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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.

An irrational number is a real number that cannot be expressed as a quotient of two integers, i.e. cannot be expressed in the form $\dfrac{a}{b}$, with $a,b\in\mathbb{Z}$. We write $\mathbb{I}=\mathbb{R}\setminus\mathbb{Q}$.

Some examples of irrational numbers are $\sqrt{2}, e, \pi$ and $\zeta(3)$.

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Irrational number and real number definition

Real number - A number which can be represented uniquely by a point on axis. And real number = rational + irrational number By this definition, if you take an irrational number such as sqrt(2) ...it's value is non terminating non repeating…
lancer
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Attempt to view irrational number as a fraction

I am wondering if an irrational number can be represented as a fraction in this way: For example (to represent $\pi$): $$\pi= 3.14159265359...=\frac{314159265359...}{100000000000...}$$ In the fraction $\frac{314159265359...}{100000000000...}$, the…
Danny
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Proof that the sqrt of a Natural Number is irrational outside of square numbers.

I'm reading a book on Abstract Algebra and I'm covering Rings at the moment. Now I'm fairly new to this so my terminology may be incorrect, but we are looking at examples in which the set of Integers can be composed into sets with different…
user150203
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Particular property of a function that admits a Taylor expansion and $f(2/3)=3/2$

I'm trying to prove this original statement : Let suppose that $f$ is a function that admits a Taylor expansion such that $$ \forall x\in \mathbb{R}, \ > f\left(x\right)=\sum_{n=0}^{+\infty}a_nx^n $$ with $a_n \in \left\{0,1\right\}$ and that…
Atmos
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Can the sequence of states of a non-halting Turing machine describe an irrational / transcendental number?

Let $T$ be a Turing machine (or another type of more suitable machine, I am not very confident with this field) with $n$ states and assume that, when started on a blank tape, $T$ does not halt. Interpret the sequence $(x_k)$ of its states (not the…
57Jimmy
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Proof on Rational Numbers

I am trying to determine whether the following structure forms a Ring under the Real Number Definition of Addition and Multiplication Consider the set of Real Numbers of the form: $A = \{a + bp \:|\: a,b \in \mathbb{Q}, p \in \mathbb{R} -…
user150203
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A Rational Game (Question on Real Analysis)

This post is to set forth a little game that attempts to demonstrate something that I find to be intriguing about the real numbers. The game is one that takes place in a theoretical sense only. It starts by assuming we have two pieces of paper. …
AplanisTophet
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Transcedental number or not?

I need help to classify this number. The solutions to $3^y-y^3=0$ are 3 and 2.47805268.... it is irrational But since it is a root for this equation is the number transcedental or not? The original problem comes from $3^y=y^3$ where one sees it is…
Latin Wolf
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Irrationality of a number represented by an infinite product, whose partial products are irrational

It is known that infinite sequences of irrational numbers can converge to rational numbers. For instance, the sequence: $$ \{x_n=\sqrt[n]{n}\}_{n=1}^\infty $$ equals $1$ when $n\to\infty$, however it is irrational for all $n>1$. But what about…
Klangen
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Positive rational raised to positive irrational is always irrational?

I've been puzzling over this question about number theory. Ignoring complex numbers, and working only with positive numbers: Can one say that a positive rational number raised to a positive irrational exponent is always irrational? Thank you, Hein
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Proof for the irrationality of $eπ$ ('complex'ly?).*

Some of you may have seen the video: https://youtu.be/DLWpj34UNRk It is a proof of the irrationality of $eπ$ by Ron (14years) , presented by 'blackpenredpen' Youtuber. The proof goes something like this: We know that $e^{iπ}=-1$ ("Euler's famous…
Ryan Scott
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Cluster points of the sequence $a_n(x):=nx-\lfloor nx \rfloor$

I have $a_n(x):=nx-\lfloor nx \rfloor$ where $x$ is real. I want to show that if $x$ is rational, then $a_n(x)$ has finitely many cluster points, if $x$ is irrational, then every real $a$ with $0\leq a \leq 1$ is cluster point of $a_n(x)$. I don't…
doniyor
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a question on dense subsets of the (0,1) interval in number theory

A well-known fact: If $\alpha$ is irrational, then $\{n\alpha\bmod{1}|n\in\mathbb{N}\}$ is dense in $(0,1)$. In particular, $\{n\sqrt{2}\bmod{1}|n\in\mathbb{N}\}$ is dense in $(0,1)$. What if we replace 1 in "mod 1" with something else, e.g. is…
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How to show $a^{\frac{1}{b}}-c^{\frac{1}{d}}$ is rational if and only if $a^{\frac{1}{b}}$ and $c^{\frac{1}{d}}$ are rational and not equal.

Given $a,d,c,d\in\mathbb{N}$, define $$ X = a^\frac{1}{b} $$ $$ Y = c^\frac{1}{d} $$ $$ Z = X-Y $$ I'm pretty sure the following is true: For any $X\neq Y$ that satisfy the above relationships, $Z$ is rational if and only if $X$ and $Y$ are…
Samuel
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irrationality of $e$ and series

I'm trying to understand a passage in the proof that e is an irrationale number. We assume that $e$ is a rational number, $e= \frac{p}{q}$ in which $p$ and $q$ are two coprime numbers. $$e= \sum_{k=0}^\infty \frac{1}{k!} \implies…
Anne
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