# Tagged Questions

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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### Is $\sqrt2+\pi$ irrational?

From this, as a layman I wonder if the same goes for $\sqrt2+\pi$? How about $\pi+\log2$?
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### Can a change of basis modify irrationality/transcendence?

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 ...
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### 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 ...
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### 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 $\mathbb{Q}$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $2+\sqrt{3}$. ...
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### 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 $0$?
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### Are all hypotenuses irrational if the shorter sides are integers?

Is it sufficient to say that providing the shorter two sides of a right triangle can be expressed as integers that work out to equal the value of the hypotenuse, then the value of the hypotenuse must ...
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### 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 ...
2k views

### when product of irrational numbers = rational number?

let $a$ and $b$ be irrational numbers. when do we have $a \cdot b$ = rational number? for example $\sqrt{2} \cdot \sqrt{2}=2$. I was wondering if there some conditions for the product to be a ...
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### For irrational real number $r$, find $n \in \mathbb{Z}$ such that $|nr - [nr]| < 10^{-10}$.
This problem is from the book "A Walk Through Combinatorics" by Richard Bona. For any irrational number $r$, there exists a positive integer $n$ such that the distance of $nr$ from the nearest ...
### Show that there are at most two rational points on $(x - a)^2 + (y - b)^2 = r^2$ for $a, b$ irrational.
For any given irrational numbers $a, b$ and real number $r \gt 0$, show that there are at most two rational points (points whose coordinates are both rational numbers) on the circle \$(x - a)^2 + (y - ...