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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15 views

irrationality of $\sum_{k=1}^{n} k^{\frac{1}{m}}$

For arbitrary $n \geq 2$ and $m \geq 2$, is $$\sum_{k=1}^{n} k^{\frac{1}{m}}$$ an irrational number?
2
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3answers
115 views

proving $ \sqrt 2 + \sqrt 3 $ is irrational [duplicate]

I need to proof that $\sqrt{3} + \sqrt{2}$ is irrational, without using the fact that an irrational number plus a rational number equals irrational. also, i can't use the rational root theorem. that's ...
1
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1answer
45 views

Exchangeability of union and intersection of open balls around all rational numbers in $[0,1]$

Let $X:=[0,1]$ and $V:= X \cap \mathbb{Q}= \{v_1,v_2,...\}$. For $n,k \ge1$ set $I_{n,k}:= X \cap (v_n-2^{-(n+k)},v_n+2^{-(n+k)}) $. Is it true that $$ \bigcup_{n\ge1} \bigcap_{k\ge1} I_{n,k} = ...
2
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0answers
42 views

$x-\left\lfloor x \right\rfloor +\frac { 1 }{ x } -\left\lfloor \frac { 1 }{ x } \right\rfloor =1\implies x$ is irrational. [duplicate]

If $x-\left\lfloor x \right\rfloor +\frac { 1 }{ x } -\left\lfloor \frac { 1 }{ x } \right\rfloor =1$, then $x$ is irrational. I am thinking of using the contrapositive: If $x$ is rational, then ...
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1answer
44 views

Construction of irrational numbers

Can an irrational number be constructed which is a) not any known transcendental number b) not a surd? If yes, then how can I construct one? A detailed answer regarding the theory behind this and some ...
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2answers
61 views

Show this equation has no rational solution $ (x^{39}+7)^3-2(x^{39}+6)+1=0$

I know that i can use the rational root theorem, and substitution, but I don't know how to connect with this equation. Can someone help me with that?
3
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4answers
425 views

Integral of rationals

Define $f(x)$ as $$f(x)=\begin{cases}0,&\text{if }x\in \mathbb{Q}\\ 1,&\text{if }x\notin \mathbb{Q}\;. \end{cases}$$ Considering the fact that there is a countable infinity of rationals yet an ...
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1answer
47 views

Finding irrational and complex roots of a cubic polynomial

I've got a question which shows short answers and no method so I'm trying to find a hand performed method of solving the cubic polynomial for the roots: ...
0
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1answer
32 views

Polynomial with irrational coefficients

Let $2^{q/4}=2^{(4w+1)/4}=2^{1/4} 2^w=X+f$ where q is a prime $X,w\in{N}$ and $0<f<1$. Since $2^{1/4}$ is irrational $f$ is irrational. Is there any way to prove that $f^4+4 f^3 X+6 f^2 ...
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0answers
39 views

express an irrational as the sum of a rational and irrational number

Simple question, apologies. This is from some sample high school math questions, target is age 16 pupils. I don't think any great sophistication is expected. P + Q = $\sqrt {5}$ P is a rational ...
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1answer
94 views

If $x - \lvert x \rvert + \frac{1}{x} - \lvert \frac{1}{x} \rvert = 1$ then $x$ is irrational [duplicate]

For every real number $x$, if $x - \lvert x \rvert + \frac{1}{x} - \lvert \frac{1}{x} \rvert = 1$ then $x$ is irrational If $x$ equals $\sqrt{2}$ I get an inequality... So is this claim false?
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1answer
52 views

Infinite sequence of digits without consecutive repeating subsequenes

Problem description: Suppose we use a set of digits $\{0,1,2\}$ to form a sequence, for example \begin{equation} 120210120102012102010210120212010\cdots \end{equation} The length can be finite or ...
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1answer
67 views

Prove that the proof for $\sqrt{2}$ being irrational doesn't work for $\sqrt{4}$

So I am supposed to show that the standard way of proving that $\sqrt{2}$ is irrational doesn't hold for $\sqrt{4}$. So making the assumption that $p$ and $q$ are natural numbers and any common ...
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2answers
62 views

Proving the irrationality of $\sqrt{5} = x \sqrt{7} + y$

I need to prove that there are no rational numbers $x, y$ that $$\sqrt{5} = x \sqrt{7} + y$$ We know that square root of prime is irrational so $y = 5 - 7x$ so the only number for it to be rational ...
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2answers
49 views

prove or disprove if a number is irrational

Prove or disprove : I'm pretty sure this isn't true yet i can't find a counter example. Thanks in advance !
5
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2answers
220 views

Proving that $x$ is irrational if $x-\lfloor x \rfloor + \frac1x - \left\lfloor \frac1x \right\rfloor = 1$

Prove : $$ \text{If } \; x-\lfloor x \rfloor + \frac{1}{x} - \left\lfloor \frac{1}{x} \right\rfloor = 1 \text{, then } x \text{ is irrational.}$$ I think the way to go here is to falsely assume that ...
1
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2answers
20 views

If you apply the Distributive Property to a Rational and an Irrational number, which will your solution be?

Say that "A" and "B" are Rational, and C is irrational, would the solution to "A(B+C)" be Rational or Irrational? An example for clarification would be wonderful.
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2answers
137 views

Where, if ever, does the decimal representation of $\pi$ repeat its initial segment?

I was wondering at which decimal place $\pi$ first repeats itself exactly once. So if $\pi$ went $3.143141592...$, it would be the thousandth place, where the second $3$ is. To clarify, this ...
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0answers
24 views

Digits of irrational exponentiation

Let us have positive irrational numbers $a$ and $b$ represented by functions $f_a,f_b\colon\mathbb{N}\to\mathbb{N}$ respectively such that $f_a(0)=\left \lfloor{a}\right \rfloor$ and $f_a(i)$, ...
5
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1answer
118 views

How to prove $e^{1/e}$ is irrational?

How do we prove $e^{\frac{1}{e}}$ is irrational ? Also how do we show it is transcendental ? The number $\eta = \exp(\exp(-1))$ occurs naturally in the context of tetration and power towers. Let ...
4
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1answer
117 views

Show that $a = b = c = 0$ for $a\sqrt{2} + b = c\sqrt{3}$ is

This is the following question: Suppose that $a, b, c$ are integers such that $a\sqrt{2} + b = c\sqrt{3}$ (i) By squaring both sides of the equation, show that $a = b = c = 0$ The answer says that ...
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3answers
2k views

subtraction of two irrational numbers to get a rational [duplicate]

Say you have a number like $\pi$ or e. Is it possible to subtract another number from it and end up with a rational number? I mean I guess you could write an equation like $\pi-x=3$ But could there ...
4
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1answer
147 views

Prove that $x$ is rational iff $a=b=0$

So the question goes let $x = a\sqrt3 + b\sqrt 5$ where $a,b$ rational. Prove that $x$ is rational iff $a=b=0$. I think I can prove this but I'm not sure if my proof is correct or rigorous. Well ...
8
votes
2answers
962 views

Is there any basis transformation under which all irrational numbers are rationals and vice-versa?

For example, if you change the length of your "unit scale" or basis for numbers to $\sqrt{2}$, then you may represent all fractional multiples of $\sqrt{2}$ as "rational numbers" in the new basis ...
0
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0answers
28 views

Rhetoric algebra for three common irrational ratios: $\frac{1}{\sqrt2}$, $\frac{1}{\sqrt3}$, $\frac{1}{\sqrt5}$

I know there are many ways to express these three irrational numbers as ratios: $\frac{1}{\sqrt2}$, $\frac{1}{\sqrt3}$, $\frac{1}{\sqrt5}$. I have three proposals below from my view of point. Can you ...
4
votes
2answers
103 views

Proving $\frac{\arccos\frac15}\pi\not\in\Bbb Q$

How would one prove $$\frac{\arccos\frac15}\pi\not\in\Bbb Q$$ Fiddling around with numbers hasn't led me anywhere: Suppose $\frac{\arccos\frac15}\pi\in\Bbb Q$, suppose it is equal to $\frac ab$. Then ...
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1answer
36 views

Geometrical progression from 1 to $\sqrt5$ to 3 by arithmetics

How do you formulate arithmetically this pretty simple Euclid geometric progression from one to three via square root of 5? GK = 1, AE = $\sqrt5$, GH = 3 but what is the equation behind the last ...
4
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2answers
85 views

To prove that element $\frac{3}{n}+i\frac{4}{5}$ has an infinite order in $\mathbb{C}$ for any $n\in\mathbb{Z}\backslash \{0\}$

The problem is to prove that element $z=\frac{3}{n}+i\frac{4}{5}$ has an infinite order in the group $(\mathbb{C},\, \cdot\, )$ for any non-zero integer $n$. Let's consider the case $|n|\neq 5$. ...
0
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1answer
67 views

What is the closest fraction (that isn't something like 31415…/1000…) that gets you pretty close to pi?

I'm just wondering what is the closest fraction (question for math nerds and geniuses) that isn't like pi/length-of-pi that gets you relatively close (like accurate to the 20th place) to pi? For ...
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3answers
78 views

Prove $\sqrt{2}+\sqrt[3]{5}$ is irrational [duplicate]

Prove: $\sqrt{2}+\sqrt[3]{5}$ is irrational I have tried to look at $1+\sqrt{2}+\sqrt[3]{5}$ but I can see how to continue
2
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3answers
192 views

Prove that $\sqrt{45}$ is irrational (using Euclid’s Lemma)

Prove that $\sqrt{45}$ is irrational (using Euclid’s Lemma) Assume $\sqrt{45}$ is rational. By definition of rational : $\sqrt{p}=\frac{a}{b}$= $\sqrt{45}$=$\frac{a}{b}$ for some $a,b$ integers and ...
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2answers
151 views

Proving $\sqrt[4]{4}$ is irrational

Prove: $\sqrt[4]{4}$ is irrational I know that $\sqrt{p}$ is irrational where $p$ is a prime number. So $\sqrt[4]{4}=\sqrt[4]{2*2}=16*\sqrt[4]{2}=16*2^{\frac{1}{2}^\frac{1}{2}}$ What can I say ...
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1answer
62 views

determine the frontier

determine the frontier of the set R\Q (where R is the real numbers and Q is the rational numbers). I figured R\Q is the same as saying the real line minus all the rational numbers which would just ...
0
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0answers
20 views

Check the validity of a statement concerning functional continuity and limits at infinity, perhaps also involving a bit of real-number analysis [duplicate]

I've been stuck on this problem: If $f\in C(0,+\infty)$, and $f(na)\to 0$ as $n\to +\infty$ for every $a>0$, then is $$\lim_{x\to+\infty}f(x)=0$$ also true? If we replace continuity by ...
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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 - ...
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2answers
63 views

Prove that $\sqrt{5}+1$ is irrational

I'm guessing this is pretty basic, but I've been struggling with this for a while now. Any help would be appreciated. Thanks.
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2answers
89 views

Why a segment of length $\sqrt{2}$ can be drawn but a segment of length $\pi$ cannot?

We know that both $\pi$ and $\sqrt{2}$ are irrational. Also, it has been proved that a segment of length $\pi$ can not be drawn whereas a segment of length $\sqrt{2}$ can be drawn. Why is it so, ...
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1answer
18 views

A question of rationality of integral powers

Given $a,b\in\mathbb{Z}$, is $x$ from $a^x=b$ ever rational? More specifically, is $x$ in $2^x=3$ irrational?
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0answers
26 views

Finding a 10-digit pattern in the decimal expansion of a normal number?

Is there any way to estimate how "deep" we'd need to go into the decimal expansion of a normal number to find a specific string of digits, say a 10-digit pattern? What about a 100-digit pattern?
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2answers
97 views

Does $\sin n$ have a maximum value for natural number $n$?

In formal, does there exist $k\in\mathbb{N}$ such that $\sin n\leq\sin k$ for all $n\in\mathbb{N}$?
5
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2answers
142 views

Is $\sqrt[2]{(2/7)}$ irrational?

I have to show that the $\sqrt(2/7)$ is irrational. Here is my work.
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1answer
100 views

Neighbors of Irrational Numbers on Real Number Line

I was looking at a post on MathOverflow about "What is your favorite 'strange' function?" One of the answers mentioned Thomae's function claiming that the function was "continuous at all irrationals ...
1
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1answer
35 views

Characterizing the roots of rational numbers

I am trying to prove the statement: if $n \in \mathbb{Q}$ and $\sqrt[m]{n} \in \mathbb{Q}$ for all positive integers, then $n = 1$. In my work, I have done all the work given by the top answer to ...
1
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1answer
102 views

Are irrational numbers irrational by nature? [duplicate]

I remember hearing an interesting theory once, I don't know the source. Since there are some numbers that are precisely expressible in decimal notation that repeat in a binary base, and vice versa, ...
2
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1answer
42 views

How to obtain the negative rational numbers

Is there any formula for generating negative rational numbers? Can Calkin–Wilf tree be used for negative numbers?
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3answers
99 views

why is the value of $(-1)^\frac{2}{3}$ not 1?

$(-1)^2$ equals 1 also $(-1)^\frac{1}{3}$ equals -1 any way you arrange it, it should be one. how ever, typing this into Wolfram or google,gives me an irrational complex number. I would love some ...
28
votes
10answers
4k views

Is it possible to represent every irrational number as a (limit of) an infinite sum of rational numbers? [duplicate]

For instance, we can certainly represent π in this fashion. $$ \frac{\pi}{4} \;=\; \sum_{n=0}^\infty \, \frac{(-1)^n}{2n+1} .\! $$ $\ln(2)$ is also irrational. And even that can be represented as an ...
0
votes
5answers
90 views

Prove rational sum and product of two irrational numbers

I need to prove that $$\exists a,b \in \mathbb{R} \setminus \mathbb{Q} : a + b, ab \in \mathbb{Q}$$ Any ideas? I, unfortunately, don't have one yet. The most obvious way with equations in integers ...
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votes
1answer
125 views

If $x$ is an irrational number, then $3x^2 + 2$ is an irrational number? [closed]

Prove or disprove using direct or indirect proof that if $x$ is irrational, then $3x^2 + 2$ is irrational? Thank you in advance.
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4answers
268 views

Is $\pi^0$ actually rational? How about $\pi^i$? [duplicate]

Is there a rational argument that a transcendental or irrational number raised to zero should magically turn it into an integer, beyond obtuse convention? How about $\pi^i$? Is there a reasonable ...