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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15 views

Proof of an irrationality criterion

I have attached a proposition whose proof I don't understand at two points. Here are my questions: Why do we have $|a_{0n}+\theta_{1}a_{1n}+\dots+\theta_{k}a_{kn}|<(\rho-\varepsilon)^{-n}$ for ...
0
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1answer
44 views

Nearest neighbor of an irrational number

I am confused in my thoughts about the irrational numbers in real line. My confusion is: If $x\in$$\mathbb R$$-\mathbb Q$ then for $\epsilon>0$ as small as you please, the element ($x+\epsilon$) ...
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0answers
48 views

Irrationality of a number [on hold]

Is there any proof for $\sqrt3$ being an irrational number where we are not forced to conclude that $\sqrt3$ is irrational number?
3
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1answer
112 views

Is the infinite decimal fraction $1.23456…n$ irrational?

How to prove that the number $ 1.23456\dots n$ is an irrational number? The number consist, of course, of natural numbers in increasing sequence.
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1answer
21 views

Confusion about irrational numbers

Irrational numbers is defined as something that cannot be expressed as a fraction . Now I got a question . So is "120%" an integer or irrational number ? Do I take 120% as 1.2 or just 120% as an ...
2
votes
1answer
96 views

How can I prove $\sqrt{2} ^{\sqrt{2}}$ is irrational? [duplicate]

I am learning proofs and a question was posed which asked us to prove that $\sqrt{2}^{\sqrt{2}}$ is irrational. They mentioned this - Hint: try using the log10 function... I tried my hand at the ...
0
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1answer
41 views

Show that $E \subset \Bbb Q$ is closed in $(\Bbb Q, d)$

Assume $(\Bbb Q, d),$ $d(p, q):= |p -q|$ is a metric space and $E := \{p \in \Bbb Q : 2 < p^2 < 3\} \subset \Bbb Q.$ I have to show that $E$ is closed. I see two ways of proving ...
12
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6answers
2k views

Visual representation of the fact that there are more irrational than rational numbers.

Would anybody know of a visual or even (preferably) geometric representation of this? To make it more specific: Text, symbols and written numbers are predominantly used as labels, and and less to ...
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2answers
27 views

Proving by contradiction (6/9)

I have been given a statement that I need to prove using the contradiction method and I am just a little unsure of how to go about setting this up and executing. Here is the statement: If x is any ...
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0answers
68 views

Polynomial taking irrationals to irrationals

Problem: Find all polynomials from $\mathbb{R}\to \mathbb{R}$ $f$ with integer coefficients taking irrationals to irrationals. My attempt: It is clear that the problem statement is equivalent to ...
-1
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1answer
126 views

Proof that $\sqrt[n]{a+1}$ and $\sqrt[n]{a-1}$ cannot be both rationals [duplicate]

Let $a \neq 0$ be a natural number. How can be proved that $\sqrt[n] {a+1}$ and $\sqrt[n]{a-1}$ cannot be both rational numbers?
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3answers
95 views

Prove $2^{1/2}+3^{1/3}$ is irrational using Galois theory.

So, I want to prove that $2^{1/2}+3^{1/3}$ is irrational, and I need to prove it using Galois theory. To start, let's forget about the sum and deal with the individual numbers and $F_1 = \mathbb{Q}(...
1
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1answer
38 views

Can we apply binomial theorem for $\quad(a+b)^\ell\quad$ if $\ell\;$ irrational.

Let be$\quad a,b\;\in\mathbb R\quad, \ell\;\in\mathbb {(R\backslash Q)} \quad $ ($\ell:$irrational) Can we apply binomial theorem for $\quad(a+b)^\ell$
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1answer
25 views

A sum of irrational numbers ending rational

Let $x$ be a positive irrational number I know that there exists $y$ such that: $$\begin{cases} y>0 \\ x+y\in \mathbb Q.\end{cases}$$ How would you construct explicitly such $y$ ? For instance ...
0
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0answers
37 views

Difference between rationalizing factor and conjugate surd

I have some confusion regarding rationalizing factor and conjugate surd. For binomial surds for example $2+\sqrt{3}$ is conjugate of $2-\sqrt{3}$ and it is also rationalizing factor of $2+\sqrt{3}$. ...
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3answers
73 views

Is there any $\alpha$ for which $e^{\alpha}$ is an integer?

Is there any $\alpha$ which gives $e^{\alpha}$ an integer. $\alpha=0$ is the trivial one. But is there any other than $0$?
0
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1answer
47 views

Are all hypotenuses irrational if the shorter sides are integers?

Is it sufficient to say that providing the shorter two sides of a right triangle can be expressed as integers that work out to equal the value of the hypotenuse, then the value of the hypotenuse must ...
0
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0answers
29 views

Show that $θ_0 < Arg (z^α) < θ_0+\epsilon$ for infinitely many values of $z^α$, where $−π < θ_0 < π$ and $ \epsilon > 0$

For $z \neq 0$ and $α$ irrational, show that $θ_0 < Arg (z^α) < θ_0+\epsilon$ for infinitely many values of $z^α$, where $−π < θ_0 < π$ and $ \epsilon > 0$. I am trying to solve this ...
18
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6answers
2k views

when product of irrational numbers = rational number?

let $a$ and $b$ be irrational numbers. when do we have $ a \cdot b $ = rational number? for example $\sqrt{2} \cdot \sqrt{2}=2$. I was wondering if there some conditions for the product to be a ...
3
votes
1answer
53 views

Which elements of $\mathbb{R}$ make sense as representatives for cosets of $\mathbb{Q}$ in the group $\mathbb{R/Q}$

I am trying to better understand the group $\mathbb{R/Q}$. It's unclear to me when two irrational numbers will give the same coset of $\mathbb{Q}$, but I know that this must happen since, for example $...
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2answers
187 views

Prove $\pi+e$ or $\pi e$ is transcendental. [closed]

I understand to prove at least one of them irrational you would compose a function by which $\pi$ and $e$ are roots $((x-\pi)(x-e))$, and show that at least one coefficient cannot be rational because $...
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2answers
27 views

For irrational real number $r$, find $n \in \mathbb{Z}$ such that $|nr - [nr]| < 10^{-10}$.

This problem is from the book "A Walk Through Combinatorics" by Richard Bona. For any irrational number $r$, there exists a positive integer $n$ such that the distance of $nr$ from the nearest ...
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2answers
37 views

Show that there are at most two rational points on $(x - a)^2 + (y - b)^2 = r^2$ for $a, b$ irrational.

For any given irrational numbers $a, b$ and real number $r \gt 0$, show that there are at most two rational points (points whose coordinates are both rational numbers) on the circle $(x - a)^2 + (y - ...
3
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2answers
80 views

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

Is $\log_27$ a rational number?
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2answers
75 views

Prove that $(√5 - 1)/2$ is irrational.

Please help me prove that $(√5 - 1)/2$ is irrational. I know how to prove √5 is irrational: Assume that √5 is rational meaning √5 = $p/q$ $p,q$ $are$ $Z$ $and$ $q≠0$ $p^2/q^2 = 5$ $q^2 = p^2/...
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1answer
71 views

Is there a way to prove that $\sqrt[7]{129}$ is irrational using the following theorem?

I want to prove that $\sqrt[7]{129}$ is irrational using the following theorem: Let $n,k$ be natural numbers. Then, $\sqrt[n]{k}$ is rational iff $k$ is the $n\text{-th}$ power of a (natural) number....
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1answer
63 views

Proof of transcendence of $\ln (\pi)$

From Wikipedia $\ln (\pi) $ is unknown to be transcendental. $e^{(ie^{(\ln(\pi)})}=-1$ $i(e^{(\ln(\pi)})=i\pi$ is transcendental. Due to the Lindemann–Weierstrass theorem any transcendental ...
2
votes
2answers
55 views

Rational Distance Problem triple — irrational point

Many points with rational coordinates are known with rational distances to three vertices of a unit square. For example, the following points are rational distances from $a=(0,0)$, $b=(1,0)$, and $c=(...
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0answers
21 views

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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3answers
116 views

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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2answers
33 views

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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0answers
52 views

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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1answer
147 views

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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2answers
51 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
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1answer
391 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 factors....
21
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2answers
373 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
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1answer
77 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 $[0,1]$?...
18
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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 ...
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0answers
37 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, ...
4
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1answer
89 views

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?
2
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3answers
103 views

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 ...
12
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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 ...
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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 $-0.685<\pi^e-e^\pi<-0.675$?
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0answers
34 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 ...
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3answers
111 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
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0answers
19 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 ...
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3answers
51 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 ...
64
votes
3answers
696 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
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1answer
55 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 ...