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
77 views

Let a,b be rationals and x irrational. Show that if $\frac{x+a}{x+b}$ is rational, then $a=b$.

I'm trying to solve the following problems: Let $a$,$b$ be rationals and $x$ irrational. Show that if $\frac{x+a}{x+b}$ is rational, then $a=b$ Let $x$,$y$ be rationals such that ...
1
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1answer
73 views

Is there any attempt to explain irrational numbers from a geometrical point of view?

I'm trying to understand irrational numbers as the result of comparing different referential symmetries, and I'd like to know if there have been any attempt to explain irrationality from any ...
1
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1answer
83 views

Is there a fixed integer $n$ for which ${\pi}^{n}$ is prime number?

I would like to know the relationship between $\pi$ and prime numbers distribution ,then I would like to ask if there is a fixed integer for which ${\pi}^{n}$ can be prime or how do i disproof that ...
0
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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 ...
0
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0answers
26 views

Proving that $z_{k_1,k_2}= k_1r_1+ k_2r_2, k_1, k_2$ is a generator for the real numbers

Prove the following statement: If $r_1$ and $r_2$ are real numbers whose quotient is irrational, then any real number $x$ can be approximated arbitrarily well by numbers of the form $z_{k_1,k_2}= ...
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2answers
53 views

Prove that all real numbers can be formed

Prove the following statement: If $r_1$ and $r_2$ are real numbers whose quotient is irrational, then any real number $x$ can be approximated arbitrarily well by numbers of the form $z_{k_1,k_2}= ...
5
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2answers
463 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 ...
2
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0answers
66 views

Paper of Paul Erdös

I'm trying to understand On Arithmetical Properties of Lambert Series by Erdös, but am stuck on the first page. He states: Put $k=\left[(\log n)^{1/10}\right]$ and let $p_1,p_2,\ldots$ be the ...
2
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1answer
94 views

Show $e^x$ is irrational for rational $x \neq 0$ [duplicate]

I want to show that if $x$ is rational and nonzero then $e^x$ is irrational. Clearly $e^{\frac{r}{s}} = \frac{p}{q} \Rightarrow q^s e^r = p^s$, but this doesn't seem helpful. The usual proof that $e$ ...
10
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7answers
1k views

How can a Cauchy sequence converge to an irrational number?

I am a physics major and would like to clear a confusion regarding complete metric spaces. I am quoting the definition of a Cauchy sequence from wikipedia below Formally, given a metric space $(X, ...
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0answers
31 views

the summation goes to the integration

Let $f$ be a continuous function on $\mathbb R$ with period $1$. Show that for any irrational number $x$, $\frac { \sum_{k =1 }^ N f (k x ) } { N} \to \int _ 0 ^1 f(t) \, d t $ as $N \to \infty$. I ...
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0answers
37 views

How do I know if this will be rational or irrational? ($a^b$)

Usually, when I have $a^b$ when $a$ and $b$ are both irrational, I assume that it will be irrational. But that is not always true, I assume, so when is the result irrational? How will I know? Take ...
11
votes
1answer
231 views

Is $\sum_{n \ge 1}{\frac{p_n}{n!}}$ irrational?

Is $\sum_{n \ge 1}{\frac{p_n}{n!}}$ irrational, where $p_n$ is the $n^{\text{th}}$ prime number? This question is spurred by the comment thread on this question where I presented a rough idea of a ...
4
votes
4answers
179 views

Why are there more Irrationals than Rationals given the density of $Q$ in $R$?

I'm reading "Understanding Analysis" by Abbott, and I'm confused about the density of $Q$ in $R$ and how that ties to the cardinality of rational vs irrational numbers. First, on page 20, Theorem ...
6
votes
3answers
243 views

$\sqrt[31]{12} +\sqrt[12]{31}$ is irrational

Prove that $\sqrt[31]{12} +\sqrt[12]{31}$ is irrational. I would assume that $\sqrt[31]{12} +\sqrt[12]{31}$ is rational and try to find a contradiction. However, I don't know where to start. Can ...
5
votes
1answer
107 views

Is $\frac{1}{2^{2^{0}}}+\frac{1}{2^{2^{1}}}+\frac{1}{2^{2^{2}}}+\frac{1}{2^{2^{3}}}+…$ algebraic or transcendental?

Inspired by this question, the series $\dfrac{1}{2^{2^{0}}}+\dfrac{1}{2^{2^{1}}}+\dfrac{1}{2^{2^{2}}}+\dfrac{1}{2^{2^{3}}}+\dots$ is clearly irrational. But is it algebraic or transcendental? I ...
0
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0answers
85 views

How to prove $e^{n}$ is irrational? [duplicate]

How to prove $e^{n}$ is irrational for $n=2,3,4,5,...$ where $e$ is natural logarithm constant?
2
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3answers
54 views

How is this limit being solved? I can't grasp it

I am going over limits for my finals as I notice this example in my schoolbook discribing limits of the undefined form $0\over0$ in the shape of an irrational fraction. $$\lim\limits_{x \to 1} ...
2
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2answers
214 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 ...
5
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1answer
59 views

Can finite sums of two numbers come arbitrarily close to zero?

Given two real numbers $a$ and $b$, define an $a$-$b$-sum as a finite sum of $a$'s and $b$'s, i.e. a sum: $$m\cdot a + n\cdot b$$ where $m,n$ are non-negative integers. Is there a pair of numbers ...
6
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4answers
912 views

Proof: Is there a line in the xy plane that goes through only rational coordinates?

Question: Is there a line in the XY plane that has all rational coordinates. Prove your answer. Idea: There is most certainly not. I believe it can be shown that between any 2 rational points that ...
8
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1answer
179 views

Prove that $\sqrt{2} + \sqrt[3]{3}$ is irrational [duplicate]

$\sqrt{2} + \sqrt[3]{3}$ is irrational ? These are my steps - $\sqrt{2} + \sqrt[3]{3} = a$ $3 = (a-\sqrt{2})^{3}$ $3 = a^{3} -3a^{2}\sqrt{2} + 6a -2\sqrt{2}$ $3a^{2}\sqrt{2}+2\sqrt{2} = ...
2
votes
3answers
102 views

Proving f(x)=0 for all x in [a,b] when we only know that f is continuous and f(x)=0 when x is rational. [duplicate]

The question is as follows a.) Let $f(x)$ be continuous function on an interval [a,b] and suppose that $f(x)=0$ for each rational value $x$ in [a,b]. Prove that $f(x) = 0$ for all $x \in [a,b]$. b.) ...
6
votes
3answers
285 views

Dense set in the unit circle- reference needed

For $x \notin \pi\mathbb Q$, that is, a real $x$ that is not a rational multiple of $\pi$, consider the set $$\{(\cos nx,\sin nx):n = 0,1,2,...\}.$$ It is known that this set is dense in the unit ...
0
votes
1answer
125 views

Prove that $\sum \frac{1}{n^2} = \frac{\pi^2}{6}$ [duplicate]

In this answer two sequences are mentioned. In particular, I would like to prove that $$\sum_{n = 1}^{+ \infty} \frac{1}{n^2} = \frac{\pi^2}{6}$$ If I knew that the sequence converges to ...
0
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0answers
71 views

Decimal digits in $\pi$

Around ten years ago I had read somewhere that there was a question in an exam for application for software engineer position in a big company which states: "What is the one billionth digit of $\pi$?" ...
5
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11answers
242 views

Why is $e$ the number that it is? [closed]

Why is $e$ the number that it is? Most of the irrational number that we learn about in school have something to do with geometry, like $\pi$ is the ratio of a circle's diameter to its circumference. ...
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3answers
61 views

Is there a pattern to the golden ratio number figures?

The golden ratio or phi is 1.6180339887498948482045... I am wondering if there is a pattern in the numbers so given a certain set of figures, you are able to figure out the rest of the figures ...
-1
votes
2answers
98 views

Irrational numbers are non-terminating/non-repeating decimals [closed]

Why is it true that all irrational numbers are non-terminating/non-repeating decimals? By definition, an irrational number is one that can't be expressed as a ratio of integers.
10
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6answers
778 views

Can I guess an irrational number formula from its digits?

Let us say I have 10,000 digits started from some point (lets say the 16th digit) of the decimal expansion square root of some arbitrary number, like 13. Is there any way I can get back the original ...
0
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3answers
99 views

How are irrational numbers, fixed points on the number line?

Please, while answering/reading this question, only keep in mind my point of view only. The question is, that how come an irrational number on a number line is a fixed point. To make things more ...
-3
votes
1answer
484 views

(22/7) is a rational number and (π) is irrational number [closed]

Why (22/7) is a rational number and (π) is irrational number. please explain. Edit: How can you say that $22/7=\pi$, when one number if rational and the other is irrational?
4
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2answers
119 views

Prove that $\sqrt{10} - \sqrt6 - \sqrt5 + \sqrt3$ is irrational

I tried the methods shown in Can $\sqrt{n} + \sqrt{m}$ be rational if neither $n,m$ are perfect squares? but I cannot extend them well into 4 numbers.
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4answers
111 views

Is $a\sqrt[3]{2} + b\sqrt[3]{4}$ irrational?

I need to prove that $$ a\sqrt[3]{2} + b\sqrt[3]{4}$$ is irrational, while $a$,$b $ are non zero rationals. I know that $\sqrt[3]{2} + \sqrt[3]{4}$ is irrational and I also know how to prove it, ...
1
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0answers
33 views

Tricky proof involving limit points [duplicate]

Show that for each irrational number $x$ the set of limit points of the sequence $(a_n)_{n\in\mathbb{N}}=nx-[nx]$ is the interval $[0,1]$. ($[x]$ is the largest integer $\leq x$) Any ideas how to ...
2
votes
2answers
58 views

Can we construct three irrational numbers $a,b,c$ such that $a+b+c \in \mathbb Q$?

This is rather easily shown to be possible if no constraint is put on $a,b,c$. However, is it also possible under the following constraint: $a, b$ and $c$ can not be rational multiples of each other. ...
0
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1answer
116 views

Cube root of $5$ is irrational

How to prove cube root of $5$ is irrational? I know this question was asked many time before but I can't understand the way the people explained the question so is it possible for someone to ...
2
votes
4answers
108 views

Can all irrational numbers be written in the form $u + v\sqrt{2}$, with $u$ and $v$ rational? [closed]

I am curious to know whether all irrational numbers can be written in the form $u + v\sqrt{2}$, with $u$ and $v$ rational. (Almost similar to how all complex numbers can be written as $x + iy$, ...
1
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1answer
76 views

Is the Cartesian square of the set of irrational numbers path connected?

Let $X=\mathbb{R}\setminus \mathbb{Q}$. Is $X\times X$ path-connected? I don't know where to start I think we need some number theory knowledge.
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1answer
36 views

Proof That all Positive Irrational Sqaure Roots Can be Raised to an Irrational Power to Get a Whole Number

Recently I have found out about a proof through a video. This proof shows that an irrational number can be raised to a irrational power to get and irrational number, but this proof only requires one ...
2
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1answer
29 views

Finding transcendental roots to an algebraic equation

So for equations with rational roots, there's a theorem that lists all the possible roots (Rational Root Theorem). If an equation has imaginary or irrational roots, their respective theorems say ...
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0answers
138 views

Interesting facts/ proofs about rational and irrational numbers

We got set some work to find some interesting facts or proofs regarding rational and irrational numbers. I wonder if anyone could offer some insight or recommend a good book/ website to look at.
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0answers
68 views

Am I pretty close to proving that e is irrational?

Show that $e=1+1/1!+1/2!+1/3!+…$ is an irrational number. Hint: show that, for all positive integers $p$, $0<p![e−(1+1/1!+…+1/p!)]<1$. Then conclude that $e$ cannot be a ratio of two integers ...
7
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1answer
110 views

Is there a function, continuous on the irrationals, with rational values, nowhere locally constant?

Question. Let $\mathbb A=\mathbb R\!\smallsetminus\!\mathbb Q$ be the irrational numbers. Is there a continuous function $\,f:\mathbb A\to\mathbb Q$, which is nowhere locally constant? – i.e., for ...
2
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0answers
55 views

Irrationality of $\pi+c$

How to prove that $\pi+c$ is irrational? where $c$ is the Champernowne Constant.
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0answers
83 views

Elementary proof that finite sums of square roots of primes is irrational

It is relatively easy to show that if $p_1$, $p_2$ and $p_3$ are distinct primes then $\sqrt{p_1}+\sqrt{p_2}$ and $\sqrt{p_1}+\sqrt{p_2}+\sqrt{p_3}$ are irrational, but the only proof I can find that ...
6
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0answers
72 views

The irrationality of Pi [duplicate]

Pi is defined as circumference/diameter, but it is an irrational number. And by definition an irrational number can't be defined by a fraction. So how is it that pi is circumference/diameter and on a ...
1
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1answer
29 views

Explanation of this (strong) induction statement

This might seem pretty simple and stupid to some but I am really not able to get it.... I was reading the proof that $\sqrt{2}$ is irrational (using induction) here. I could understand most of it ...
2
votes
2answers
57 views

Continuity question: Show that $f(x)=0, \forall x\in\mathbb{R}$. [duplicate]

Assume $f:\mathbb{R}\rightarrow\mathbb{R}$ is continuous on $\mathbb{R}$ and such that $f(r)=0$ for every rational number $r$. Show that $f(x)=0, \forall x\in\mathbb{R}$ using the $\varepsilon-\delta$ ...
2
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0answers
22 views

A binary irrational with bits defined by primes

Define a number $q$ in binary notation whose $n$-th bit is $1$ for $n$ prime, and $0$ for $n$ composite. So its 2nd, 3rd, 5th, 7th, 11th, etc. bits are $1$, with all other bits $0$. Here is $q$ out to ...