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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6
votes
4answers
892 views

Proof: Is there a line in the xy plane that goes through only rational coordinates?

Question: Is there a line in the XY plane that has all rational coordinates. Prove your answer. Idea: There is most certainly not. I believe it can be shown that between any 2 rational points that ...
8
votes
2answers
129 views

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

$\sqrt{2} + \sqrt[3]{3}$ is irrational ? These are my steps - $\sqrt{2} + \sqrt[3]{3} = a$ $3 = (a-\sqrt{2})^{3}$ $3 = a^{3} -3a^{2}\sqrt{2} + 6a -2\sqrt{2}$ $3a^{2}\sqrt{2}+2\sqrt{2} = ...
2
votes
3answers
74 views

Proving f(x)=0 for all x in [a,b] when we only know that f is continuous and f(x)=0 when x is rational. [duplicate]

The question is as follows a.) Let $f(x)$ be continuous function on an interval [a,b] and suppose that $f(x)=0$ for each rational value $x$ in [a,b]. Prove that $f(x) = 0$ for all $x \in [a,b]$. b.) ...
6
votes
3answers
228 views

Dense set in the unit circle- reference needed

For $x \notin \pi\mathbb Q$, that is, a real $x$ that is not a rational multiple of $\pi$, consider the set $$\{(\cos nx,\sin nx):n = 0,1,2,...\}.$$ It is known that this set is dense in the unit ...
0
votes
1answer
106 views

Prove that $\sum \frac{1}{n^2} = \frac{\pi^2}{6}$ [duplicate]

In this answer two sequences are mentioned. In particular, I would like to prove that $$\sum_{n = 1}^{+ \infty} \frac{1}{n^2} = \frac{\pi^2}{6}$$ If I knew that the sequence converges to ...
0
votes
0answers
58 views

Decimal digits in $\pi$

Around ten years ago I had read somewhere that there was a question in an exam for application for software engineer position in a big company which states: "What is the one billionth digit of $\pi$?" ...
2
votes
11answers
210 views

Why is $e$ the number that it is? [closed]

Why is $e$ the number that it is? Most of the irrational number that we learn about in school have something to do with geometry, like $\pi$ is the ratio of a circle's diameter to its circumference. ...
1
vote
3answers
55 views

Is there a pattern to the golden ratio number figures?

The golden ratio or phi is 1.6180339887498948482045... I am wondering if there is a pattern in the numbers so given a certain set of figures, you are able to figure out the rest of the figures ...
-1
votes
2answers
61 views

Irrational numbers are non-terminating/non-repeating decimals [closed]

Why is it true that all irrational numbers are non-terminating/non-repeating decimals? By definition, an irrational number is one that can't be expressed as a ratio of integers.
10
votes
6answers
753 views

Can I guess an irrational number formula from its digits?

Let us say I have 10,000 digits started from some point (lets say the 16th digit) of the decimal expansion square root of some arbitrary number, like 13. Is there any way I can get back the original ...
0
votes
3answers
88 views

How are irrational numbers, fixed points on the number line?

Please, while answering/reading this question, only keep in mind my point of view only. The question is, that how come an irrational number on a number line is a fixed point. To make things more ...
-3
votes
1answer
123 views

(22/7) is a rational number and (π) is irrational number [closed]

Why (22/7) is a rational number and (π) is irrational number. please explain. Edit: How can you say that $22/7=\pi$, when one number if rational and the other is irrational?
4
votes
1answer
103 views

Prove that $\sqrt{10} - \sqrt6 - \sqrt5 + \sqrt3$ is irrational

I tried the methods shown in Can $\sqrt{n} + \sqrt{m}$ be rational if neither $n,m$ are perfect squares? but I cannot extend them well into 4 numbers.
1
vote
4answers
104 views

Is $a\sqrt[3]{2} + b\sqrt[3]{4}$ irrational?

I need to prove that $$ a\sqrt[3]{2} + b\sqrt[3]{4}$$ is irrational, while $a$,$b $ are non zero rationals. I know that $\sqrt[3]{2} + \sqrt[3]{4}$ is irrational and I also know how to prove it, ...
1
vote
0answers
32 views

Tricky proof involving limit points [duplicate]

Show that for each irrational number $x$ the set of limit points of the sequence $(a_n)_{n\in\mathbb{N}}=nx-[nx]$ is the interval $[0,1]$. ($[x]$ is the largest integer $\leq x$) Any ideas how to ...
2
votes
2answers
58 views

Can we construct three irrational numbers $a,b,c$ such that $a+b+c \in \mathbb Q$?

This is rather easily shown to be possible if no constraint is put on $a,b,c$. However, is it also possible under the following constraint: $a, b$ and $c$ can not be rational multiples of each other. ...
0
votes
1answer
74 views

Cube root of $5$ is irrational

How to prove cube root of $5$ is irrational? I know this question was asked many time before but I can't understand the way the people explained the question so is it possible for someone to ...
2
votes
4answers
99 views

Can all irrational numbers be written in the form $u + v\sqrt{2}$, with $u$ and $v$ rational? [closed]

I am curious to know whether all irrational numbers can be written in the form $u + v\sqrt{2}$, with $u$ and $v$ rational. (Almost similar to how all complex numbers can be written as $x + iy$, ...
1
vote
1answer
55 views

Is the Cartesian square of the set of irrational numbers path connected?

Let $X=\mathbb{R}\setminus \mathbb{Q}$. Is $X\times X$ path-connected? I don't know where to start I think we need some number theory knowledge.
0
votes
1answer
29 views

Proof That all Positive Irrational Sqaure Roots Can be Raised to an Irrational Power to Get a Whole Number

Recently I have found out about a proof through a video. This proof shows that an irrational number can be raised to a irrational power to get and irrational number, but this proof only requires one ...
2
votes
1answer
26 views

Finding transcendental roots to an algebraic equation

So for equations with rational roots, there's a theorem that lists all the possible roots (Rational Root Theorem). If an equation has imaginary or irrational roots, their respective theorems say ...
1
vote
0answers
64 views

Interesting facts/ proofs about rational and irrational numbers

We got set some work to find some interesting facts or proofs regarding rational and irrational numbers. I wonder if anyone could offer some insight or recommend a good book/ website to look at.
1
vote
0answers
64 views

Am I pretty close to proving that e is irrational?

Show that $e=1+1/1!+1/2!+1/3!+…$ is an irrational number. Hint: show that, for all positive integers $p$, $0<p![e−(1+1/1!+…+1/p!)]<1$. Then conclude that $e$ cannot be a ratio of two integers ...
7
votes
1answer
97 views

Is there a function, continuous on the irrationals, with rational values, nowhere locally constant?

Question. Let $\mathbb A=\mathbb R\!\smallsetminus\!\mathbb Q$ be the irrational numbers. Is there a continuous function $\,f:\mathbb A\to\mathbb Q$, which is nowhere locally constant? – i.e., for ...
2
votes
0answers
53 views

Irrationality of $\pi+c$

How to prove that $\pi+c$ is irrational? where $c$ is the Champernowne Constant.
5
votes
0answers
63 views

Elementary proof that finite sums of square roots of primes is irrational

It is relatively easy to show that if $p_1$, $p_2$ and $p_3$ are distinct primes then $\sqrt{p_1}+\sqrt{p_2}$ and $\sqrt{p_1}+\sqrt{p_2}+\sqrt{p_3}$ are irrational, but the only proof I can find that ...
5
votes
0answers
66 views

The irrationality of Pi [duplicate]

Pi is defined as circumference/diameter, but it is an irrational number. And by definition an irrational number can't be defined by a fraction. So how is it that pi is circumference/diameter and on a ...
0
votes
1answer
27 views

Explanation of this (strong) induction statement

This might seem pretty simple and stupid to some but I am really not able to get it.... I was reading the proof that $\sqrt{2}$ is irrational (using induction) here. I could understand most of it ...
2
votes
2answers
49 views

Continuity question: Show that $f(x)=0, \forall x\in\mathbb{R}$. [duplicate]

Assume $f:\mathbb{R}\rightarrow\mathbb{R}$ is continuous on $\mathbb{R}$ and such that $f(r)=0$ for every rational number $r$. Show that $f(x)=0, \forall x\in\mathbb{R}$ using the $\varepsilon-\delta$ ...
2
votes
0answers
20 views

A binary irrational with bits defined by primes

Define a number $q$ in binary notation whose $n$-th bit is $1$ for $n$ prime, and $0$ for $n$ composite. So its 2nd, 3rd, 5th, 7th, 11th, etc. bits are $1$, with all other bits $0$. Here is $q$ out to ...
1
vote
0answers
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
votes
3answers
105 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
vote
1answer
44 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
votes
0answers
40 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 ...
1
vote
1answer
41 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 ...
1
vote
2answers
57 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
votes
4answers
412 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 ...
0
votes
1answer
39 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
votes
1answer
24 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 ...
0
votes
0answers
31 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 ...
0
votes
1answer
93 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?
1
vote
1answer
34 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 ...
1
vote
1answer
62 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 ...
1
vote
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 ...
0
votes
2answers
47 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
votes
2answers
193 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
vote
2answers
19 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.
6
votes
2answers
126 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 ...
0
votes
0answers
23 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
votes
0answers
97 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 ...