Numbers not expressible as a ratio of two integers. Examples: $\sqrt{2},\phi,e,\pi,\zeta(3)$. Some of them are algebraic ($\sqrt{2},\phi$) and some transcendental ($e,\pi$).

learn more… | top users | synonyms

0
votes
2answers
20 views

How can I prove that the square root of two prime numbers multiplied is non-rational number?

$P$ and $Q$ are two distinct prime numbers. How can I prove that $\sqrt{PQ}$ is an irrational number?
1
vote
0answers
20 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
32 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
64 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
6answers
155 views

Proving the irrationality of $\sqrt{5}$

I am working on proving that $\sqrt{5}$ is irrational. I think I have the proof down, there is just one part I am stuck on. How do I prove that $x^2$ is divisible by 5 then x is also divisible by 5. ...
-2
votes
2answers
69 views

If$\ f(n)-g(n)$ tends to a rational number as$\ n \to \infty$, might we say that either both have an “irrational growth” or a “rational” one? [closed]

Let me explain myself.$\ x-y=\frac{a}{b}$ if and only if either$\ x$ and$\ y$ are both rational or irrational. Does this also mean that with$\displaystyle\lim_{n \to +\infty} f(n)=\lim_{n \to +\infty} ...
1
vote
4answers
56 views

Prove or disprove the rationality of $ x^y $

Prove or disprove: "If $x$ is a rational number, and $y$ is an irrational number then $x^y$ is irrational" I am stuck with this, these are my steps. let $x=2$ and $y=\sqrt{2}$ ...
0
votes
1answer
19 views

Quotient of two rational sequences and the nature of its limit

Suppose we have two sequences of rational numbers, $(p_i)_{i=1}^\infty$ and $(q_i)_{i=1}^\infty$, and suppose $$\lim_{i\to\infty}\frac{p_i}{q_i}=c<\infty,$$ where $c$ is known. Are there any ...
1
vote
2answers
65 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 ...
3
votes
2answers
62 views

Proof of $\sqrt{n^2-4}, n\ge 3$ being irrational

Is the proof of $n\ge 3$, $\sqrt{n^2-4} \notin \mathbb{Q} \ \text{correct}$? $\sqrt{n^2-4} \in \mathbb{Q} \\ \sqrt{n^2-4} = \frac{p}{q} \\ (\sqrt{n^2-4})^2 = \left(\frac{p}{q}\right)^2 \\ ...
1
vote
5answers
278 views

Show that an expression is irrational

Show that for all $n\in \mathbb{N}$ the number $(\sqrt{2}-1)^n$ is irrational. I do not get the idea of the proof at all, any help appreaciated. edit: I am also thinking whether it will be possible ...
0
votes
1answer
30 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 ...
1
vote
2answers
26 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 ...
8
votes
6answers
810 views

Is $x^{1-\frac{1}{n}}+ (1-x)^{1-\frac{1}{n}}$ always irrational?

Let $x$ be rational with $0<x<1$ and let $y$ be the rational defined by $y = 1 - x.$ Let $n$ be any natural number with $n>2.$ Then I want to prove that $$x^{(1-1/n)}+ y^{(1-1/n)}$$ will ...
0
votes
1answer
41 views

Irrational power of root

Let $a$ and $b$ be rational numbers, such that $\sqrt{a}$ and $\sqrt{b}$ are irrational. Can $\sqrt{a}^\sqrt{b}$ be rational? I found examples, where the irrational power of an irrational number is ...
0
votes
3answers
101 views

Proving that the square root of 5 is irrational

Prove that $\sqrt{5}$ is irrational. I begin with the identity $(\sqrt{5} + 2 )(\sqrt{5} - 2 ) = 1$. Then I am told to extract $\sqrt{5}$ from the first or second factor and consider it to be ...
9
votes
4answers
179 views

Show that $({\sqrt{2}\!+\!1})^{1/n} \!+ ({\sqrt{2}\!-\!1})^{1/n}\!\not\in\mathbb Q$

How could we prove that there does not exist a positive integer $n$, so that the number $$({\sqrt{2}+1})^{1/n} + ({\sqrt{2}-1})^{1/n}$$ is rational? I think it could be done inductively from a more ...
0
votes
1answer
17 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
53 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?
1
vote
1answer
68 views

What can be said about $\pi+e$ and $\pi e$? Are these numbers rational or irrational? [duplicate]

"homework" What can be said about $\pi+e$ and $\pi e$? Are these numbers rational or irrational? I know that both $\pi$ and $e$ are irrational. What can be said about $\pi+e$, and $\pi e$?
1
vote
3answers
34 views

Prove that if $x$ and $y$ are rational numbers and $y\ne 0$, then $x/y$ is a rational number

Prove that if $x$ and $y$ are rational numbers and $y\ne 0$, then $\frac{x}{y}$ is a rational number. How do I prove this, and also which proving method would I use? I'm confused between that and ...
0
votes
3answers
55 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
32 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 ...
3
votes
2answers
96 views

Prove that $\sqrt[n]{2}+\sqrt[n]{3}$ is irrational for every natural $n \ge 2$.

I want to prove that that $\sqrt[n]{2}$ + $\sqrt[n]{3}$ is irrational for every natural $n \ge 2$. I tried to use some theorem of minimal polynomials, but I get nothing. Also i tried to assume that ...
0
votes
1answer
29 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 ...
2
votes
2answers
47 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
154 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 ...
0
votes
1answer
54 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 ...
7
votes
4answers
875 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
417 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
6answers
262 views

Prove that the square root of 3 is irrational [duplicate]

I'm trying to do this proof by contradiction. I know I have to use a lemma to establish that if $x$ is divisible by $3$, then $x^2$ is divisible by $3$. The lemma is the easy part. Any thoughts? How ...
0
votes
0answers
38 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
73 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
56 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
105 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
36 views

Calculate digits of pi without needing to reuse them

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

Verify if the following demonstration is correct

THEOREM: "There are $a, b$ irrationals, such that $a^b$ is rational." PROOF: " If $\sqrt{2} ^ \sqrt{2}$ is rational, $a = b = \sqrt{2}$, otherwise, $a = \sqrt{2} ^ \sqrt{2}$ and $b = \sqrt{2}$, so ...
1
vote
2answers
26 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? ...
3
votes
1answer
38 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 ...
3
votes
0answers
50 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 ...
1
vote
1answer
31 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
82 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 ...
6
votes
3answers
135 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?
7
votes
3answers
420 views

Help with determining irrationality of a number?

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
24 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
56 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
49 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
57 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 ...
13
votes
4answers
568 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 ??
3
votes
2answers
125 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 ...