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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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
735 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
4k views

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?
0
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1answer
595 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 ...
7
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3answers
695 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 ...
36
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9answers
6k 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
396 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 ...
8
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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
855 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 ...
27
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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 ...
27
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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?