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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13
votes
5answers
204 views

Expansion of $(1+\sqrt{2})^n$

I was asked to show that $\forall n\in \mathbb N$ there exist a p $\in \mathbb N $* such that $$(1+\sqrt{2})^n = \sqrt{p} + \sqrt{p-1}$$ I used induction but it wasn't fruitful,so I tried to use the ...
1
vote
0answers
27 views

Can a change of basis modify irrationality/transcendance?

Fix a real number $x$. We can consider its binary expansion, for instance $x = (0.01101001100101101001011\ldots)_2$. Now we consider the real number $y = (0.01101001100101101001011\ldots)_{10}$ : we ...
3
votes
2answers
253 views
+50

The relationship between tan(x) and square roots

Please note: I am working in DEGREES I think the easiest way to illustrate my point is by showing some examples: $ \tan(0^\circ) = \sqrt 0 = 0$ $ \tan(22.5^\circ) = \sqrt 2 -1$ $ 3 \cdot \tan(30 ^...
3
votes
2answers
87 views

Writing continued fractions of irrational numbers as infinite series

Infinite sums have been formulated for famous irrational numbers, such as $\pi, \phi,e,\sqrt2$ and a few others that can be listed here and here: Here are some examples: (There are more examples ...
0
votes
1answer
21 views

Pugh's exercise on Dedekind cuts addition

I am trying to solve the following exercise: Let $x=A|B$ and $x'=A'|B'$ be cuts in $\mathbb{Q}$. Show that although $B+B'$ is disjoint from $A+A'$, it may happen in degenerate cases that $\mathbb{Q}$ ...
5
votes
1answer
32 views

Intuitive reconciliation between Dedekind cuts and uncountable irrationals

I've looked around, haven't found a good explanation of this one. Basically, I'm looking for the simplest route to get from these starting points: The set of all rational numbers is countably ...
3
votes
1answer
59 views

Proving $\sqrt{2}$ is irrational: why $ q = p - \frac{p^2 -2}{p+2}$ [duplicate]

I've just begun self-studying Rubin's Principals of Mathematical Analysis. I'm having difficulty understanding a specific line in example 1.1 (proving $\sqrt{2}$ is irrational). Specifically, I'm ...
0
votes
3answers
150 views

For $\pm\sqrt 1\pm\sqrt 2 \pm\sqrt 3 \pm\cdots\pm\sqrt {2009}$, show there is a choice of signs such that it is irrational [on hold]

Considering $$\pm\sqrt 1\pm\sqrt 2 \pm\sqrt 3 \pm\cdots\pm\sqrt {2009}$$ where you can replace each $\pm$ with $+$ or $-$. Prove that there is at least one choice of signs such that the number is ...
7
votes
0answers
67 views
+50

Greedy algorithm Egyptian fractions for irrational numbers - patterns and irrationality proofs

This is related to another question on this site, but it's not a duplicate, because the actual questions I ask are completely different. In one of the answers Jeffrey Shallit provided a very useful ...
3
votes
2answers
152 views

$\pi$ and $e$ as coded trajectories

Question about the number $\pi$ and $e$ and their unpredictability. We know that $\pi=3.141592653589793238462643383279502884...$ Suppose that we are in the origin of the plane i.e. at the point $(0,0)...
5
votes
4answers
152 views

Are there any natural proofs of irrationality using the decimal characterization?

Mathematicians typically define rational number to mean quotient of two integers. It is not hard to show that a number is rational by that definition if and only if its decimal expansion terminates ...
385
votes
13answers
106k views

Does Pi contain all possible number combinations?

I came across the following image, which states: $\pi$ Pi Pi is an infinite, nonrepeating (sic) decimal - meaning that every possible number combination exists somewhere in pi. Converted ...
8
votes
4answers
867 views

Understanding non-solvable algebraic numbers

Background We know from Galois theory that the zeros of a polynomial with rational coefficients whose Galois group is solvable can be expressed in a formula that involves rational powers of the ...
1
vote
5answers
116 views

Any real number has at most two decimal representations [closed]

Here, Tao says that any real number has at most two decimal representations. Is this really true? I always thought $\pi$ has only one decimal representation.
0
votes
0answers
55 views

Is $\pi e$ irrational? [duplicate]

During our ongoing research, we need to prove that $\pi e<\lceil \pi e\rceil$. Is $\pi e$ irrational? How to prove it? Thanks- mike
1
vote
2answers
70 views

Notation for representing ANY number?

i'm working on a mathematics/number-manipulation program, and i was wondering if you could practically have a representation that could holds the value of any number. This would need to include ...
6
votes
1answer
177 views

Addition, subtraction, multiplication and division of irrational numbers, correct to $n$ decimal places

Suppose we want to do one of the four basic arithmetic operations on two irrational numbers, and we want some confidence that our answer is correct to $n$ significant figures/decimal places. Doing ...
0
votes
2answers
57 views
2
votes
2answers
96 views

Proof of $\pi^e$ and $e^\pi$ Being Irrational

By contradiction, if $\pi^e$ were rational, then we could write $\pi^e=\frac{a}{b}$ where $a,b\in\mathbb{I}^+$ and $b\neq0$. So: $$\begin{align} \\ \pi^e&=\frac{a}{b} \\ e\ln(\pi)&=\ln(a)-\...
1
vote
1answer
22 views

What is the term for 'PI-indexing'?

As a teenager - the concept of irrational numbers fascinated me. The idea that all possible numbers existed in PI. From that I reasoned that any piece of data you have now also existed in PI ...
7
votes
1answer
479 views

Chinese estimate for $\pi$. Were they lucky?

The famous chinese estimate $\pi\approx\frac{355}{113}$ is good. I think that is too good. As a continued fraction: $$\pi=[3:7,15,1,292,\ldots]$$ That $292$ is a bit too big. Is there a reason for a ...
0
votes
0answers
11 views

Irrationality of a real number $b$ that is related to a real number $a$ by the umbral Taylor series.

Given the sequence $\{a_0,a_1,...a_x\}$ then we may represent $a_n=\sum_{i=0}^{\infty} \Delta^i (0) {n \choose i}$. Where $\Delta^i(0)$ represents the operation mapping $a_n$ to $a_{n+1}-a_n$ iterated ...
5
votes
1answer
146 views

Is the infinite decimal fraction $1.23456…n$ irrational?

How to prove that the number $ 1.23456\dots n$ is an irrational number? The number consist, of course, of natural numbers in increasing sequence.
8
votes
12answers
5k views

Are there any irrational numbers that have a difference of a rational number?

Are there any irrational numbers that have a difference of a rational number? For example, if you take $\pi - e$, it looks like it will be irrational ($0.423310\ldots$) - however, are there any ...
3
votes
2answers
133 views

Help needed in proof that $\sqrt2$ is irrational

In the following proof that $\sqrt2$ is irrational, I cannot make sense of why $\frac{q a_n + p b_n}{q} \geqslant \frac{1}{q}$. The sums corresponding to $a_n$ and $b_n$ are alternating sums, so why ...
2
votes
1answer
46 views

Integer polynomial on integer values: dense mod $\pi$?

I was trying to answer the question at How to show that this limit does not exists? I was able to show that $\sin(n^2)$ does not tend to a limit, but am still unable to show that $n^2$ itself is ...
1
vote
1answer
58 views

Is $\zeta(2n+1)\notin (2\pi)^{2n+1}\mathbb{Q}$ already known?

Is it already shown or at least conjectured that $$\zeta(2n+1)\notin (2\pi)^{2n+1}\mathbb{Q}?$$ You have any names and years who proved or conjectured it?
3
votes
2answers
46 views

What is already known for $\zeta(n)$, $n\in 2\mathbb{N}+1$

Apéry showed $\zeta(3)\notin\mathbb{Q}$. What is also known or conjectured for $\zeta(n)$ with n odd? Is for example something known for $\zeta(5)$? Is there a theorem that says 'at least one of $\...
2
votes
0answers
50 views

Why are we interested in such things as $\zeta(3)\notin\mathbb{Q}$? [closed]

In 17xx Euler gived a formula for the real numbers $\zeta(2n),~n\ge 1,$ which showed the irrationality of $\zeta(2n)$. In 1975 Apéry showed $\zeta(3)\notin\mathbb{Q}$. Why are we interested in such ...
2
votes
1answer
71 views

Question on a proof of $\zeta(3)\notin\mathbb{Q}$

I have a question on this article proving $\zeta(3)\notin\mathbb{Q}.$ by using modular forms. This is theorem 1 at page 275 (page 5 in the pdf). Most things in the proof are clear but I don't get the ...
2
votes
2answers
29 views

For $\forall m \in \mathbb{N}, m \ge 3$ there are m elements of S in arithmetic progression.

Let $S=\{[n\pi], n \in \mathbb{N}\}$. Prove $\forall m \in \mathbb{N}, m \ge 3$ there are $m$ elements of $S$ in arithmetic progression. I don't know how to prove it, but I have the feeling the ...
2
votes
1answer
36 views

Proof of an irrationality criterion

I have attached a proposition whose proof I don't understand at two points. Here are my questions: Why do we have $|a_{0n}+\theta_{1}a_{1n}+\dots+\theta_{k}a_{kn}|<(\rho-\varepsilon)^{-n}$ for ...
4
votes
0answers
78 views

$\{\{(2^n+3^m)\alpha\}:n,m\in\mathbb{N}\}$ is dense in [0,1]??

Let $\alpha$ be an irrational real number. I wonder whether $\{\{(2^n+3^m)\alpha\}:n,m\in\mathbb{N}\}$ is dense in [0,1] in which $\{x\}$ means the fractional part of x. This is equivalent to the ...
0
votes
1answer
47 views

Nearest neighbor of an irrational number

I am confused in my thoughts about the irrational numbers in real line. My confusion is: If $x\in$$\mathbb R$$-\mathbb Q$ then for $\epsilon>0$ as small as you please, the element ($x+\epsilon$) ...
-1
votes
1answer
22 views

Confusion about irrational numbers

Irrational numbers is defined as something that cannot be expressed as a fraction . Now I got a question . So is "120%" an integer or irrational number ? Do I take 120% as 1.2 or just 120% as an ...
2
votes
1answer
160 views

How can I prove $\sqrt{2} ^{\sqrt{2}}$ is irrational? [duplicate]

I am learning proofs and a question was posed which asked us to prove that $\sqrt{2}^{\sqrt{2}}$ is irrational. They mentioned this - Hint: try using the log10 function... I tried my hand at the ...
0
votes
1answer
43 views

Show that $E \subset \Bbb Q$ is closed in $(\Bbb Q, d)$

Assume $(\Bbb Q, d),$ $d(p, q):= |p -q|$ is a metric space and $E := \{p \in \Bbb Q : 2 < p^2 < 3\} \subset \Bbb Q.$ I have to show that $E$ is closed. I see two ways of proving ...
12
votes
6answers
2k views

Visual representation of the fact that there are more irrational than rational numbers.

Would anybody know of a visual or even (preferably) geometric representation of this? To make it more specific: Text, symbols and written numbers are predominantly used as labels, and and less to ...
22
votes
3answers
7k views

irrationality of $\sqrt{2}^{\sqrt{2}}$.

The fact that there exists irrational number $a,b$ such that $a^b$ is rational is proved by the law of excluded middle, but I read somewhere that irrationality of $\sqrt{2}^{\sqrt{2}}$ is proved ...
-2
votes
7answers
3k views

$0.333333$ - a recurring or non-terminating decimal?

I have read like, 1.All terminating and recurring decimals are RATIONAL NUMBERS. 2.All non-terminating and non recurring decimals are IRRATIONAL NUMBERS. if the statements are right, then here ...
-1
votes
2answers
28 views

Proving by contradiction (6/9)

I have been given a statement that I need to prove using the contradiction method and I am just a little unsure of how to go about setting this up and executing. Here is the statement: If x is any ...
1
vote
0answers
74 views

Polynomial taking irrationals to irrationals

Problem: Find all polynomials from $\mathbb{R}\to \mathbb{R}$ $f$ with integer coefficients taking irrationals to irrationals. My attempt: It is clear that the problem statement is equivalent to ...
-1
votes
1answer
128 views

Proof that $\sqrt[n]{a+1}$ and $\sqrt[n]{a-1}$ cannot be both rationals [duplicate]

Let $a \neq 0$ be a natural number. How can be proved that $\sqrt[n] {a+1}$ and $\sqrt[n]{a-1}$ cannot be both rational numbers?
2
votes
3answers
96 views

Prove $2^{1/2}+3^{1/3}$ is irrational using Galois theory.

So, I want to prove that $2^{1/2}+3^{1/3}$ is irrational, and I need to prove it using Galois theory. To start, let's forget about the sum and deal with the individual numbers and $F_1 = \mathbb{Q}(...
1
vote
1answer
39 views

Can we apply binomial theorem for $\quad(a+b)^\ell\quad$ if $\ell\;$ irrational.

Let be$\quad a,b\;\in\mathbb R\quad, \ell\;\in\mathbb {(R\backslash Q)} \quad $ ($\ell:$irrational) Can we apply binomial theorem for $\quad(a+b)^\ell$
0
votes
1answer
26 views

A sum of irrational numbers ending rational

Let $x$ be a positive irrational number I know that there exists $y$ such that: $$\begin{cases} y>0 \\ x+y\in \mathbb Q.\end{cases}$$ How would you construct explicitly such $y$ ? For instance ...
0
votes
0answers
54 views

Difference between rationalizing factor and conjugate surd

I have some confusion regarding rationalizing factor and conjugate surd. For binomial surds for example $2+\sqrt{3}$ is conjugate of $2-\sqrt{3}$ and it is also rationalizing factor of $2+\sqrt{3}$. ...
9
votes
5answers
4k views

Sum of two periodic functions is periodic?

I have following paragraph taken from the Stanford's study material. Question: Is the sum of two periodic functions periodic? Answer: I guess the answer is no if you are Mathematician, yes ...
-1
votes
3answers
77 views

Is there any $\alpha$ for which $e^{\alpha}$ is an integer?

Is there any $\alpha$ which gives $e^{\alpha}$ an integer. $\alpha=0$ is the trivial one. But is there any other than $0$?
0
votes
0answers
30 views

Show that $θ_0 < Arg (z^α) < θ_0+\epsilon$ for infinitely many values of $z^α$, where $−π < θ_0 < π$ and $ \epsilon > 0$

For $z \neq 0$ and $α$ irrational, show that $θ_0 < Arg (z^α) < θ_0+\epsilon$ for infinitely many values of $z^α$, where $−π < θ_0 < π$ and $ \epsilon > 0$. I am trying to solve this ...