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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rational angles with sines expressible with radicals

An angle x is rational when measured in degrees. sin(x) is can be written using radicals. What are the conditions on x? If nested square roots are allowed? What I know so far: If sin(x) can be ...
2
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7answers
2k views

How to prove that $\sqrt 3$ is an irrational number? [duplicate]

Possible Duplicate: $a^{1/2}$ is either an integer or an irrational number I know how to prove $\sqrt 2$ is an irrational number. Who can tell me that why $\sqrt 3$ is a an irrational ...
3
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3answers
612 views

Is the constant $e$ infinitely long?

The number $e = 2.718281828...$ is the base of the natural logarithm. Its decimal representation is infinitely long. Why does this mathematical constant contain an infinite number? What is the reason ...
5
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1answer
396 views

Is it ever really Pi Time?

Walking with my son at 3:14pm the other day, I mentioned to him, "Hey, it's Pi Time". My son knows 35 digits of $\pi$ (don't ask), and knows that it's transcendental. He replied, "is it exactly ...
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6answers
12k views

Proving Irrationality

How is it possible to prove a number is irrational? First part of that question: How it possible to know that a number will go on infinitely? Second part: How is it possible to know that no ...
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1answer
368 views

How to express an irrational as a continued fraction in computer with high precision?

Background I'm writing a C++ library for continued fraction using MPIR (Multiple Precision Integers and Rationals) library http://www.mpir.org/ due to the limitation of built-in ...
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3answers
186 views

What is $\{y\in\mathbb Q\mid y=\cos(x),\quad x\in[0,2\pi]\cap\mathbb Q\}?$

Given a range of the rational numbers, $x$, between $0$ and $2\pi$\, what is the set of rational numbers $ y = \cos(x) $? I was inspired by the stackoverflow question Can $\cos(a)$ ever equal $0$ in ...
12
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0answers
376 views

Irrationality of e

I have a question about the irrationality of $e$: In proving the irrationality of $e$, one can prove the irrationality of $e^{-1}$ by using the series $$e^x = 1+x+\frac{x^2}{2!} + \cdots + ...
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3answers
5k views

Can you raise a number to an irrational exponent?

The way that I was taught it in 8th grade algebra, a number raised to a fractional exponent, i.e. $a^\frac x y$ is equivalent to the denominatorth root of the number raised to the numerator, i.e. ...
4
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2answers
222 views

Extrapolating properties of rational numbers to irrational/transcendental numbers

I've had this idea in my head for a while, but I've never told anybody because... well, I really don't know. I just never thought that it might even be remotely correct, but here goes. Here is just an ...
6
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2answers
394 views

For any irrational number such as pi, would any sequence of length n appear in its decimal places?

If pi is an irrational number that goes on infinitely forever, does it mean that I can get any sequence of numbers of any length, and somewhere in the decimals of Pi, this sequence will exist. Eg. ...
4
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4answers
269 views

Relationship between degrees of continued fractions

I'm trying to compute the values of differing degrees of continued fractions like $\sqrt 2$, e and other similar fractions. My theory was to take the reduced fraction at an arbitrary depth and the ...
3
votes
1answer
415 views

Can we have a mirror image of an irrational decimal?

Is it possible to have a number that extends to the left of the decimal point in mirror image of an irrational number? Such as <...95141.30000...>, to write pi as a mirror image.
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6answers
7k views

Prove that the product of a rational and irrational number is irrational

Could you please confirm if this proof is correct? Theorem: If $q \neq 0$ is rational and $y$ is irrational, then $qy$ is irrational. Proof: Proof by contradiction, we assume that $qy$ is rational. ...
7
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1answer
272 views

What bizarrities lurk within the decimal expansion of an irrational number?

i.e. can we write pi = 3.14159...X... where X consists of (say) 10^100 consecutive zeroes? [Originally asked on reddit without response :-( ]
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2answers
551 views

How come such different methods result in the same number, $e$?

I guess the proof of the identity $$ \sum_{n = 0}^{\infty} \frac{1}{n!} \equiv \lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x $$ explains the connection between such different calculations. How ...
3
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3answers
748 views

If $n$ is any positive integer, prove that $\sqrt{4n-2}$ is irrational

If $n$ is any positive integer, prove that $\sqrt{4n-2}$ is irrational. I've tried proving by contradiction but I'm stuck, here is my work so far: Suppose that $\sqrt{4n-2}$ is rational. Then we ...
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2answers
121 views

-$\frac{2\sqrt{2}-6}{7}$ = $\frac{6-2\sqrt{2}}{7}$ correct?

When asked to rationalize the denominator for $\frac{2}{\sqrt{2}+3}$, I came up with $\frac{6-2\sqrt{2}}{7}$ but my algebra book gives -$\frac{2\sqrt{2}-6}{7}$ as the answer. I think we're both ...
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4answers
421 views

why does $\sqrt2 = \frac{2}{\sqrt2}$?

I noticed just now that $\sqrt2 = \frac{2}{\sqrt2}$ I'm suprised because isn't this like saying $x = \frac{2}{x}$?
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1answer
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Is there a proof that $\pi \times e$ is irrational?

A little reading suggests: It is known that either $\pi + e$ or $\pi \times e$ is transcendental (or possibly both), but no proof is known that one of those two numbers in particular is ...
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2answers
1k views

Is the area of a circle ever an integer?

Is the area of a circle ever an integer? I was trying to answer someone else's question on yahoo answers today and I got thumbs down from people on my answer and have come here to get a thorough ...
3
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1answer
237 views

How can I determine a number is irrational?

I have a hypothesis about regular polygons, but in order to prove or disprove it I need a way to determine whether an expression is rational. Once I boil down my expression the only part that could be ...
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2answers
849 views

Is this number transcendental?

My son was busily memorizing digits of $\pi$ when he asked if any power of $\pi$ was an integer. I told him: $\pi$ is transcendental, so no non-zero integer power can be an integer. After tiring of ...
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0answers
76 views

Are limits on exponents in moduli possible, if the modulus is relatively prime?

I asked a similar question to this recently. Here, I consider an arbitrary, but fixed, modulus m, which is relatively prime to x and y. Can anybody extend the answer given in the previous question? ...
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1answer
155 views

Are limits on exponents in moduli possible?

Suppose I show that: $$x^{f(z)/g(z)} = y \pmod{4}$$ is impossible for some given positive integers $x$ and $y$, where, \begin{align*} f(z) &= \phi(4) k_1(z) + 1 \\ &= 2 k_1(z) + 1\\ g(z) ...
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1answer
172 views

Estimating $\#\{\{\alpha k\} < 1/\sqrt{k} : k \leq n\}$ for irrational $\alpha$

Suppose $\{\alpha n\}$ is the fractional part of $\alpha n$. Put $$A_{\alpha}(n) = \#\{\{\alpha k\} < 1/\sqrt{k} : k \leq n\}.$$ If $\alpha$ is irrational, can I find some constant $K$ such that ...
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2answers
2k views

Is there a proof that $\pi$ is an irrational number?

Most math texts claim that $\pi$ is an irrational number. However, I'm having a little bit of trouble understanding that. Since nobody has calculated all of the digits of $\pi$, how can we know that ...
12
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5answers
2k views

Irrationality proofs not by contradiction

Per now, I have basically come upon proofs of the irrationality of $\sqrt{2}$ (and so on) and the proof of the irrationality of $e$. However, both proofs were by contradiction. When thinking about ...
8
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4answers
838 views

Irrational$^\text{Irrational}$

How do I compute $\text{(irrational)}^{\text{(irrational)}}$ up to a required number of decimals say m, in the fastest way ? (one way is of course compute both the irrational numbers to a precision ...
6
votes
3answers
686 views

Proving that a series converges to an irrational number

How do we show that if $g \geq 2$ is an integer, then the two series $$\sum\limits_{n=0}^{\infty} \frac{1}{g^{n^{2}}} \quad \ \text{and} \ \sum\limits_{n=0}^{\infty} \frac{1}{g^{n!}}$$ both converge ...
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8answers
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Designing an Irrational Numbers Wall Clock

A friend sent me a link to this item today, which is billed as an "Irrational Numbers Wall Clock." There is at least one possible mistake in it, as it is not known whether $\gamma$ is irrational. ...
6
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1answer
750 views

Sine values being rational

Can $$\sin r\pi $$ be rational if $r$ is irrational? Either a direct or existence proof is fine.
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5answers
2k views

Sum of rational numbers

The sum of a finite number of rational numbers is of course a rational number, but the sum of an infinite number of rational numbers might be an irrational number. Can someone give me some intuition ...
3
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2answers
1k views

Is $n^{th}$ root of 2 an irrational number? [duplicate]

Possible Duplicate: $a^{1/2}$ is either an integer or an irrational number. Will every $n^{th}$ root of $2$ be an irrational number? If yes, how can I prove that?
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8answers
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Proof that dividing irrational number by an irrational number can result in an integer?

How can I prove that dividing an irrational number by an irrational number (besides himself) can result in an integer?
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1answer
610 views

About irrational logarithms

Could someone provide, please, a proof of the theorem below? "Being $x$ and $b$ integers greater than $1$, which can not be represented as powers of the same basis (positive integer) and integer ...
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3answers
702 views

Why are some mathematical constants irrational by their continued fraction while others aren't?

Catalan's Constant and quite a few other mathematical constants are known to have an infinite continued fraction (see the bottom of that webpage). On wikipedia (I'm sorry, I can't post anymore ...
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9answers
7k views

$\sqrt a$ is either an integer or an irrational number.

I got this interesting question in my mind: How do we prove that if $a \in \mathbb N$, then $\sqrt a$ is an integer or an irrational number? Can we extend this result? That is, can it be shown ...
3
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1answer
1k views

Solving an equation with irrational exponents

Is there any theory (analogous to Galois theory) for solving equations with irrational exponents like: $ x^{\sqrt{2}}+x^{\sqrt{3}}=1$ ?
3
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2answers
397 views

Nature of the series: $ \sum\limits_{k=1}^{\infty} \frac{2^{n_k}}{(n_{k})!}$

Prove that if $\{n_k\}$ is a strictly increasing sequence of positive integers, then the sum of the series $$\sum_{k=1}^{\infty} \frac{2^{n_k}}{(n_{k})!}$$ is an irrational number. This is just a ...
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3answers
2k views

Real Numbers to Irrational Powers

In a related question we discussed raising numbers to powers. I am interested if anybody knows any results for raising numbers to irrational powers. For instance, we can easily show that there ...
4
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2answers
870 views

About powers of irrational numbers

Square of an irrational number can be a rational number e.g. $\sqrt{2}$ is irrational but its square is 2 which is rational. But is there a irrational number square root of which is a rational ...
28
votes
5answers
9k views

Proof that the irrational numbers are uncountable

Can someone point me to a proof that the set of irrational numbers is uncountable? I know how to show that the set $\mathbb{Q}$ of rational numbers is countable, but how would you show that the ...
28
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8answers
3k views

How can you prove that the square root of two is irrational?

I have read a few proofs that $\sqrt{2}$ is irrational. I have never, however, been able to really grasp what they were talking about. Is there a simplified proof that $\sqrt{2}$ is irrational?