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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4
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2answers
212 views

Exponentiation when the exponent is irrational

I am just curious about what inference we can draw when we calculate something like $$\text{base}^\text{exponent}$$ where base = rational or irrational number and exponent = irrational number
2
votes
0answers
128 views

Irrationality of $\pi$ from the spigot algorithm?

The spigot algorithm for BPP formula gives hexadecimal digits of $\pi$ one at a time. Is it possible to prove directly that this algorithm cannot be computed with bounded-memory? (From R J Lipton It ...
3
votes
1answer
246 views

Field containing all square roots of rational numbers

What is the smallest field which contains all square roots of positive rational numbers? I guess I mean “smallest” in terms of set inclusion, i.e. the minimal one with regard to the “$\subseteq$” ...
9
votes
5answers
630 views

Algorithms for “solving” $\sqrt{2}$

The very first words out of my mouth need to be this... "Solving" is the wrong term since I am speaking about irrational numbers. I just don't know which word is the correct word... So that can be ...
4
votes
2answers
5k views

Is a non-repeating and non-terminating decimal always an irrational?

We can build $\frac{1}{33}$ like this, $.030303$ $\cdots$ ($03$ repeats). $.0303$ $\cdots$ tends to $\frac{1}{33}$. So,I was wondering this: In the decimal representation, if we start writing the ...
4
votes
5answers
245 views

Will we get all real numbers if we add all limits?

Consider a set of all rational numbers from 0 to 1 inclusive. If we add to this set all limits of all convergent sequences of these numbers, will we obtain a set of all real numbers from 0 to 1?
3
votes
2answers
296 views

Difference between irrational numbers with and without a pattern.

I'm not sure how to talk about what I want to talk about, so I'll give some examples. The number $\pi$ is irrational and has no repeating pattern, but is computable by an easy rule; divide the ...
3
votes
0answers
146 views

Must be rational number

Let $a$, $b$ positive rational number. Suppose that there exist two odd positive integers $p$, $q$ such that $\sqrt[p]{a}+\sqrt[q]{b}$ is rational. Prove that both $\sqrt[p]{a}$ and $\sqrt[q]{b}$ are ...
5
votes
1answer
208 views

Irrationality of reciprocal Fibonacci constant

I read that it was proved that reciprocal Fibonacci constant $$\sum_{n} \frac{1}{F_n} = \frac{1}{1} + \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{5} + \frac{1}{8} + \frac{1}{13} + \frac{1}{21} ...
3
votes
2answers
191 views

Irrational root of a function

DISCLAIMER: I apologize in advance if this question is naive. Every suggestion on how to approach the following problem will be very much appreciated. I'm interested in the root of the following ...
5
votes
0answers
99 views

Irrationality proof by fast converging series?

I read here http://www.mathpages.com/home/kmath455.htm that $\sum_{n=1}^\infty \frac{1}{d_n}$ is irrational if $d_{n+1} > d_{n}^2$ for all $n > N_0$. Can we prove $\pi$, $e$ or some other ...
7
votes
0answers
238 views

How does one prove that $\zeta(3)$ is irrational?

How does one prove that $\zeta(3)$ is irrational ? I would like to know how Apery did it. In particular how a recursion gives rise to irrationality !?
1
vote
3answers
398 views

If $a+b$ is an irrational number, is $a-b$ an irrational number, too?

Question 1: If $a+b$ is an irrational number. Is $a-b$ an irrational number, too? Question 2: If $\cos(a)-\sin(a)$ is irrational, Is $\sin(a)-\cos(a)$ irrational, too?
5
votes
1answer
177 views

Are these numbers $h_{r,s}$ irrational?

I came across these numbers in my work some time ago. This type of expressions do not exist in closed form (not to confuse with Vandermonde convolution), I already know that. To simplify I denote ...
1
vote
2answers
466 views

Proof $ \sqrt{1 + \sqrt[3]{2}} $ is irrational using the theorem about rational roots of a polynomial

I'm having trouble with this specific problem at the moment. The theorem states that if $n/m$ is a rational root of a polynomial with integer coefficients, the leading coefficient is divisible by m ...
5
votes
2answers
422 views

Irrational Numbers Containing Other Irrational Numbers

Does $ \sqrt{2} $ contain all the digits of $ \pi $ in order? Does it contain all the digits of $ \pi $ in order an infinite number of times? Does $ \pi $ contain all the digits of $ \sqrt{2} $ in ...
10
votes
2answers
1k views

Multiples of an irrational number forming a dense subset

Say you picked your favorite irrational number $q$ and looking at $S = \{nq: n\in \mathbb{Z} \}$ in $\mathbb{R}$, you chopped off everything but the decimal of $nq$, leaving you with a number in ...
4
votes
0answers
155 views

Direct proof that $\sqrt{2}$ is irrational? [duplicate]

I've been puzzled for some days now, and I can't come up with an answer. I'm trying to come with a direct proof that $\sqrt{2}$ is irrational. Can somebody check of this is a valid proof ? Let ...
0
votes
1answer
137 views

$a_n(x):=nx-\lfloor nx \rfloor$

i have $a_n(x):=nx-\lfloor nx \rfloor$ where $x$ is real. i want to show that if $x$ is rational, then $a_n(x)$ has finitely many cluster points, if $x$ is irrational, then every real $a$ with $0\leq ...
4
votes
5answers
155 views

irrationality of numbers with rational sum

Assume that $x_1, \dots, x_n$ are non-negative real numbers such that $$ x_1 + \dots + x_n \in \mathbb Q~~~~~~~~~~~~~~ \text{ and } ~~~~~~~~~~~~~~~x_1 + 2x_2 + \dots + nx_n\in \mathbb Q. $$ Does ...
0
votes
2answers
97 views

Decimal Representaion

A rational number can be represented in the form p/q. prove that the period of the the repeating decimal should at the most q-1.
1
vote
2answers
1k views

The logarithm of 3 base 10 is irrational

Prove that the logarithm of 3 base 10 is irrational The Fundamental Theorem of Arithmetic is that every integer is a product of primes. So far I have, Suppose $\log_{10}(5)$ is rational. Then ...
23
votes
6answers
3k views

Why must we distinguish between rational and irrational numbers?

The difference between rational and irrational numbers is always stated as: rational numbers can be written as the ratio of two integers, and irrational numbers can't. However, why do mathematicians ...
-1
votes
2answers
130 views

Irrationals can be separable by finding a countable dense subset. [duplicate]

Possible Duplicate: Is the set of irrationals separable as a subspace of the real line? Prove the irrationals are separable directly by finding a countable dense subset.
4
votes
0answers
67 views

Digits of two irrational numbers, given their power with fixed number of digits

I have $a, b \in \mathbb{R} \setminus \mathbb{Q}$, I want to know the result of $a^b$, but I don't know exact $a, b$ because I write them in numeric form. My question is how many digits of $a, b$ have ...
6
votes
1answer
153 views

General Continued Fractions and Irrationality

A while back I came across a result about non-simple continued fractions that allows proving that some numbers are irrational. The result in modern terminology is: If, in the continued fraction ...
11
votes
1answer
253 views

A hole puncher that hates irrational distances [duplicate]

Possible Duplicate: Irrational painting device I have a special hole puncher that does the following: When applied to any point $ x \in \mathbb{R}^{2} $, it removes all points in $ ...
11
votes
1answer
366 views

Is the difference of the natural logarithms of two integers always irrational or 0?

If I have two integers $a,b > 1$. Is $\ln(a) - \ln(b)$ always either irrational or $0$. I know both $\ln(a)$ and $\ln(b)$ are irrational.
3
votes
5answers
148 views

Proof of Easy Theorem?

I was reading the proof of this theorem and have a little trouble understanding one part of it: Theorem: If $k > 2$ and $n$ are natural numbers, then $n^{\frac{1}{k}}$ is irrational unless $n$ is ...
3
votes
5answers
191 views

Please explain this step in proving the square root of 3 is irrational

Assume that $$3 = \frac{p^2}{q^2}$$ So, $$ 3 q^2 = p^2$$ So $p^2$ is divisible by $3$. How we can conclude this?
0
votes
2answers
319 views

Are the digits of irrational/transdental numbers random?

If I were to look at the decimal representation of some irrational or even transdental number, and if I choose a natural number at random can I expect that it is some digit with probability $0.1$ ?
1
vote
1answer
408 views

how to prove cubic root of 25 is irrational

how to prove cubic root of 25 is irrational using mathematical induction? I've been trying to do it for hours but can't get it, help plz guys :S
3
votes
2answers
286 views

Prove or disprove: For all positive integers $ n$ , $\sqrt[3]{n}+\sqrt[3]{n+1}$ are irrational numbers.

Prove or disprove: For all positive integers $ n$ , $\sqrt[3]{n}+\sqrt[3]{n+1}$ are irrational numbers.
3
votes
2answers
336 views

Is there any sans-calculus proof of irrationality of $\pi$?

Is there a proof that will convince someone who doesn't understand calculus, of $\pi$'s irrationality .
5
votes
1answer
123 views

A rope and Pi's irrationality

Here is a question which has been puzzling me for some time. You have a thin rope of an integer length $L$. You can bend it to create a rectangle of perimeter $L$. Fine so far. Next, through some ...
3
votes
1answer
90 views

Why is this proof on rational set valid?

Prove that if $y$ is rational and $x$ is irrational, then $y + x$ and $yx$ (assume y $\neq 0 $) is irrational. I kinda guessed the proof and it turned out to be right with the key, but it doesn't ...
6
votes
3answers
312 views

Is this number irrational?

Is the following (decimal) number irrational? 0.10100100010000100000100000010000000100... etc. My intuition tells me it is irrational. My informal "proof" is ...
2
votes
3answers
706 views

(How to/Can I) show irrational numbers?

This might sounds stupid, but I really don't know can I show Irrational numbers in proves? And if so, how to show it? For example, when I want to show Rational numbers, I do this: $\frac{m}{n} $ , ...
3
votes
3answers
214 views

Closed form representation of an irrational number

Can an arbitrary non-terminating and non-repeating decimal be represented in any other way? For example if I construct such a number like 0.1 01 001 0001 ... (which is irrational by definition), can ...
3
votes
3answers
2k views

Prove that if $n$ is not the square of a natural number, then $\sqrt{n}$ is irrational. [duplicate]

Possible Duplicate: $\sqrt a$ is either an integer or an irrational number. I have this homework problem that I can't seem to be able to figure out:Prove: If $n\in\mathbb{N}$ is not the ...
0
votes
1answer
7k views

Online tool to check if number is rational or irrational?

I am new to this forum. I've been programing for some time, and now starting my engineering degree. I am trying to find an online utility that will help me grasp the concept of irrational numbers ...
1
vote
1answer
189 views

Arbitrary Sequence of Digits in Irrational Number

What are numbers in which we can find arbitrary sequence of digits (in a certain base-$n$ expansion)? I know that $0.123456789101112131415\cdots$ does (and its analogues in other bases), but does this ...
17
votes
2answers
999 views

Why is $\pi$ irrational if it is represented as $c/d$?

$\pi$ can be represented as $C/D$, and $C/D$ is a fraction, and the definition of an irrational number is that it cannot be represented as a fraction. Then why is $\pi$ an irrational number?
2
votes
1answer
71 views

Approximate rational dependence

After seeing question Why is $10\frac{\exp(\pi)-\log 3}{\log 2}$ almost an integer? I wonder if there is an algorithm that can find approximate rational dependence?! I pick any irrational numbers ...
5
votes
1answer
1k views

$\log_7 n$ is either an integer or an irrational number

Show that $\log_7 n$ is either an integer or an irrational number where n is a positive number. I assumed that it is rational and tried to get a contradiction for $\log_7 n = a/b$, where b does ...
2
votes
3answers
2k views

Proving that for each prime number $p$, the number $\sqrt{p}$ is irrational [duplicate]

Possible Duplicate: $\sqrt a$ is either an integer or an irrational number. I'm a total beginner and any help with this proof would be much appreciated. Not even sure where to begin. ...
6
votes
1answer
617 views

Deciding whether $2^{\sqrt2}$ is irrational/transcendental

Is $2^\sqrt{2}$ irrational? Is it transcendental?
8
votes
1answer
234 views

Irrationality of a series

Here is a series in which $m \geq 2 $. I want to ask how to prove the below series is irrational: $$\sum _{n=1}^{\infty} \frac{1}{m^{n^2}}$$
3
votes
0answers
92 views

how many numbers of irrationality measure $x$

Does there exist $x>2$ such that uncountably many reals have irrationality measure x? Must there exist at least one number of irrationality measure $x$? related question on sets of constant ...
20
votes
3answers
1k views

What is the simplest way to prove that the logarithm of any prime is irrational?

What is the simplest way to prove that the logarithm of any prime is irrational? I can get very close with a simple argument: if $p \ne q$ and $\frac{\log{p}}{\log{q}} = \frac{a}{b}$, then because ...