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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Roots of polynomials: Vieta's Formula

Let $p_n(t) = c_0 + c_1 t + c_2 t^2 + \ldots + c_n t^n$ with $c_i \in \mathbb{Q}$ and let the roots of $p_n(t) = 0$ be $R = \{r_1, r_2, \ldots r_n \}$. Vieta's formula states that $\sum_{i=1}^n r_i = ...
2
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3answers
124 views

Irrationality of $n$-th root of positive rationals other than $1$

If $a,b \in \mathbb Z^+ , a \ne b$ then is it true that $\sqrt[a+b]{\dfrac ab}$ is irrational ? This question actually popped up from seeing whether there exists a non-trivial homomorphism from ...
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0answers
33 views

Conclusions about addition and multiplication of rational and irrational numbers.

Let $r_1,r_2 \in \mathbb{R}\setminus\mathbb{Q}$, $q\in \mathbb{Q}$ , does the folliwng values are rational, irrational , or we can't decide? $r_1 + q$ $r_1 + r_2$ $r_1 r_2$ $r_1q$ my answers: ...
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1answer
40 views

Prove that √m is irrational by showing that n√m is empty

Let m∈N be such that m≠k^2 for all k∈N. Prove that √m is irrational by showing that {n∈N: n√m∈N} must be empty.
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2answers
58 views

Cardinality of set of Dedekind cuts (elementary)

Under the Dedekind construction the irrationals are defined as those cuts $(A,B)$ where $B$ has no least element ($A$ not having a greatest element by definition), for example the $q^2=2$ case. I can ...
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2answers
30 views

$\sqrt{m}$ irrational

Thinking about it, I think I found the following criterion for irrationality of $\sqrt{m}$ if $m$ is a positive integer. Let $p_1^{a_1}\cdots p_k^{a_k}$ be the prime factorization of $m$. Then ...
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2answers
49 views

Difficult denomiator rationalization questions

These are two questions from a competitive exam involving irrationals where I am supposed to simplify it to match one of the given options. QUESTION 1: The value of $$ \frac {2 (\sqrt 2+ ...
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2answers
53 views

Find the values of k for which $\sqrt{1+\frac{k}{n}}$ is irrational.

I would like to find the positive integers $k$ for which $\sqrt{1+\frac{k}{n}}$ is irrational for all $n\in\mathbb{N}$. I was led to this question when I was making up an example for my class, and I ...
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1answer
44 views

Irrationality of Decimal Expansion of Primes

I've heard the proof that this number is irrational is accessible to even a novice to number theory: $\alpha = 0.2 \ 3 \ 5 \ 7 \ 11 \ 13 \ 17 \ldots$ The proof may utilize that a number is ...
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5answers
547 views

How do you solve a logarithm with a non-integer base?

How would one calculate the log of a number where the base isn't an integer (in particular, an irrational number)? For example: $$0.5^x = 8 \textrm{ (where } x = -3\textrm{)}$$ $$\log_{0.5}8 = -3$$ ...
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2answers
55 views

Is such a number necessarily irrational?

Suppose $(q_{n})_{n\in\mathbb{Z}_{\gt 0}}$ is a decreasing sequence of positive rational numbers such that $Q:=\displaystyle{\sum_{n>0}q_{n}}$ is finite. Let's denote by $n_{i}$ and $d_{i}$ the ...
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2answers
121 views

Is $\mathbb{R}\setminus\mathbb{Q}$ a union of countable family of closed sets?

Can we represent set of irrational numbers as union of countable family of closed sets?
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1answer
148 views

How to prove that the problem cannot be solved by the four Arithmetic Operations?

The original prolbem is as in the figure: Suppose the square has unit side length, find the area of blue region. The exact solution is: $$\begin{aligned}S=&\frac{\pi-\sqrt{7}}{4}+2 ...
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4answers
136 views

Show that $\sqrt[3]{2} + \sqrt[3]{4}$ is irrational

Our professor asked us this to prove that $$ \sqrt[3]{2} + \sqrt[3]{4} \notin \Bbb Q. $$ I know how to prove each one separately that it is irrational, but when it comes to summing two irrational ...
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3answers
105 views

subtraction of two repeating decimals rationals

When I was looking at the proof that every repeating decimal is rational, I came across this example: $x=5.33333333\ldots$ ($3$ repeat indefinitely) $10x=53.3333333\ldots$ ($3$ repeat indefinitely) ...
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1answer
47 views

Does it exist a function that is continuous at every rational point and discontinuous at every irrational point?And vice versa?

Actually there are 2 questions, but they are closely related. Does it exist a function that is: 1. Continuous at every rational point and discontinuous at every irrational point? 2. Continuous at ...
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2answers
38 views

Approximation of numbers [closed]

How could we approximate an irrational number by rationals?? Could you give me some hints?? I don`t have any idea how we could approximate them by rationals...
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1answer
54 views

A density question

Let $\theta \in \mathbb{R} \setminus \mathbb{Q}$. Is the set $\{ (2n+1) \theta \bmod 1: n \in \mathbb{N} \}$ dense in $[0,1]$?
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1answer
52 views

Identify irrational basis of $\mathbb{Q}$-vector space

A real sequence $\mathbf{x}=(x_k)_{k\in\mathbb{N}_0}$ is of the form $$ x_k=\alpha r_k,\quad \alpha\in\mathbb{R}\backslash\mathbb{Q},\quad r_k\in\mathbb{Q},\tag{*} $$ if and only if all of its terms ...
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3answers
93 views

Prove that $(√3+2)^{m}$ is not a natural number for all natural numbers $m≥1$

The aim of this question is to show this lemma: Prove that $(√3+2)^{m}$ is not a natural number for all natural numbers $m≥1$.
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0answers
42 views

Sum of 2 different irrational logarithms = Irrational?

I am having some problems proving that the following sum is irrational or rational: $\log_2(3)+\log_3(2)$ = irrational. This is all I've got for now: $\log_2(3)=\frac mn \iff 2^{\frac mn}=3 \iff ...
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1answer
47 views

Nature of the range of $e^x$

Apart from the trivial cases, $x=\log a$ where $a\in\mathbb{Q}$, are all values of $e^x$ irrational? Are some transcendental?
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7answers
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How to show that the product of two irrational numbers may be irrational?

Show that the product of two irrational numbers may be irrational. You may use any facts you know about the real numbers. All we know is that $\sqrt{2}$ is irrational and that $\sqrt{2}\cdot ...
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3answers
80 views

How do irrational numbers lie on the number line? [closed]

If we construct a square with side length 1, take its diagonal length : $\sqrt{2}$ However I still don't understand HOW it can lie on the number line. Imagine another irrational number $\pi = ...
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2answers
70 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?
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2answers
434 views

Irrationality of sum of two logarithms

I try to prove that the number $$\log_2 5 +\log_3 5$$ is irrational. But I have no idea how to do it. Any hints are welcome.
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1answer
73 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
43 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
72 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
188 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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4answers
124 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
27 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 ...
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2answers
74 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 ...
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2answers
65 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
288 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
34 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
37 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
840 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
54 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
113 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 ...
11
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5answers
210 views

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

How could we prove that for every positive integer $n$, the number $$({\sqrt{2}+1})^{1/n} + ({\sqrt{2}-1})^{1/n}$$ is irrational? I think it could be done inductively from a more general ...
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1answer
21 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 ...
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1answer
87 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
80 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
50 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
60 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
50 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 ...
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2answers
105 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
32 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
63 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 ...