# 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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### Represent 0.101100111000... as a infinite series.

I want to prove that 0.101100111000... is irrational. I found a formula for a similar number: $$\sum_{m=1}^{\infty} \frac{1}{10^{\frac{1}{2}(m^2 + m)}}=0.10100100010000...$$ But the infinite series ...
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### For continuous $h:\mathbb{R\longrightarrow R}$, prove $h(x)=0$ for x in $\mathbb{Q}$ implies $h(x)=0$ for x in $\mathbb{R}$

The problem statement is as follows: For continuous $h:\mathbb{R\longrightarrow R}$, prove $h(x)=0$ for $x$ in $\mathbb{Q}$ implies $h(x)=0$ for $x$ in $\mathbb{R}$. My attempt feels a bit awkward and ...
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### Does irrational numbers real numbers [closed]

I started to learn math, and the book say that a.d1d2d3…. with infinity numbers after is a number. But how can it be if it’s will never reach end? Why is this the definition of irrational number? I ...
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### How to prove $\sqrt{7}$+$\sqrt{5}$ is irrational?

We are learning the rational root theorem right now. It is pretty clear how to prove a number is irrational when we know how to construct a polynomial that only includes integer and has the irrational ...
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### Criteria for irrationality of Euler's constant

Define for $n\in\mathbb{N}$, $$I_n=\int_0^1\int_0^1 -\frac{(x(1-x)y(1-y))^n}{(1-xy)\log xy}dx dy$$ In this article it is proved that $$I_n=\binom{2n}{n}\gamma+L_n-A_n$$ where $L_n=d^{-1}_{2n}\log S_n$,...
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### Buffon's Needle Problem

Buffon's Needle Problem "Suppose we have a floor made of parallel strips of wood, each the same width, and we drop a needle onto the floor. What is the probability that the needle will lie across ...
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2k views

### What's the "simplest" equation with only rational coefficients that produces a graph with no two rational coordinates

What's the "simplest" equation with only rational coefficients that produces a graph with no rational coordinates? Obviously I haven't precisely defined "simple" so any answer will ...
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### Intersection of translations of irrationals [closed]

Let $l>0$ and $\mathbb{Q}^c$ denotes set of irrationals. We define $$E=\cap_{x\in(-l,l)}( \mathbb{Q}^c+x),$$ where $\mathbb{Q}^c+x=\{q+x:q\in\mathbb{Q}^c\}.$ Is it easy to show that $E$ cannot ...
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### Is $\sum_{n=1}^\infty\Gamma(n)/n^n$ demonstrably irrational?

The Question What do we know about $$\xi(z)=\frac{1}{\Gamma(z)}\sum_{n=1}^\infty \frac{\Gamma(n+z-1)}{n^n}= \int_0^1 \frac{dx}{(1+x\ln x)^z}$$ Specifically, what can be said about $\xi$ regarding ...
1 vote
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1 vote
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### How can I prove that $\pi+\pi^2$ is irrational?

I am trying to prove that $\pi+\pi^2$ is irrational assuming that $\pi$ is transcendental. My work: I noticed that at least one of $\pi+\pi^2$ and $\pi-\pi^2$ is transcendental. (Because algebraic ...
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1 vote
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### Infinite Series of Rational Terms

Does anyone have a counterexample to the following conjecture? Conjecture: Suppose that the terms $a_{n}$ is a sequence of rational numbers and further that for all real numbers $r$, there are ...
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### Proof by contradiction - square root 2 is irrational [duplicate]

Click Here For Image To Proof With reference to the above image, why is it for this proof that sqrt(2) is irrational, after making the first assumption that sqrt(2) is rational, we can also make what ...
### Why square root of $1$ is $1$ but square root of $0.1$ is $0.316$, again square root of $0.01$ is $0.1$?
I would like to understand the reason behind this pattern: \begin{align} \sqrt 1 &= 1 \\[4pt] \sqrt{0.1} &= 0.31622 \\[4pt] \sqrt{0.01} &= 0.1 \\[4pt] \sqrt{0.001} &=0.03162 \\[...