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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2answers
802 views

Lambert's Original Proof that $\pi$ is irrational.

I am trying to find Lambert's original proof that $\pi$ is irrational. Wikipedia has a little description but it is quite lacking. Can someone direct me to Lambert's original proof or post his proof ...
10
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14answers
2k views

Irrational numbers in reality

I have a square stone slab 1 metre by 1 metre, by the Pythagorean identity the diagonal from one corner to another is given by $\sqrt 2$. However $\sqrt 2$ is an irrational number, could someone ...
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0answers
73 views

Archimedes' Apprxomation 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 ...
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2answers
64 views

Help With a proof (Irrational Number)

Prove the following statement by proving its contrapositive: if $r$ is irrational, then $r^\frac{1}{5}$ is irrational. Its contrapositive will be: If $r^\frac{1}{5}$ is not irrational, then $r$ is ...
4
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1answer
495 views

Proof that $\sqrt[m]{a} + \sqrt[n]{b}$ is irrational

Is there a way to prove that $\sqrt[m]{a} + \sqrt[n]{b}$ ($\sqrt[m]{a}$ and $\sqrt[n]{b}$ are irrational); $a, b, m, n \in \mathbb{N}$; $m, n \neq 2$; is irrational without using the theorem mentioned ...
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2answers
140 views

$x+y\sqrt{2}$ infimum ($x,y\in \mathbb{Z}$)

I've looked for help with this question but I have not found anything, I hope this is not a duplicate. Define the set $A=\{\mid x+y\sqrt{2}\mid \ : x,y\in \mathbb{Z}\ \mbox{and} \mid ...
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1answer
60 views

The minimum number of digits after the floating-point, which uniquely identify every irrational square root

Let the following: $B:$ a natural number larger than $1$ $S:$ a set of irrational numbers in the range $(0,1)$ represented in base $B$ $L:$ the minimal prefix length which uniquely identifies every ...
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1answer
82 views

How do we prove sqrt2 is irrational? [duplicate]

What kind of number is sqrt2, rational or irrational? And what are the analytic and non analytic ways of proving it?
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2answers
1k views

Deciding whether a number is rational (2 examples)

1) Prove that number irrational $\sqrt{7-\sqrt{2}}$ I created a polynomial $x=\sqrt{7-\sqrt{2}}$ so $P(x)=x^4-14x^2+47$ and since $47$ is prime we check $P(x)$ for $ {1,-1,47,-47}$ and since all of ...
4
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2answers
164 views

The set $E= \{x\in [0,1]: \sum_{j=1}^\infty t^j|x−q_j|^{-r} <\infty\}$ does not contain all irrational numbers in $[0,1]$

Let $q_1,q_2,q_3,...$ be an enumeration of $\mathbb{Q}\cap[0,1]$ and let $r,t \in (0,1).$ Consider the set $$E= \{x\in [0,1]: \sum_{j=1}^\infty t^j|x−q_j|^{-r} <\infty\} $$ (a) Show that $E\neq ...
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2answers
1k views

The sum of two irrational square roots

This is very similar to this question, but I was wondering if there was a simpler proof. In particular, a proof that would prove that $\sqrt{x}+\sqrt{y}$ is an irrational number if both $\sqrt{x}$ ...
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2answers
142 views

Definition of Rational/ Irrational Numbers reguarding denominators

The definition of a Irrational number is "Irrational numbers don't include integers OR fractions. However, irrational numbers can have a decimal value that continues forever WITHOUT a pattern." So ...
7
votes
2answers
923 views

Dimension of R over Q without cardinality argument. [duplicate]

I am looking for the easiest (elementary) proof that $\mathbb R$ is infinite dimensional as a $\mathbb Q$-vector space, without using cardinality. It should be understandable at highschool level. So ...
3
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2answers
132 views

Are there any non-trivial counterexamples to the non-closure of the irrational numbers over addition?

It is trivial to show that the set of irrational numbers is not closed under addition. Just choose an irrational number $p$ and add it to its additive inverse $-p$ to get $0\in\mathbb{Q}$. However, I ...
7
votes
3answers
232 views

Using decimals of $\pi$ to store data

I read recently about an idea to, instead of storing actual data, converting the data to a string of digits and then store the index of where this pattern occurs in some number, for example $\pi$. The ...
0
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2answers
64 views

How to find irrational approximates

Say I have a rational number, $n$, that approximates an irrational number of the form: $$n \approx {a+\sqrt b \over c}$$ in terms of being irrational. What is a good way of finding the unknown ...
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2answers
57 views

Does the absence of horizontal lines shows that there are no $n,m\in \mathbb{N}$ such that $n^2=2m^2$?

When I was learning about the proof of the irracionality of $\sqrt{2}$, I remember of trying to visualize it by ploting the graphs of $f(n)=n^2$ and $g(m)=2m^2$, but at the time I got confused and ...
2
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2answers
173 views

Conjecture: if $a+b$ and $ab$ are rational, $a$ and $b$ are rational

I can't find a rigorous proof but I have a feeling it's true. Informal argument: Suppose $a+b$ and $ab$ are rational, $a$ and $b$ are irrational (since just one can't be irrational). Then $a$ and ...
2
votes
1answer
96 views

Show that $\arctan(n)$ is irrational for all $n \in \mathbb{N}$

Question : Show that $\arctan(n)$ is irrational for all $n \in \mathbb{N}$. Hint: My solution doesn't use continued fraction. I am interested in other possible proofs for this question.
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4answers
1k views

Is the fact that there are more irrational numbers than rational numbers useful?

Although it is known that the cardinality of the set of irrational numbers is greater than the cardinality of the set of rational numbers, is there any usefulness/applications of this fact outside of ...
0
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2answers
103 views

Prove that if $n \geq 2$, then $\sqrt[n]{n}$ is irrational. Hint, show that if $n \geq 2$, then $2^{n} > n$.

Prove that if $n \geq 2$, then $\sqrt[n]{n}$ is irrational. Hint, show that if $n \geq 2$, then $2^{n} > n$. So, my thought process was that I could show that $2^{n} > n$ using induction, but ...
2
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1answer
156 views

Does the limit of a sequence with floor function exist?

Question : Let $a_n=n\alpha-\lfloor n\alpha\rfloor\ (n=1,2,\cdots)$ where $\alpha$ is an irrational number. Then, does the limit $n\to\infty$ of $(a_n)^n$ exist? I know that ...
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1answer
119 views

How to prove that $\cos(n)$ is irrational?

We know that $\cos(1)$ is real and transcendental (1). Then by using the fact that for every $n \in \mathbb{N}$ there exists a polynomial $P_n$ of degree $n$ with integer coefficients such that ...
3
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1answer
101 views

Irrationality measure.

I would like someone to give me a definition of what irrationality measure is, I have stumbled over several definitions which may be equivalent but as I lack understanding I cant see this correlation. ...
15
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1answer
336 views

Can we prove that the solutions of $\int_0^y \sin(\sin(x)) dx =1$ are irrational?

Can we prove that the solutions of $$\int_0^y \sin(\sin(x)) dx =1$$ are irrational? Wolfram Alpha gives two approximate sets of solutions as $\{4.58+2\pi k|k\in\mathbb{Z}\}$ and $\{1.69+2\pi ...
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votes
3answers
179 views

Real numbers that are not the roots of any polynomial equation with algebraic coefficients

An algebraic number is a number which is a root of some non-zero polynomial equation with rational coefficients. A transcendental number is a number which is not a root of any non-zero polynomial ...
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3answers
59 views

negative powers $(x^{-2} = 1/x^2)$

I need clarification for negative power of a number. I understand $x$ to the power of $2$ is equal to $x\cdot x$ But how $x$ to the power of $-2$ is equal to $\dfrac{1}{x^2}$ ?
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0answers
55 views

“Building blocks” for computable functions

In an (otherwise very enlightening) answer to another question of mine the question came up What functions are allowed as building blocks for computable functions? I was astonished that there ...
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4answers
626 views

Why there are irrational numbers?

I do not quite get it. Why can't we represent all real numbers as a sum of rational numbers? Why do we need irrational numbers? For example, ...
2
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1answer
92 views

is $(\mathbb{Q} \times (\mathbb{R}\setminus\mathbb{Q}))\cup((\mathbb{R}\setminus\mathbb{Q})\times\mathbb{Q})$ connected? path connected?

let $$X=(\mathbb{Q} \times (\mathbb{R}\setminus\mathbb{Q}) ) \cup ((\mathbb{R}\setminus\mathbb{Q})\times\mathbb{Q}) $$ and let $$\tau=\tau (\text{euclid})$$ what are the connected components of ...
2
votes
1answer
50 views

Solving surds without compairing

Question: Let $a + \sqrt{2b} = 3 - 2\sqrt{2}$ .Find the value of $a - \sqrt{2b}$ What I did: I compared the whole numbers and the irrational numbers in both sides and calculated the answer $3 + ...
3
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2answers
353 views

A dense set on $[0,1)$

Let $x\in \mathbb{R}$ an irrational number. Define $X=\{nx-\lfloor nx\rfloor: n\in \mathbb{N}\}$. Prove that $X$ is dense on $[0,1)$. Can anyone give some hint to solve this problem? I tried ...
2
votes
1answer
43 views

Need help to simplify irrational equation

I have faced a problem simplifying this equation. . I tried to solve it this way: , but I just can't get the correct answer. This equation is from high school course, so it must have quite a simple ...
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3answers
49 views

A pretty much simple number theory problem

Let $x$ be an irrational number, and $n$ be a positive integer. Will there ever be a set of $(n,x)$ which satisfies $x(n-x) \in \mathbb{Z}$ ? If so, could you suggest those numbers? And, if not, ...
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1answer
78 views

Square root of an odd composite being irrational

Is there an odd composite number $n$ such that $\sqrt{n}$ is irrational?
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6answers
709 views

Is $i$ irrational?

On the one hand, $i(=\sqrt{-1})$ cannot be expressed as a ratio of integers, so, by definition, $i$ is not rational $\iff i$ is irrational. However, the set of irrational numbers, ...
5
votes
2answers
195 views

A question about decimal representation of irrational numbers.

Is this true that any finite word of the alphabet $\mathcal{A_9}=\{0,1,2, \ldots,8,9\}$ appears somewhere in the decimal representation of $\sqrt{2}$ ? Thanks !
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votes
0answers
140 views

The “trick” functions in the “$\pi$ is transcendental” proofs

I was reading this paper and I wondered how did Hermite decide to define a function $$f(x)=\frac{x^{p-1}(x-1)^p\cdots (x-m)^p}{(p-1)!}$$ Are these functions only tricks or there is a deeper meaning?
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10answers
3k views

Visualizing the square root of 2

A junior high school student I am tutoring asked me a question that stumped me - I was wondering if anyone could shed some light on it here. We were talking about how the square root of 2 is an ...
3
votes
3answers
356 views

Prove $\sqrt6$ is irrational

Suppose $\sqrt6 = \frac pq$ where $p$ and $q$ have no common factors. $$6 = \frac {p^2}{q^2}$$ $$6p^2 = q^2$$ So $q^2$ and therefore $q$ is divisible by $6$. $$p^2 = \frac {q^2}{6}$$ So $p^2$ ...
7
votes
1answer
282 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 ...
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4answers
204 views

Prove that $(4/5)^{\frac{4}{5}}$ is irrational

Prove that $(4/5)^{\frac{4}{5}}$ is irrational. My proof so far: Suppose for contradiction that $(4/5)^{\frac{4}{5}}$ is rational. Then $(4/5)^{\frac{4}{5}}$=$\dfrac{p}{q}$, where $p$,$q$ are ...
3
votes
1answer
70 views

Euclidean geometry and irrational numbers.

I was wondering, given a square that is $1 \times 1$, how can we know that the diagonal is an irrational length geometrically??? We could use the Pythagorean Theorem to see that the diagonal of a ...
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1answer
54 views

Looking for irrational Numbers Proof [duplicate]

$a,b,c,~$and $d$ are rational numbers. $b>0$ and $d>0$ the $\sqrt{b}$ and the $\sqrt{d}$ are both irrational. if $a+\sqrt{b}=c+\sqrt{d}$ show that $a = c$ and $b = d$. I know that a=c and ...
3
votes
3answers
316 views

Is it possible that $\pi$ is finite in other numerical bases?

In base $\pi$, the number $\pi$ is $1\cdot \pi^1 + 0\cdot \pi ^ 0 $, which is equal to $10$. So, is $\pi$ an irrational number in all bases or not?
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votes
1answer
47 views

Is there any combination of numbers which upon division gives the exact number of P?

In other words, there are (probably) infinite combination of numbers/operations which leads to irrational numbers. So I wonder, if there is one which gives exact number representation of P(π)? Do we ...
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1answer
61 views

$\lim \{r^n\}$ exists, Is $r$ an integer?

$r\in\Bbb R$, $|r|\gt1$ and $\lim\limits_{n\to\infty}\{r^n\}$ exists. Can one conclude that $r$ is an integer? Here, $\{x\}=x-[x] $ is the fractional part of $x\in\Bbb R$ If $r\in\Bbb Q$, the ...
0
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1answer
54 views

Is Dirichlet's function enough to prove constants like $\gamma$ irrational?

This function appears without any reference in the book The Irrationals : $$\lim_{m\to\infty}\lim_{n\to\infty} \cos^{2n}(m!\pi x)=\left\{ \begin{array}{lr} 1 & : x \in \mathbb{Q}\\ ...
8
votes
3answers
330 views

If $\sum\frac1{a_n}$ is convergent, then irrational?

$\{a_n\}$ is a strictly increasing sequence of positive integers such that $$\lim_{n\to\infty}\frac{a_{n+1}}{ a_n}=1$$ If $\sum\limits_{n=1}^\infty\frac1{a_n}$ is convergent, can one conclude ...
2
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1answer
120 views

Do circles exist

So I was wondering about circles today and if they really do exsist. If you graph a circle in function mode, your equation looks like$$y=\sqrt{1-x^2}$$ Now for simple purposes lets take a portion of ...