# 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 , ...
183 views

### 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 ...
39 views

### 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?
206 views

### $ֿ\sqrt{2}$ 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 ...
1 vote
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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 ,...
1 vote
91 views

### 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 ...
71 views

### 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 ...
1 vote
52 views

### 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 ...
47 views

### 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 $$T_k T_l = T_{k+l} + T_{|k-l|}$$ ...
182 views

### 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 ...
1 vote
143 views

### 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 vote
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1 vote
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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: $$\frac{b}{\sqrt{a^4}-\sqrt{c^2}}$$ Simply multiplying by the conjugate won't work, even with multiple iterations. Is ...
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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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### 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 ...
1 vote
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)(... 5 votes 2 answers 711 views ### 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{2} is irrational. Hint: Start by working out the first ... 2 votes 0 answers 197 views ### 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 ... 0 votes 2 answers 145 views ### 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 0s interspersed with 2,4,6,8,... consecutive 1s. A friend sent me this recently. The ... 1 vote 1 answer 204 views ### 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)... 0 votes 0 answers 35 views ### 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 ... 0 votes 0 answers 39 views ### 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 ... 0 votes 2 answers 141 views ### 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 ... 9 votes 2 answers 461 views ### 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 ...
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 ...