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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2
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0answers
24 views

Hilbert curve: how can I find the image of an irrational point in $[0,1]$?

I have a question on the construction of the Hilbert curve: how can I find the image of an irrational point in $[0,1]$? Consider the Hilbert curve $$ f_h:[0,1]\rightarrow [0,1]^2 $$ Consider a ...
1
vote
1answer
57 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
25 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 ...
0
votes
2answers
30 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
vote
2answers
70 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 = ...
1
vote
1answer
69 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) ...
2
votes
2answers
54 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 ...
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
110 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
415 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
450 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
561 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: $$ ...
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
0answers
79 views

Show that a sum of consecutive radicals is irrational $\forall n$ [closed]

I need to show that the number $\sqrt 2+ \sqrt[3]{3}+\sqrt[4]{4}+\sqrt[5]{5}+\cdots+\sqrt[n]{n}$ is irrational for any $n\ge2$. I don't have a clue about how I could show that. Thank you!
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
51 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
102 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 ...
2
votes
1answer
143 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
12k 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
259 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
130 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
467 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
355 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 ...
1
vote
1answer
75 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 ...
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 ...
1
vote
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, ...
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 ...
4
votes
1answer
85 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?
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 ...
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 $-0.685<\pi^e-e^\pi<-0.675$?
-8
votes
3answers
107 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
0
votes
1answer
111 views

Archimedes' Approximation 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 ...
1
vote
0answers
18 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 ...
61
votes
3answers
651 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
votes
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 ...
5
votes
5answers
133 views

Convergent sequence of irrational numbers that has a rational limit.

Is it possible to have a convergent sequence whose terms are all irrational but whose limit is rational?
2
votes
2answers
215 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 ...
1
vote
1answer
52 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 ...
1
vote
2answers
60 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 ...
1
vote
4answers
105 views

Proof that this is an irrational number

Prove that $2\sqrt2 + \sqrt7 $ is an irrational number. I am trying to use contradiction to show that this is irrational. Also I am using the fact that $ 2\sqrt2 + \sqrt7 = \frac{1}{2\sqrt2 - ...
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 ...
29
votes
4answers
4k views

Proving that $m+n\sqrt{2}$ is dense in R

I am having trouble proving the statement: Let $S = \{m + n\sqrt 2 : m, n \in\mathbb Z\}$. Prove for every $\epsilon > 0$, The intersection of $S$ and $(0, \epsilon)$ is nonempty.
1
vote
0answers
31 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 ...