# Questions tagged [irrational-numbers]

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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### Proof that $\sqrt2$ is irrational using the Fundamental Theorem of Arithmetic [duplicate]

I'm working through Lang's Basic Mathematics, which gives an idea of my level. The text includes the common proof that $\sqrt2$ is irrational using reduction of the rational number to least terms plus ...
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### Is ${}_3 F_2(a,b,c;d,e;x)$ irrational for $a,c,d,e,x \in \mathbb{Q} / \{0 \}$

My understanding of G-functions is simply nonexistent but I do know that they can assume algebraic values at nonzero rational arguments. But could those assumed values be rational? Specifically I was ...
1 vote
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### A relationship between the initial precision of an irrational number with the computation of its continued fraction?

Warning: this might be trivial. Suppose I want to compute the simple continued fraction $$[a_0;a_1,a_2,a_3,\ldots]$$ for a (non quadratic) irrational number $x$. I can do this numerically very ...
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### Proving that sine is irrational at rational arguments with infinite fractions

In his proof of irrationality of $\pi$, Lambert uses the following continued fraction for tangent: $$\tan x=\cfrac{x}{1- \cfrac{x^2}{3-\cfrac{x^2}{5-\cfrac{x^2}{7-\cdots}}}}$$ He notes that for any ...
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### Is the area enclosed by p(x,y) always irrational?

Take a polynomial $p \in \mathbb{Q}[X,Y]$. Now draw the graph of $p(x,y)=0$. If, like $X^2-Y^2-1$, this turns out to enclose a finite area, is the area enclosed always irrational? There are some ...
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### Irrational numbers as "periods" in discrete sequences based on complex exponentials

Main string description To keep the core of the question short, I leave the context introduction at the end of this text. Here I describe a construction that is related to that introduction, but to ...
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### The sum of a sequence where any term is equal to the absolute value of the difference between the prior two terms.

In a sequence, any term is equal to the absolute value of the difference between the prior two terms. Will the sum of the sequence converge? It's known that if the first two given terms are rational, ...
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### Prove that $\sqrt{18} - \sqrt{12}-\sqrt{45} + \sqrt{6}$ is irrational [duplicate]

Prove that $\sqrt{18} - \sqrt{12}-\sqrt{45} + \sqrt{6}$ is irrational I tried to let $x = \sqrt{18} - \sqrt{12}-\sqrt{45} + \sqrt{6}$, and assume for the sake of contradiction that $x$ is rational. ...
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### Hard time grasping Dedekind cuts and real numbers [duplicate]

I have been trying to understand some elementary set theory recently, and am trying to understand how the real number line can be defined using the set of rational numbers. In particular, I am trying ...
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