# Tagged Questions

Questions about real numbers not expressible as the quotient of two integers. For questions on determining whether a number is irrational, use the (rationality-testing) tag instead.

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### Is the value of $\log_27$ a rational number?

Is $\log_27$ a rational number?
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### Proof that the union of rational and irrational numbers sets is a set of real numbers [duplicate]

I see it all the time but is there a nice way to show that this is true? Or is this just a definition? I know that $\mathbb{Q} \subset \mathbb{R}$ and $\mathbb{I} \subset \mathbb{R}$, but how do we ...
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### Show that $7^{\sqrt {5}}>5^{\sqrt {7}}.$

Show that $7^{\sqrt {5}}>5^{\sqrt {7}}.$ I am stuck in this problem. Any help in solving this problem will be appreciated.
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### Rationalising a fraction with a surd

The given fraction is: $$\frac{2}{1+\sqrt5}$$ Can someone explain to me how to rationalise this (in steps - GCSE Level)? My only idea is to mutliply the top and bottom by $1+\sqrt5$ ?? TIA.
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### Understanding a medieval approximation

A medieval text (Maimonides's commentary to chapter 2 of Eruvin in my retranslation from the Hebrew) discusses a rectangle whose area is $5000$ square cubits. It reads in relevant part: … that the ...
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### Which numbers are necessary?

The Greeks were initially convinced that all numbers were rational until upon pain of contradiction were forced to accept that $\sqrt{2}$ was irrational and needed to be included in our number system ...
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### Rational Question for $a + b$ and Irrationality of $a^2 + b^2$

I have looked into the question and need help. Find some $a,b$ ${\in}$ $\mathbb{R}$ such that $a + b$ ${\in}$ $\mathbb{Q}$, $a^2 + b^2 \not\in \mathbb{Q}$, and $\frac{a}{2} < b < a$. Or prove ...
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### Is this a valid argument for proving that a sum of reciprocals is irrational?

Suppose we have a strictly increasing sequence of natural numbers. Suppose that the sum of the reciprocals of the elements converges. And suppose that the elements have infinitely many prime factors....
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### Is $\frac{1}{11}+\frac{1}{111}+\frac{1}{1111}+\cdots$ an irrational number?

Obviously: $$\frac{1}{10}+\frac{1}{100}+\frac{1}{1000}+\cdots=0.1111\dots=\frac{1}{9}$$ is a rational number. Now, if we make terms with demoninators in the form: $$q_n=\sum_{k=0}^{n} 10^k$$ Then ...
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### Powers-of-10-multiples of $\pi$ (or any irrational) are dense

Very related, but not the same, to this question Multiples of an irrational number forming a dense subset, is the next one: Is the sequence $(\{10^n\pi\})_{n=1}^\infty$ dense in the interval $[0,1]$?...
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### Is there a way to write an infinite set that contains only irrational numbers without integer multiples?

Is there a way to write an infinite set that contains only irrational numbers without integer multiples? The infinite set must not contain integer multiples of any other members of that set. For ...
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### Can we evaluate the alternating sum of the digits of an irrational number?

Suppose you had a summation $\sum(-1)^na_n$, where $a_n$ is the $n$th digit of $e$ and $a_0=2$. I know it diverges, but I want to know if its possible to evaluate anyways. Since it is alternating, ...
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### Deleting digits from an irrational number [closed]

Is it true that by deleting infinitely many appropriate digits out of the decimal representation of any positive irrational number, we can always get back the original number?
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### Prove that there is no largest irrational number

I have to prove that there is no largest irrational number from the result of the a previous proof: Prove that if $x$ is rational and $y$ is irrational then, $x+y$ is irrational. I was able to prove ...
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### Is there an explicit irrational number which is not known to be either algebraic or transcendental?

There are many numbers which are not able to be classified as being rational, algebraic irrational, or transcendental. Is there an explicit number which is known to be irrational but not known to be ...
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### How one can approximate irrational raised to irrational power?

How one can evaluate irrational number raised to irrational power? Like is there an easy way to prove that $-0.685<\pi^e-e^\pi<-0.675$?
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### On the limit $\lim_{n \to +\infty} n \{ n \xi \}$

Assume that $\xi \in \mathbb{R} \setminus \{Q\}$ is a given irrational number. I am trying to draw some conclusion about the limit $$\lim_{n \to +\infty} n \{ n \xi \}$$ where $\{\cdot\}$ denotes ...
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### The dilemma of Pi [closed]

Is Pi rational or irrational ? Pi can be represented as 22/7 which is a rational number. Whereas 3.14 is a non terminating and non recurring number which is a irrational number
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### General Techniques - Number sets

There are many problems involving, proving numbers are irrational or not an integer and so forth (e.g roots of polynomials, size of an angle) What are some general techniques/tricks that I can use in ...
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### Irrational Numbers and their sequence

I have a question about irrational or just long sequences of rational numbers. My question is that, what method/algorithm is used to determine what digit will come next in the sequence, I mean how do ...
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### All real numbers in $[0,2]$ can be represented as $\sqrt{2 \pm \sqrt{2 \pm \sqrt{2 \pm \dots}}}$

I would like some reference about this infinitely nested radical expansion for all real numbers between $0$ and $2$. I'll use a shorthand for this expansion, as a string of signs, $+$ or $-$, with ...
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### Why there are real numbers with infinite digits, but no such natural numbers (or another reason why real numbers are uncountable)

This question is me trying to understand (again) why there can be no one-to-one correspondence between the sets of natural and real numbers. The source of confusion is this: if we abstract completely ...
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### Irrational numbers generated by a deterministic cellular automaton?

If we consider a simple 1D cellular automaton (acting on a binary string) and record a value at a fixed position in the string, we can interpret the recorded sequence as a binary number. Most simple ...
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### show that this statement is false (counterexample) if $a,b \in \mathbb R \backslash \mathbb Q$ then $a \cdot b \in \mathbb R \backslash \mathbb Q$

if $a,b \in \mathbb R \backslash \mathbb Q$ then $a \cdot b \in \mathbb R \backslash \mathbb Q$ Okay so the question asks to show, with a counter example, that the above statement is false. Here ...
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### Proof by Contradiction relating to rational and irrational numbers

I've been given the question: given $x,y\in\mathbb{R}\setminus\mathbb{Q}$ and $x+y =\frac{m}{n}$, prove $x-y$ is irrational. I tried solving this using a proof by contradiction but I feel like I got a ...
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### Prove $\cos\frac{\pi}{2^{n+1}}$ is irrational

Prove that for every number $n\in\mathbb N$,number $\cos\frac{\pi}{2^{n+1}}$ is irrational. I really don't know where to start.
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### $\pi \not\in \mathbb{Q}$?

I've taken this fact for granted; some thinking tells me that indeed, I cannot express it with fractions. So it's not rational. But well, if $p,q \in \mathbb{Q}$ then $p+q \in \mathbb{Q}$ since it is ...
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### Rational numbers as angles - where do irrationals fit in?

If we make a rectangular grid with integer coordinates, it's possible to assign a unique angle to any rational number, using the definition $\tan \phi=y/x$ for $\phi \in (-\pi/2, \pi/2)$. For ...
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### How to make continued fractions of any number?

I recently found an continued fraction representation of $\pi$, and I wondered how can I make an continued fraction that converges into a number? The MAIN question is: how do you make a continued ...
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### Prove: $\tan\frac{\pi}{24}=2\sqrt{2+\sqrt{3}}-\sqrt{3}-2$

How to prove that $$\tan\frac{\pi}{24}=2\sqrt{2+\sqrt{3}}-\sqrt{3}-2$$ I get $$\tan\frac{\pi}{24}=\sqrt\frac{2\sqrt{2}-\sqrt{3}-1}{ 2\sqrt{2}+\sqrt{3}+1}$$ but i can't transform it.
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### When is a finite sum of powers of non-integer a rational number? [closed]

Concretely, is there $b \in \mathbb R, n,k \in \mathbb N$ such that $\sum_{i = n}^{n+k} b^i \in \mathbb Q$ ?
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### Bijection between $[0,1)$ and the space of binary sequences

My question deals with the problem of showing that the set $$\Omega = \{ \omega \colon \omega =(a_1,a_2, \ldots ), a_i =0,1\}$$ has the same cardinality as the interval $[0,1)$. In a textbook I read ...
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### Show that a certain number defined via its decimal expansion is not rational

For each function $f:\mathbb{N}\to \mathbb{N}$ we define the real number, in decimal notation $A(f)=0.f(0)f(1)f(2)f(3)\ldots$. Show that, if $f(x) =x^2$, then $A(f)=.0149162536\ldots$ is irrational....
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### Cubic polynomial with three (distinct) irrational roots

I am looking for an equation $$x^3+ax^2+bx+c=0, \qquad a, b, c \in \Bbb Z,$$ of degree $3$ that has $3$ different roots. For an equation of degree $2$ it is easy---for example $x^2-2=0$---but I ...
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### Rational or Irrational number [closed]

we know that "$a$" is a Irrational number .But "$a^2+a$" is Rational. Can You find "$a$"? (more than one answer is available)
I have 1 right triangle of dimensions $\sqrt75$$, 11, 14. I'd like to know how to quickly obtain the other right triangles with \sqrt75 as a leg, and two integers as the hypotenuse and the other ... 2answers 43 views ### Irrational Numbers and their squares If s is irrational is s^2 irrational? Looking at example (a) s= \sqrt 2 then s^2= 2, which is rational but looking at example (b) s= 5^{1/3}, then s^2= 5^{2/3} which is irrational or \... 2answers 45 views ### Irrationality of 1/a + 1/b I have thought about this and was wondering if anyone could provide an example of real numbers a and b such that a + b is rational but 1/a + 1/b is irrational or prove the statement false. 2answers 45 views ### Why must a and b both be coprime when proving that the square root of two is irrational? Suppose we wish to prove that the square root of two is irrational. We begin by assuming that it is rational. Namely, where both a and b are integers$$\frac{a}{b} = \sqrt 2 % MathType!MTEF!2!1!+... 6answers 144 views ### Proving that$2\sqrt 3+3\sqrt[3] 2-1$is irrational Prove that$2\sqrt 3+3\sqrt[3] 2-1$is irrational My attempt: $$k=2\sqrt 3+3\sqrt[3] 2-1$$ Suppose$k\in \mathbb Q$, then$k-1\in \mathbb Q$. $$2\sqrt 3+3\sqrt[3] 2=p/q$$ I'm stuck here and don'... 1answer 40 views ### Division of Square Root of Primes are Irrational Prove that for any distinct primes$p$and$q$, the ratio$\frac{\sqrt p}{\sqrt q}$is irrational. I know that separately$\sqrt p$and$\sqrt q$are irrational, so my initial thought process was to ... 2answers 1k views ### Reversing the digits of an infinite decimal Let$x$be a real number in$[0,1)$, with decimal expansion $$x = 0.d_1 d_2 d_3 \cdots d_i \cdots \;.$$ If the decimal expansion is finite, ending at$d_i$, then extend with zeros:$d_k = 0$for all ... 6answers 240 views ### How can never ending decimal numbers represent finite lengths? e.g. pi(π),$\sqrt{2}$Recently, I was in a discussion with a colleague that, whether the πd really can represent the accurate perimeter of a circle or not. To clarify that doubt, I came ... 0answers 122 views ### Infinitely nested radical expansions for real numbers Conjecture. For any real number$x \in (0,1]$there exists a unique expansion in the form$x=-2+\sqrt{a_1+\sqrt{a_2+\sqrt{a_3+\cdots}}}$with$a_k$being natural numbers from the set$(2,3,4,5,6)$. ... 3answers 58 views ### Rationalize a surd$\frac{1}{1+\sqrt{2}-\sqrt{3}}\$
How can I rationalize the following surd $$\frac{1}{1+\sqrt{2}-\sqrt{3}}$$ What would be the conjugate of the denominator