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.
2,421
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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 ...
0
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1
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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 $\sqrt[3]{7}$+$\sqrt{5}$ 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 ...
4
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2
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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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Sum of fractional parts greater not equal to $1$
I was trying to solve Putnam's B6 from 1995:
"For any $a>0$,set $S(a)=\{\lfloor{na}\rfloor|n\in \mathbb{N}\}$. Show that there are no three positive reals $a,b,c$ such that
$$S(a)\cap S(b)=S(b)...
5
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1
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Do permutations on the decimal expansions of irrational numbers retain the property of irrationality?
Suppose we have an irrational number with the following decimal expansion:
$$A = a_0 \ a_1 \ a_2 \ a_3 \ a_4 \ a_5 \ a_6 \dots $$
Now, construct a new real number through a permutation on the decimals ...
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Where will irrationals fit on the number line?
Now this might be a very dumb question but this has been bothering me from some days.
Imagine I want to create the real number line and for that I start with the rational numbers. So I start to put ...
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Can the irrationality of $\sqrt3$ be proven geometrically by infinite descent, similarly to Tom Apostol's proof of the irrationality of $\sqrt2$?
A geometric proof of the irrationality of $\sqrt{2}$ works by constructing two right isosceles triangles with legs $n$ and hypotenuse $m$, and finding in the construction similar triangles with legs $...
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Irrational Numbers and Surjection from $[0,1)$ to $[0,1)^2$
I am searching for some clues or solutions of the question below:
For $\sqrt{2}=1.41421356\cdots$,
Is $1.1236\cdots$ irrational?
To say more formally:
Let $f$ a function from $[0,1)$ to $[0,1)^2$ ...
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Why is root three an irrational number? [closed]
Why is $\sqrt{3}$ an irrational number since it can be expressed as ratio of two numbers $(2\sqrt{3}+3)$ and $(2+\sqrt{3})$ ?
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Limitations of rational numbers [closed]
I have been looking at the popular proofs that rational numbers have limitations when trying to define real-world lengths.
For instance, there does not exist a rational $c$ such that $c^2=2$.
...
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Necessary and sufficient condition for a number to be algebraic
I need a necessary and sufficient condition for a number to be algebraic over integers.
I got this on wikipedia see here
If $\alpha$ is an irrational number which is the root of an irreducible ...
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3
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Proving that the ratio of the hypotenuse of an isosceles right triangle to the leg is irrational
In an isosceles right angled triangle, vertex opposite to the hypotenuse which has $90^\circ$ is always on the perpendicular bisector of the hypotenuse.
So my question is, if we consider a sides of ...
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2
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Is this a valid way of showing that there are uncountably many irrational numbers?
The definition of a countably infinite set is that each element can be paired up with a natural number. The basic goal of the proof is to show that an infinite number of distinct irrational numbers ...
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Kodaira Kunihiko's exercise problem.
I once heard that in the freshman year of university of the mathematician Kodaira Kunihiko, he solved a problem stating that "the base e of the natural exponential function is not an irrational ...
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Problem in the proof of root 2 is irrational
The standard way we prove that the square root of $2$ is irrational is the following:
Let us assume the square root of $2$ is rational and is equal to $\frac{a}{b}$ where $a$ and $b$ are co-prime.
∴$\...
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Is the decimal $0.1....1$ with $1$ in every $10^i$ an irrational or a rational number?
I came across this question: "Is the decimal $0.1....1$ with $1$ in every $10^i$ an irrational or a rational number?" and I am trying to figure it out, but I don't know where to start. The ...
2
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Let $a$ and $b$ be rationals. Prove that if $a<b$ then there exists an irrational $x$ such that $a<x<b$.
Let $a$ and $b$ be rationals. Prove that if $a<b$ then there exists an irrational $x$ such that $a<x<b$.
Before anyone starts linking other posts, I have gone through the following posts.
...
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Does an algebraic irrational number always have all digits from the base?
If you take $\sqrt{2}$ in base $10$ and remove all digits $2$-$9$ you will get something like $1.1100010010101$... which i believe to be transcendental, and
it got me curious, could there be an ...
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Name for rational connection between irrational numbers
I'm trying to find the established name, if there is one, for what I'll call "rational connection" in this post.
So we would say some irrational number $x$ is "rationally connected&...
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How can two irrational numbers add to a rational number when each doesn't have an irrational part that can simply cancel out? [duplicate]
I think this is asking the same question as here except that I don't quite understand the accepted answer. For instance, I can produce(*) what I think is a counterexample:
$$\sqrt[3]{26+15\sqrt{3}} + ...
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3
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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 ...
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Does simple proof of the irrationality of $\sqrt{2}$ extend to $\sqrt{n}$, where $n$ is square-free?
This question is about extending a common proof of the irrationality of $\sqrt{2}$ to $\sqrt{p}$ for prime $p$, and then asking which other kinds of number $n$ does the same proof tell us that $\sqrt{...
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Can a logarithm be written as a power using only rational numbers?
I was thinking about the fact that we can write certain irrational numbers as the result of an exponentiation of rational numbers, for example $\sqrt2=2^{0.5}$. My question is, can a irrational ...
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Proving there are no rational cubic roots between two $f^3$ and $(f+1)^3$
Consider the difference between two consecutive positive cubes $f^3$ and $(f+1)^3$ is given by $3f^2 + 3f + 1$. If $f$ is a positive integer, there are no integer solutions for $\sqrt[3]{3f^2 + 3f + 1}...
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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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What are integers $a$ and $b$ if $a\sqrt2 + \sqrt3=0$? And its generalisation. [duplicate]
What are the integers $a$ and $b$ if $a\sqrt2 + b\sqrt3 = 0$?
I suppose answer should be $(a, b) = (0, 0)$. But I am unable to justify it.
Generalisation: What are integers $a_i$'s if
$$
a_1\sqrt{p_1} ...
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1
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Irrational root of polynomial problem (Putnam 1976/A4)
Let $p(x) ≡ x^3 + ax^2 + bx - 1$, and $q(x) ≡ x^3 + cx^2 + dx + 1$ be polynomials with integer coefficients. Let $α$ be a root of $p(x) = 0$. $p(x)$ is irreducible over the rationals. $α + 1$ is a ...
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Can a fraction of irrational numbers be an integer (specific case)? [duplicate]
Can the fraction $\frac{\pi b}{a - e b}$ where $e$ is the usual Euler constant, and $a$,$b$ are integers, be equal to an non-zero integer ?
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Can one establish the irrationality of $\varphi$ and other numbers with aperiodic tilings?
Background
The Penrose tiling P2 is aperiodic and consists of two quadrilateral shapes: kites and darts. It is depicted in the image below, with kites in blue and darts in red:
...
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2
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Why this reductio ad absurdum works to show $\sqrt7$ is irrational but it does not work for $\sqrt[3]8$?
I'm trying to refresh my mathematics after 10 years and I'm having some problems understanding this.
The following is used to show how the square root of 7 is not a rational number:
$\sqrt{7} = \frac{...
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Prove that there is a irrational number between two real numbers. [duplicate]
I wanna prove this statement:
$\forall a, b \in \mathbb{R}, a < b \exists x \in \mathbb{R} \setminus \mathbb{Q} : a < x < b$
So basically I want to prove, that there is a irrational number ...
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0
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Proving the divergence of a number-theoretic sequence
I have recently been trying to solve a problem involving infinite series, and there is one which I have been able to reduce to a problem in Number Theory, which, however, still gives me trouble. The ...
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Is the following function eventually positive?
Consider the function $$E(m)={\sqrt [3] {3}}{m} - [{\sqrt [3] {3}}{m}]-{\frac{1}{m}}$$
Here $[x]$ denotes the integer part of the real number $x$ and $m$ runs through the positive integers.
Is the ...
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2
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Difference betwwen $(\sqrt{2}, \pi)\cap \mathbb Q$ and $[\sqrt{2}, \pi]\cap \mathbb Q$ in $\mathbb Q$
Consider two metric spaces $X,Y$ such as $Y\subseteq X$. Let $A\subseteq Y$. $A$ is open in $Y$ $\iff$ $\exists B\subseteq X$ open in $X$ such that $ A=B\cap Y$
According to this theorem, the set $(\...
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Dedekind cuts: How to prove irrationality and how to compute decimal expansion in practice
When defining irrational numbers like $e$ and $\pi$ via Dedekind cuts (like for example here),
how can one prove that these numbers are in fact irrational? All irrationality proofs of $e$ that I know ...
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1
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Is this proof that $\sqrt{6}$ is irrational valid?
I was doing some grading and came across the following proof that a student gave for $\sqrt 6$ being irrational:
Assume $\sqrt{6} \in \mathbb Q$. Then $\sqrt{6} = \frac{a}{b}$ for some $a,b \in \...
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Suppose $n \in \Bbb N$ has a root that isn't whole, prove $\sqrt n$ is irrational
Suppose $n \in \Bbb N$ has a square root that isn't whole, prove $\sqrt n$ is irrational.
The exercise wants me to prove it in the following steps:
Suppose $\sqrt n$ is rational, then there exists a ...
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0
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Another proof of e is irrational?
I find some proof here, but can I prove it in the following way?
Assume $e$ is rational, then $e=\frac{p}q$, both $p$ and $q$ are positive integers. By Lagrange's Remainder Theorem,
$$e^x=1+x+\frac{1}{...
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1
answer
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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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1
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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 ...
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1
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40
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Creating an inequality for the numerator significance
An open mathematical problem is if the Euler-Mascheroni constant is irrational and if so transcendental. There has been some progress in (dis)proving this. It is known that:
$\gamma\in\mathbb{Q}\...
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2
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63
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Refute the assertion that $a^b$ is irrational for all irrational real numbers $a$,$b$.
I found this exercise in the first tome of Ken Binmore's Foundations of Analysis.
I know that similar questions exist in MSE so I will try to avoid the duplicate. I know also Eugenia Cheng (2004) ...
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1
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Bijective correspondence from all irrational numbers to irrational numbers of any open interval?
Can you example a Bijective correspondence from all irrational numbers to irrational numbers of any open interval (for example (0, 1) interval)? Is it even possible?
\begin{equation}
f: \mathbb{Q}^{c} ...
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0
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121
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On the irrationality of $\zeta(\frac{3}{2})$
It is known that $$\zeta(s)=\sum_{n=1}^\infty\frac{1}{n^s}$$Where $\zeta$ is riemann's zeta function. Usually people make $s$ an integer. But I thought of non integer values of $s$ and started with $s=...
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3
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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 \\[...
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