Mathematics Stack Exchange
Mathematics Stack Exchange

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 $\zeta(3)$.

2322 questions
1
vote
2 answers

How do we prove that the irrational numbers have no upper bound

From Calculus to Apostol I know that real numbers do not have upper bound, I also know that irrational numbers belong to real numbers. Would the mathematical proof be different? I quote the theorems to determine that the real numbers are not upper…
P.Saira
  • 21
  • 6
1
vote
2 answers

Numbers raised to irrational numbers

I realized something the other day that's got my mind in a knot again. Given some constant $ n $, I know that: $$ n^2 = n \cdot n$$ $$ n^3 = n \cdot n \cdot n$$ And so on so forth. I also know that: where, $ n^{1/2} = k $, then: $ k \cdot k = n $…
npengra317
  • 97
  • 6
1
vote
2 answers

Rationalizing an expression; when to stop, what is the higher level purpose?

I'm reviewing Algebra in an attempt to review Calculus and came upon a question in the Algebra Diagnostic that asked to rationalize an expression and simplify. To my understanding, rationalization is the process of rewriting a given expression so…
mburke05
  • 113
  • 3
1
vote
1 answer

Why can't imaginary numbers be irrational

Edit: To those voting to close as 'opinion based', the core question (which should satisfy the 'opinion based'-criteria) is this: Is there any specific reason why an imaginary number can't be classified irrational; additionally, are the properties…
Albert Renshaw
  • 1,644
  • 2
  • 17
  • 34
1
vote
1 answer

Density of a real subset $A$ such that $\forall (a,b) \in A^2, \ \sqrt{ab} \in A$

Suppose that $A$ is a non empty subset of the positive reals such that $$\forall (a,b) \in A^2, \ \sqrt{ab} \in A \tag{*}$$ How to prove that $A \cap (\mathbb R \setminus \mathbb Q)$ is dense in $(\inf A, \sup A)$? I'm trying to find sequences of…
mathcounterexamples.net
  • 64,237
  • 4
  • 35
  • 95
1
vote
4 answers

Find an equation where all 'y' is always irrational for all integer values of x

Intuitively it appears to me that if $x$ is an integer, $y$ has to be an irrational number for the following equation. $10y^2-10x-1 = 0$ Can someone prove me right or wrong?
balki
  • 113
  • 4
1
vote
4 answers

How to prove the density of irrational numbers in $\mathbb{R}$ without proving density of rationals first

I am asked to prove the density of irrationals in $\mathbb{R}$. I understand how to do this by proving the density of $\mathbb{Q}$ first, namely, adding a known irrational number such as $\sqrt{2}$ to $x,y \in \mathbb{R}$ ($x
14X24X351356
  • 11
  • 2
1
vote
1 answer

Which irrational numbers turned out to have descriable digit sequences?

So, I've wondered if there are irrationals where I can calculate some digit $n\in\mathbb{N}$ when given only the $k_n\in \mathbb{N}_0$ ($k_n
SK19
  • 2,785
  • 7
  • 32
1
vote
3 answers

Proper subset of the set of irrationals such that it is countable and dense in $\Bbb R$

We know that $\Bbb R$ is separable i.e. it contains a dense subset which is countable. We have $\Bbb Q$ and ${\Bbb R} - {\Bbb Q}$ to be dense subsets respectively countable and uncountable. I was looking for a countable dense subset of $\Bbb R$…
user422112
1
vote
2 answers

Marcin Mazur proof on irrationality of $\sqrt2$

I have a sketch of proof of irrationality of $\sqrt2$ due to Marcin Mazur. It is asupposed that $\sqrt2=\frac ab$ where $a$ is the smallest positive numerator that $\sqrt 2$ can have as a fraction. Since $\sqrt2=\frac{-4\sqrt2+6}{3\sqrt2-4}$, we…
user486600
1
vote
0 answers

Finding parameters for which a sequence is dense

I want to find the set of parameters $\alpha\in\mathbb{R}$ such that the sequence $$ x_n=(\alpha \lambda^nv)\,\mathrm{mod}\,1 $$ is dense in the unit square $\lbrack 0,1\rbrack^2$ where $$ \lambda=\frac{3+\sqrt{5}}{2}\qquad\text{and}\qquad…
lbf_1994
  • 439
  • 2
  • 13
1
vote
0 answers

Density of sequence in the unit square

I want to show that the sequence $$ x_n=\left(\frac{3+\sqrt{5}}{2}\right)^n\cdot\begin{pmatrix}\frac{1+\sqrt{5}}{2}\\ 1\end{pmatrix}\,\mathrm{mod}\,1\in\mathbb{R}^2 $$ is dense in the unit square $\lbrack 0,1\rbrack^2$ (The $\mathrm{mod}\,1$…
lbf_1994
  • 439
  • 2
  • 13
1
vote
1 answer

Does getting $\pi$ as the result of an expression mean the expression must involve an irrational (transcendental?) number?

Should we have an irrational (transcendental?) number in an expression to result in number $\pi$? As an example: $$\pi = \frac{\text{circle circumference}}{\text{an integer diameter}}.$$ If the statement is true, then for any given integer diameter…
RickSanch3z
  • 13
  • 3
1
vote
1 answer

For $x \in [0, 1]$, find explicit form for the $n_k$ such that $\{n_k\sqrt{2}\} \to x$.

This is inspired by the comment in Minimum values of the sequence $\{n\sqrt{2}\}$ that, by Kroneckers Approximation Theorem, the fractional part of $n\sqrt{2}$ is dense in $[0, 1]$. My question is that, given an $x \in [0,1]$, is there an explicit…
marty cohen
  • 101,285
  • 9
  • 66
  • 160
1 2 3
…
99
100