaps I'm in way over my head asking this; regardless, considering the abundance of known formulas and properties of $\gamma$, seemingly comparable in quantity to the likes of $\pi$ or $e$, why is ...
- 4,448
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votes
2
answers
145
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How To Show The Irrationality Of $0.0110001111...$
Show that $F=0.0110001111...$ is irrational, with $F$‘s decimal expansion being $1,3,5,7,9,...$ consecutive $0$s interspersed with $2,4,6,8,...$ consecutive $1$s.
A friend sent me this recently. The ...
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1
answer
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Determining the parity of the coefficients of a quadratic given that there is a rational solution
Que. If the quadratic $ax^2 + bx + c$ has a rational root, and $a$, $b,$ and $c$ are integers, then
A) at least one of $a, b, c$ is even
B) all of $a,b,c$ are even
C) at most one of $a,b,c$ is odd
D)...
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0
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35
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Prove rationality of a number by proof by contradiction
So I was going through proof of irrationality of $\sqrt2$ and tried to construct a proof whether $\sqrt4$ is rational by proof of contradiction.
Proof: (By Contradiction)
For the purposes of ...
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0
answers
39
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Showing some decimal representations are not rational
I have solutions for the following two problems and I am hoping to obtain some feedback on my solutions.
Does the number $0.123456789101112l314\ldots$ which is obtained by writing successively all ...
user675768
0
votes
2
answers
141
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Rational Zero Test of Polynomials
Say I'm trying to find the factors of
$r^3 - 3r^2 + 3r - 4 = 0$
I thought I could do the rational zero test by finding the factors of p and q and then doing $\frac{p}{q}$
So, in this case, $p = \pm ...
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votes
2
answers
461
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Proof that $\log_23 +\log_52$ is irrational number
Problem is to prove that
$$\log_23 +\log_52$$
is irrational number.
My attempt:
I try to write number like $$\log_23 +\frac{1}{\log_25}$$ but I didn't get anything(proof by contradiction). I also try ...
- 1,297
1
vote
0
answers
56
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Find all rational angles in degrees whose cosine is also rational [duplicate]
I know it is called Niven's Theorem, but I don't know how to prove it. I am looking for a clue or a direction for where to begin. Here is what I do know:
$\cos{\theta^
{\circ}}$ can be expanded to ...
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