# 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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### For all natural numbers $n$, if $n$ is odd, then $\sqrt{15^n}$ is irrational.

How can I show that for all natural numbers $n$, if $n$ is odd, then $\sqrt{15^n}$ is irrational? I have tried to use a proof by contradiction to no avail. I have gotten decently far with a proof by ... 23 views

### Position of specific value

Let's assume a have an arbitrarily long number, take π for example. Since we know π is infinite, there will at some point be a group of numbers like "2015201620172018...", correct? My ...
1 vote
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### If $\Big|x-\frac{p}{q}\Big|<\frac{1}{2q^2}$ then $p/q$ is necessarily one of the convergents : Extend the proof to irrational $x$

Prove that, if $x$ is any irrational number, and if $p/q$ is a rational fraction in lowest terms, with $q\geq 1$, such that $$\Big|x-\frac{p}{q}\Big|<\frac{1}{2q^2}$$ then $p/q$ is necessarily one ...
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### A pattern that leads to regular continued fractions of quadratic irrationals [closed]

The following expression can be obtained by converting the continued fraction of quadratic irrationals to single fraction. $$\sqrt{N} = \frac{b\sqrt{N}+aN}{a\sqrt{N}+b}$$ The equation holds for any ...
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### 1956 AHSME Question 23 on the quotient of an irrational number with a real number

I was solving some problems from the American High School Math Exam of 1956 (AHSME) from the Art of Problem Solving Website. Question 23 Problem 23 About the equation $ax^2 - 2x\sqrt {2} + c = 0$, ...
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### Does every line through the origin come arbitrarily close to some other lattice point?

Given positive real numbers $x$ and $\epsilon$, do there necessarily exist positive integers $p$ and $q$ such that $\left| qx - p \right| < \epsilon$? Or geometrically: in a two-dimensional ...
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### What is meant, exactly, by nonrepeating when talking about irrational numbers?

My question is referring to the exact definition mathematicians use when describing the decimal expansions of irrational numbers as "nonterminating and nonrepeating." Now, I understand, at ...
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### The set of irrationals numbers is countable?

I tried to prove this using statement using the difference of sets $\mathbb{R}-\mathbb{Q}$ and the fact that $\mathbb{R}$ is not countable and $\mathbb{Q}$ is countable. In general, is it possible to ...
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### Decimal expansion of irrational numbers?

Apologies if this is a trivial question. Is there any particular study which deals with the properties of decimal expansions of irrational numbers? Studies that deal with patterns and so on. Are there ...
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### Prove that the sum of two irrational numbers can be irrational [closed]

The question title itself is self explanatory. We need to prove that the sum of two irrational numbers can be irrational (it necessarily doesn't need to be always, rather i am trying to prove that, ...
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### Irrationality Result

I have recently derived the following result: for any odd positive integer $k$, one of $\zeta(k)$ and $\zeta(k,2/3)$ is irrational. I cannot find this result anywhere. Is it already known? Thank you ...
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### Correlation between $\sqrt{1/10}$ and length of powers of integers.

I'm searching information about this simple problem involving square roots and length of powers. It's very simple but it seems interesting, at least for me. I'm not a mathematician. Description By ...
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### Is the ratio of the perimeter of any shape with circular curves to its diameter result in an irrational number?

Is the ratio of the perimeter of any shape with circular curves to its diameter result in an irrational number? I suppose it would depend one what would be defined as "circular curves", as ...
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### Proving that a certain set is dense using a useful characterization of the irrational numbers. (Based on Chapter 4 Exercise 25 (b) Baby Rudin)

The following is exercise 25 (b) Chapter 4 in Rudin's Principle's of Mathematical Analysis: Let $\alpha$ be an irrational real number. Let $C_1$ be the set of all integers, let $C_2$ be the set of all ...
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### Spivak, Chapter 8 "Least Upper Bounds", Problem 5c: for $r<s$ rational numbers, prove there is irrational number between $r$ and $s$?

5c) Suppose that $r<s$ are rational numbers. Prove that there is an irrational number between $r$ and $s$. Hint: as a start, you know there is an irrational number between $0$ and $1$. There is no ...
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### Alternative proof that $\sqrt{2}$ is irrational. [duplicate]

If we set $\sqrt{2} = \frac{a}{b}$ where $a,b$ are positive and coprime, we get $2 = \frac{a^2}{b^2}$. However, as $a^2,b^2$ must also be coprime, this fraction can only be an integer if $b^2 = 1$, so ...
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### Non-periodic continued fraction with explicitly known convergents?

Is an irrational number with non-periodic continued fraction expansion known, for which one can give explicit formulas for the convergents $p_n/q_n$ or at least for the denominators $q_n$ (similar to ...
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### Seeking clarification about another question concerning irrational numbers.

The post A question about decimal representation of irrational numbers. asked: "Is this true that any finite word of the alphabet $\mathcal{A_9}=\{0,1,2, \ldots,8,9\}$ appears somewhere in the ...
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### Question about irrational numbers and finite subsequences of their decimal places.

A more general question than in "A question about decimal representation of irrational numbers.": Since there is an infinite amount of irrational numbers could you always find one that ...
51 views

### Limit of a piecewise function defined by x being rational or irrational

If $$f(x)=\begin{cases} x² \quad \quad \text { if x is rational} \newline 0 \ \ \ \ \ \ \ \ \ \ \text {if x is irrational} \end{cases}$$ prove that $\lim_{x \to 0}f(x) = 0$
### Given $\varphi$ is golden ratio, how do I prove this $\sum \limits_{j=1}^{\infty}\frac{(1-\varphi)^j}{j^2}\cos{\frac{3j\pi}{5}}=\frac{\pi^2}{100}$?
Given $\varphi$ is golden ratio, how do I prove this: $\displaystyle \tag*{}\sum \limits_{j=1}^{\infty}\frac{(1-\varphi)^j}{j^2}\cos{\frac{3j\pi}{5}}=\frac{\pi^2}{100}$ My approach: We can reduce ... Consider the real number in $(0,1)$ having the decimal expansion $${\alpha} = 0.{a_1}{a_2}{a_3}\cdots$$ where $a_j$ is obtained by adding up the digits in the decimal expansion of the positive ...