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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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 ...
9
votes
1answer
385 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 ...
21
votes
2answers
334 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 ...
1
vote
1answer
67 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, ...
3
votes
1answer
57 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?
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 ...
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 $-0.685<\pi^e-e^\pi<-0.675$?
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 ...
-8
votes
3answers
99 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
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 ...
-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 ...
58
votes
3answers
593 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
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 ...
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 ...
1
vote
2answers
57 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 ...
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 ...
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)
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.
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 % ...
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 ...
4
votes
6answers
194 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 ...
9
votes
0answers
110 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 $(2,3,4,5,6)$. ...
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
1
vote
3answers
38 views

exhibit a countable set of irrational numbers with justification

So I have been given this problem and I am totally stumped on what to do...everything I have learned says the irrational numbers are uncountable but I am supposed to exhibit a countable set of ...
1
vote
1answer
41 views

Rationality of $a^2+b^2$

I have looked into this topic lately and have not found an answer to the following question. Is the following true: If $a,b\in\mathbb{R}$ and $a + b$ is rational, then $a^2 + b^2$ is rational
0
votes
1answer
62 views

The irrationality of $\pi/e$ is listed as open yet the infinite product formula for it seems to suggest a way to prove it.

And the formula of all rational products seems to suggest that taking some n as n approaches infinity, the formula will have an always increasing amount of uncancelled primes(so provably non ...
4
votes
2answers
52 views

Rationalising factor of $a+b \sqrt{2}+c \sqrt{3} + d \sqrt{6}$

I am trying to express the inverse of $a+b \sqrt{2}+c \sqrt{3} + d \sqrt{6}$ (given $a, b, c, d \in \mathbb{Q}$) in the form $e+f \sqrt{2}+g\sqrt{3}+h\sqrt{6}$ (where $e, f, g, h \in \mathbb{Q}$). I ...
4
votes
1answer
82 views

How to prove $\log_23$ is irrational?

I think using contradiction is good. Assume $\log_23$ is rational Then $\exists p\in \Bbb{Z}, q\in \Bbb{Z}^*: \log_23 = \frac{p}{q}$ ###$p, q$ has no common factors. Then $3^{q}=2^{p}$ ... Here ...
-1
votes
3answers
61 views

If $x$ is rational and $xy$ is irrational, then $y$ is irrational. [closed]

This is a statement that I need to prove. Let $x$ and $y$ be real numbers. If $x$ is rational and $x\times y$ is irrational, then $y$ is irrational. I believe you have to prove this using ...
-1
votes
2answers
63 views

Suppose that a sequence of rational fractions p/q converge to an irrational number

Suppose that a sequence of (rational) fractions p/q converge to an irrational number r. Show that q converges to infinity.
3
votes
2answers
65 views

Rational Points on $\sin x$ and $\cos x$

Are there any values for $x$ such that both $\sin x$ and $\cos x$ are rational besides $\displaystyle\frac{n\pi}{2}$ and $n\pi$, where $n$ is an integer? I also only want to include $x$ values that ...
1
vote
1answer
35 views

Continous function from $ \Bbb Q \rightarrow \Bbb R $, $ f = 1 $ for $x > \sqrt2$ and $ f = 0$ for $x < \sqrt2$

I'm not really sure how to go about this problem. Show that $h : \Bbb Q \rightarrow \Bbb R $, with $$ h(x)=\begin{cases} 0 &\text{for $|x|< \sqrt{2}$} \\ 1 &\text{for $|x|>\sqrt{2}$} ...
1
vote
1answer
81 views

Which set is more dense: set of irrational numbers or set of rational numbers? [duplicate]

Is the infinity of irrational numbers equal to the infinity of rational numbers? Or is one is greater than other? And what is the proof? I could not find out a rigorous proof about this. P.S. I am ...
1
vote
1answer
55 views

Set of Rational numbers a countable set?

How can we say that rational numbers is a countable set? I can divide a rational number by infinite different number of natural numbers so shouldn't there be infinite rational numbers. ...