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 ($e,\pi$).

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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 $f(x)=x^2-2$. ...
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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
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1answer
63 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 ...
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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. ...
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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} ...
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4answers
55 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 $y=\sqrt{2}$ ...
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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
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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
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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 \\ ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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
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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 ...
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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
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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?
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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 $\pi e$?
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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 ...
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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.
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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
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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
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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
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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
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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
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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 ...
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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 ...
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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? ($0.1234567891011121314...$)
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6answers
260 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
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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
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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
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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 ?
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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 ...
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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!
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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 ...
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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? ...
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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
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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 ...
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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 ...
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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 ...
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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?
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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 ...
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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 ...
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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 ...
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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} ...
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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 ...
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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 ??
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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 ...
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2answers
119 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 ...