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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0
votes
1answer
42 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 ($x+\epsilon$) ...
3
votes
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.
-4
votes
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?
-1
votes
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
votes
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
votes
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 ...
22
votes
3answers
6k views

irrationality of $\sqrt{2}^{\sqrt{2}}$.

The fact that there exists irrational number $a,b$ such that $a^b$ is rational is proved by the law of excluded middle, but I read somewhere that irrationality of $\sqrt{2}^{\sqrt{2}}$ is proved ...
-2
votes
7answers
3k views

$0.333333$ - a recurring or non-terminating decimal?

I have read like, 1.All terminating and recurring decimals are RATIONAL NUMBERS. 2.All non-terminating and non recurring decimals are IRRATIONAL NUMBERS. if the statements are right, then here ...
-1
votes
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 ...
1
vote
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
votes
1answer
125 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?
2
votes
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
vote
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 $\quad(a+b)^\ell$
0
votes
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
votes
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 $2+\sqrt{3}$. ...
9
votes
5answers
3k views

Sum of two periodic functions is periodic?

I have following paragraph taken from the Stanford's study material. Question: Is the sum of two periodic functions periodic? Answer: I guess the answer is no if you are Mathematician, yes ...
-1
votes
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
votes
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
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 ...
0
votes
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 ...
18
votes
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 $...
1
vote
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 - ...
-1
votes
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 $...
1
vote
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 ...
0
votes
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 ...
1
vote
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/...
1
vote
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....
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=(...
0
votes
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 ...
1
vote
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.
3
votes
3answers
421 views

Field containing all square roots of rational numbers

What is the smallest field which contains all square roots of positive rational numbers? I guess I mean “smallest” in terms of set inclusion, i.e. the minimal one with regard to the “$\subseteq$” ...
4
votes
4answers
328 views

Relationship between degrees of continued fractions

I'm trying to compute the values of differing degrees of continued fractions like $\sqrt 2$, $e$ and other similar fractions. My theory was to take the reduced fraction at an arbitrary depth and the ...
6
votes
1answer
451 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 ...
31
votes
2answers
581 views

Show that $e^{\sqrt 2}$ is irrational

I'm trying to prove that $e^{\sqrt 2}$ is irrational. My approach: $$ e^{\sqrt 2}+e^{-\sqrt 2}=2\sum_{k=0}^{\infty}\frac{2^k}{(2k)!}=:2s $$ Define $s_n:=\sum_{k=0}^{n}\frac{2^k}{(2k)!}$, then: $$ s-...
1
vote
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.
5
votes
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. ...
5
votes
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 ...
2
votes
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 ...
1
vote
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 ...
37
votes
4answers
13k 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
261 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
134 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
504 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
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 ...
21
votes
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 ...
9
votes
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....