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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37
votes
4answers
12k views

Can you raise a number to an irrational exponent?

The way that I was taught it in 8th grade algebra, a number raised to a fractional exponent, i.e. $a^\frac x y$ is equivalent to the denominatorth root of the number raised to the numerator, i.e. ...
3
votes
2answers
254 views

How to understand Apostol's proof of the irrationality of $\sqrt{n}$ if $n$ is not a perfect square?

Recently I am reading the textbook of Apostol, Mathematical Analysis, Second Edition. On page 7, there is a theorem 1.10: If $n$ is a positive integer with is not a perfect square, then $\sqrt{n}$ is ...
7
votes
2answers
128 views

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 ...
5
votes
2answers
450 views

What exactly are those “two irrational numbers” $x$ and $y$ such that $x^y$ is rational? [duplicate]

It's possible to prove nonconstructively that there exists irrational numbers $x$ and $y$ such that $x^y$ is rational, but that proof only proves that such numbers exist and does not specify what they ...
5
votes
1answer
57 views

Show that the number is irrational $\forall n$

I need to show that the number $\sqrt 2+ \sqrt[3]{3}+\sqrt[4]{4}+\sqrt[5]{5}+...+\sqrt[n]{n}$ is irrational for any n, and I don't have a clue about how I could show that. Thank you!
5
votes
2answers
47 views

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 ...
21
votes
2answers
342 views

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 ...
9
votes
1answer
386 views

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 ...
1
vote
1answer
68 views

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 ...
17
votes
6answers
3k views

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 ...
1
vote
0answers
35 views

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, ...
12
votes
2answers
1k views

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 ...
3
votes
1answer
61 views

Deleting digits from an irrational number

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?
3
votes
0answers
33 views

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 ...
0
votes
0answers
28 views

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 $-0.685<\pi^e-e^\pi<-0.675$?
-8
votes
3answers
102 views

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
6
votes
1answer
434 views

Chinese estimate for $\pi$. Were they lucky?

The famous chinese estimate $\pi\approx\frac{355}{113}$ is good. I think that is too good. As a continued fraction: $$\pi=[3:7,15,1,292,\ldots]$$ That $292$ is a bit too big. Is there a reason for a ...
0
votes
1answer
107 views

Archimedes' Approximation of Square Roots

Supposing a square root $\sqrt{X}$, let $x$ be the approximation of $\sqrt{X}$, then we get these 2 formulas to estimate $\sqrt{X}$: $x_{n+1}=\frac{x_n+\frac{X}{x_n}}{2}$ and ...
1
vote
0answers
17 views

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 ...
58
votes
3answers
602 views

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 ...
-1
votes
3answers
50 views

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 ...
5
votes
5answers
131 views

Convergent sequence of irrational numbers that has a rational limit.

Is it possible to have a convergent sequence whose terms are all irrational but whose limit is rational?
2
votes
2answers
209 views

Approximation of $\pi$ using Brahmagupta's Identity

Brahmagupta, an ancient Indian Mathematician, gave an pretty efficient algorithm for finding integer solutions to the famous Pell's Equation, far before Fermat propounded this before the European ...
1
vote
1answer
50 views

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 ...
1
vote
2answers
58 views

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 ...
1
vote
4answers
102 views

Proof that this is an irrational number

Prove that $2\sqrt2 + \sqrt7 $ is an irrational number. I am trying to use contradiction to show that this is irrational. Also I am using the fact that $ 2\sqrt2 + \sqrt7 = \frac{1}{2\sqrt2 - ...
6
votes
3answers
360 views

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 ...
1
vote
1answer
70 views

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.
1
vote
4answers
77 views

$\pi \not\in \mathbb{Q}$?

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 ...
3
votes
0answers
37 views

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 ...
29
votes
4answers
3k views

Proving that $m+n\sqrt{2}$ is dense in R

I am having trouble proving the statement: Let $S = \{m + n\sqrt 2 : m, n \in\mathbb Z\}$. Prove for every $\epsilon > 0$, The intersection of $S$ and $(0, \epsilon)$ is nonempty.
1
vote
0answers
27 views

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 ...
0
votes
2answers
59 views

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.
0
votes
2answers
42 views

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$ ?
3
votes
3answers
42 views

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 ...
1
vote
1answer
26 views

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 ...
1
vote
4answers
81 views

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 ...
2
votes
3answers
207 views

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)
9
votes
0answers
111 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 $(2,3,4,5,6)$. ...
0
votes
1answer
24 views

Pythagorean Theorem on Spiral of Theodorus Triangles

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 ...
0
votes
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 ...
0
votes
2answers
42 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.
4
votes
6answers
196 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 ...
2
votes
2answers
37 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 % ...
0
votes
0answers
17 views

Continuity - approximating an irrational number via rationals [duplicate]

If $x=p/q$, where $(p,q)=1$ are integers, then $f(x)=1/q$. If x is irrational then f(x)=0. Prove that: a) f is continuous for all irrationals b) f is not continuous for all rationals. I think ...
5
votes
6answers
135 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 ...
0
votes
1answer
36 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 ...
10
votes
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 ...
9
votes
1answer
346 views

Proof of $\pi$ not being a quadratic irrational number.

Does someone know a proof (books, articles) that $\pi$ is not a quadratic irrational? The proof should not use that $\pi$ is transcendental. Any hints would be appreciated.
1
vote
3answers
54 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