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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1answer
79 views

Constraining mathematics to a subset of $\mathbb{R}$

Let's imagine we're only using rational numbers for everything in mathematics. Problems arise quite soon when you try to calculate diagonals of squares or perhaps roots of something like $f(x)=x^2-2$. ...
0
votes
1answer
109 views

Does every plane curve contain a rational point?

Does every plane curve contain a rational point? I think the answer is yes, but I can not prove this. Please help. However, if it is possible to build a pathological curve - without rational points, ...
4
votes
1answer
160 views

Systematic way to represent any irrational number

I'm wondering if there's a way to symbolically (or is there a more lose constraint?) represent ANY irrational number in a systematic way. You can represent any rational number as two integers and I ...
1
vote
2answers
87 views

Is it possible to not have irrational numbers?

(Math noob question): Is there a base that can be used like binary that produces no irrational numbers or numbers with an infinite amount of one number after the decimal (don't know the name)? I feel ...
0
votes
1answer
107 views

$\Bbb{Q}$ is not complete: Carification regarding a proof

In class today we proved that $\Bbb{Q}$ is not complet, you used the fact that $$ \sum_{k=0}^N\frac{1}{k!}\underset{N\to+\infty}{\longrightarrow}e\notin\Bbb{Q}.$$ After that I was perplex to prove ...
2
votes
2answers
67 views

rational number plane vector space or not?

Two questions: 1. Is $\mathbb{Q}^2$ a vector space over the field $\mathbb{Q}$? 2. Is $\mathbb{Q}^2$ a vector space over the field $\mathbb{R}$? My answer to the first question is yes. Because the ...
0
votes
1answer
33 views

Application of the Rational Roots Theorem

Let f(x)=3x$^3$ - 40x$^2$ + 97x + 10 a. Find a rational number r such that f(r) = 0. (Hint: Use the rational roots theorem to narrow down possibilities for r.) So, I figured this part out. write r ...
0
votes
1answer
143 views

Monotone increasing sequence of rationals with an irrational limit

I am trying to use rationals in order to approximate irrationals. Is it possible to construct a monotonically increasing sequence of rationals the limit of which is an irrational? If so, how?
0
votes
3answers
71 views

There is at most one way to represent a number as $a+b\sqrt 2$ with rational $a,b$

If $a,b,c,d\in\mathbb Q$ and $a+b\sqrt 2= c + d\sqrt 2$, then prove $a=c$ and $b=d$ ? I don't have any idea to solve this , it's freaking me out.
3
votes
0answers
119 views

Irrational roots of unity?

Is it possible to take irrational roots of unity? For example, say I wanted to solve $f(x)=(x+1)^{\sqrt{2}}=1$. I found that one solution is the obvious $x=0$, and another one can be written nicely as ...
0
votes
1answer
41 views

For which values of $n$, the real part of the $n$-th root of unity is a quadratic irrational?

For which values of $n$, the real part of the $n$-th root of unity is a quadratic irrational? That is, when is it a root of a quadratic polynomial with integer coefficients? I believe that the answer ...
4
votes
2answers
615 views

Unit Quaternion to a Scalar Power

I'm trying to modify a physics engine for efficiency. Currently, as objects move around the world, their orientation (a quaternion) is updated every frame, by multiplying by the rotation (another ...
2
votes
1answer
192 views

New mathematical constant formed by continued fraction with prime numbers?

Notational convention: $$\bigoplus_{k=0}^{\infty}a_k=a_0+\cfrac{1}{a_1+\cfrac{1}{a_2+\cfrac{1}{a_3+\cfrac{1}{\ddots}}}}$$ Let $$ P:=\bigoplus_{k=1}^{\infty}p_k$$ where $p_k$ is the k-th prime number....
0
votes
1answer
65 views

Under what conditions can $a\sqrt{b} \pm c\sqrt{d}$ be written as $u+v\sqrt{w}$?

Let $a,b,c,d \ge 1$ be integers with $b$ and $d$ nonsquare and $a\sqrt{b} \ge c\sqrt{d}$. Now I have three related questions: Under what conditions can one find $u,v,w$ such that $a\sqrt{b} \pm c\...
6
votes
4answers
1k views

Can any two irrational numbers NOT of the form (m+A) and (n-A) be added to produce a rational number?

$m$ and $n$ being rational numbers, A being an irrational number. I was wondering if two irrational numbers when added always yield an irrational number. All the counter-examples I could find were of ...
1
vote
1answer
630 views

How to prove that a number is irrational

We write all postive whole integers after the comma, how do we prove that this is an irrational number? ($0.1234567891011121314...$)
0
votes
0answers
44 views

Are there any general results about the irrationality of $a^{\frac{p}{q}}$?

Are there any general results about the irrationality of $a^{\frac{p}{q}}$ for $a\in\mathbb{Z}^+$, $p,q\in\mathbb{Z}$, $q\neq 0$ and $a\neq 1$?
2
votes
1answer
95 views

Periodicity with irrational numbers

Recently, I invented the following theorem and found a proof, it seems strange since it is very counter-intuitive to me. The proof is long and non-conceptual. Is there a place or a branch of math ...
2
votes
1answer
66 views

How to know equation solution irrational

I would appreciate if somebody could help me with the following problem Q: Is the solution of the equation $$2\cos^2\pi x+\cos \pi x-2=0 $$ irrational ?
5
votes
1answer
138 views

Periodicity with irrational numbers

Recently, I composed the following math problem and found a solution, it seems strange since it is very counter-intuitive to me. Is there a place or a branch of math where I can read about it? Or at ...
0
votes
0answers
61 views

Calculate digits of $\pi$ without needing to reuse them

I am looking for some algorithms that can calculate the digits of $\pi$ without needing to reuse previous digits. I would like to find the most simple and fast algorithms possible. Thanks!
0
votes
2answers
86 views

Represent non-integer values on the factorial base

I want to compute the representation of the following values using the factorial number system: $\pi$ $e$ $\phi$ I know how to do it for integer values, but is it feasible for non-integer values? ...
2
votes
1answer
55 views

Are there any number systems better suited to nature?

For example, number such as $\pi$ and $e$ cannot be represented as rational numbers in our number study and extend in decimal places to infinity. QUESTION: Is there a possibility that some other ...
2
votes
0answers
63 views

Is this irrationality proof correct?

Consider a non-square integer $n$. If its square root was rational, then we would have $$\sqrt n=\frac{a}{b}$$ for some $a,b\in\mathbb{Z}$ and so $a^2=nb^2$. But this is impossible, because $n$ is non-...
1
vote
1answer
43 views

Yet another product of irrational numbers

Let $~\alpha~$ and $~\beta~$ be irrational numbers such that $$~\alpha \notin \{\beta, -\beta\}$$ and $$~\alpha \notin \left\{\frac{1}{\beta}, -\frac{1}{\beta}\right\}$$ I suppose that in this case ...
3
votes
2answers
358 views

square root of 2 irrational - alternative proof

I have found the following alternative proof online. It looks amazingly elegant but I wonder if it is correct. I mean: should it not state that $(\sqrt{2}-1)\cdot k \in \mathbb{N}$ to be able to ...
5
votes
3answers
198 views

What type of numbers are roots of $x^{2} = -1$ themselves?

Are the roots $i, -i$ themselves irrational numbers or complex numbers or left auxiliary so that undefined?
8
votes
3answers
546 views

Prove $\alpha \in\mathbb R$ is irrational, when $\cos(\alpha \pi) = \frac{1}{3}$

I am trying to prove: If $\cos(\pi\alpha) = \frac{1}{3}$ then $\alpha \in \mathbb{R} \setminus \mathbb{Q}$ So far, I've tried making it into an exponential, since exponentials are easier to ...
0
votes
1answer
35 views

How do I check if intervals can be nested?

Hi I'm in grade 10 and I'm doing nested intervals. Under my current circumstances I am unable to have a teacher, so I'm self teaching with the help of textbooks. I understand very, very basic ...
1
vote
3answers
225 views

Prove that if $x$ and $y$ are irrational numbers, there exists an irrational number $z$ such that $y < z < x$

My teacher proposed this question a few days ago along with the similar case for rational numbers. I've already figured out the proof for rational numbers (just prove that their arithmetic mean is ...
1
vote
2answers
68 views

Cantor diagonal argument; related number

I was reading another question on mse about cantors proof and I'm curious about a number that could be defined from it. Well there could be a whole heap of them, but one for now. Define $A=\Bbb{Q} \...
0
votes
2answers
100 views

Correctness of the proof that the set $\{x \in \mathbb{Q} : x>0 \text{ and } x^2>2\}$ does not have a smallest element

Let $F=\{x \in \mathbb{Q} : x>0 \text{ and } x^2>2\}$. I am asked to show that $F$ does not have a smallest element. The hint is to simply prove the claim: 'If $p$ is a rational number in $F$ ...
14
votes
4answers
696 views

Process to show that $\sqrt 2+\sqrt[3] 3$ is irrational

How can I prove that the sum $\sqrt 2+\sqrt[3] 3$ is an irrational number ??
4
votes
2answers
165 views

Can one prove existence of incommensurables without the Pythagorean theorem?

Euclid's proof that the side and the diagonal of a square have no common measure, probably going back to Pythagoreans, reduces it to proving the irrationality of $\sqrt{2}$. This reduction uses the ...
2
votes
2answers
321 views

Example of a proof using the axiom of commensurability

I'm teaching our intro to proofs course (well, one of them) and one of the classic illustrations of an overturned "axiom" is the Greek axiom of commensurability, which stated in geometric terms the ...
3
votes
4answers
117 views

If $q^n$ is irrational for all $n>1$, then $q$ is irrational.

Theorem. Let $q \in \mathbb{R}$ an arbitrary given number. If $q^n$ is irrational for all $n>1$ integer, then $q$ is irrational. My Questions. What is a the name of this statement and what is the ...
-1
votes
1answer
79 views

How is circle closed?

I have this thought that circle in 'real' is not a closed figure. We all know that 'pi' is irrational.And integers are nodes in a 'monstrous' line of real numbers. Irrational numbers are non-...
5
votes
4answers
227 views

Is $\ln\sqrt{2}$ irrational?

I know that the natural log of any positive algebraic number is transcendental, as a consequence of the Lindemann-Weierstrass theorem, but what about the natural log of the square root of two (which ...
3
votes
1answer
102 views

Rational values of $\sin(\log(x))$

Apart from the trivial solution $\sin(\log(1))=0$, is $$\sin(\log(x))$$ ever rational if $x$ is rational?
0
votes
2answers
139 views

How do you multiply infinite quantities?

Out of curiosity I was watching this video from njwildberger on youtube: https://www.youtube.com/watch?v=4DNlEq0ZrTo Where he says that you can't define associativity between irrational numbers ...
2
votes
1answer
27 views

Existence of a sequence of integers $\lbrace a_k\rbrace_{k\geq 1}$ so that the first $k$ digits of $a_k\alpha$ are $0$ where $\alpha$ is irrational.

Let $\alpha$ be an irrational number. Is there a sequence $\lbrace a_k\rbrace_{k\geq 1}$ of integers so that the first $k$ digits of the fractional part of $a_k·\alpha$ are $0$? (in base $2$, for ...
6
votes
5answers
2k views

For an irrational number $a$ the fractional part of $na$ for $n\in\mathbb N$ is dense in $[0,1]$ [duplicate]

How to prove that the $\{$ fractional part of $n\alpha\mid n \in \mathbb{N}$ $\}$ is dense in $[0,1]$ for an irrational number $\alpha$. NOTICE that $n$ is in $\mathbb{N}$ Also notice that this is ...
0
votes
2answers
100 views

$1$ is not congruent because of Fermat's Last Theorem?

I would like someone to explain something I did not understand. I was reading a page called "nuking the mosquito" where they give very complex proofs for very simple results. The proof I want to talk ...
3
votes
3answers
435 views

Irrational number “test”?

Suppose we have a finite quantity $a$, which we would like to prove to be irrational, supposing that it is indeed irrational. Then, would it be enough to show that $$a=\lim_{n\to\infty}\frac{u_n}{v_n}...
2
votes
2answers
208 views

Are numbers like $\left ( -2 \right )^{\sqrt{2}}$ real or complex?

I know that numbers with rational power can be converted to radicals and based on the degree of the radical we can say that whether they are real or complex. But what about numbers like $\left ( -2 \...
0
votes
2answers
1k 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
votes
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 ...
0
votes
1answer
117 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 $x_{n+2}=\frac{x_{n}x_{...
0
votes
2answers
75 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
votes
1answer
505 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 ...