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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25
votes
11answers
1k views

Computing irrational numbers

I am genuinely curious, how do people compute decimal digits of irrational numbers in general, and $\pi$ or nth roots of integers in particular? How do they reach arbitrary accuracy?
3
votes
2answers
87 views

Showing $r\in\mathbb Q\setminus \{0\}\implies e^r\notin \mathbb Q$

For a given $n>0$, let $\displaystyle J_n:x\to \frac{1}{n!}\int_{-x}^x(x^2-t^2)^ne^tdt$ a. Prove that there exists $A_n,B_n\in \mathbb R_n[X]$ such that $\forall x\in \mathbb R^+, ...
2
votes
1answer
429 views

Irrationality of $\pi$ and circumference to diameter ratio.

How is $\pi$ actually defined? If it is defined as the ratio of the circumference of a circle to its diameter then from this definition itself either of the circumference and diameter has to be ...
16
votes
4answers
183 views

Homework 8th grader: $\pi^2$ is irrational

I'm tutoring a girl in 8th grade (so she is 14 years old) and she recently had a mathematics chapter about numbers. In the last paragraph they introduced the difference between rational and irrational ...
7
votes
2answers
7k views

Is a non-repeating and non-terminating decimal always an irrational?

We can build $\frac{1}{33}$ like this, $.030303$ $\cdots$ ($03$ repeats). $.0303$ $\cdots$ tends to $\frac{1}{33}$. So,I was wondering this: In the decimal representation, if we start writing the ...
1
vote
2answers
35 views

Show: t(x) = x√2 + √3 is irrational. Hint: consider t(x)²

Earlier in the question we were asked to show that the square root of 6 is irrational, which I did. But I can't seem to figure the last part out. I have included an image for reference. Help is ...
-5
votes
0answers
76 views

Irrational + Irrational = Rational? [closed]

Is it possible for two irrational numbers, when added, to be a rational number? The numbers should not be such that the irrational part can be cancelled. Without the criteria, I can obviously get ...
39
votes
9answers
7k views

Is an irrational number odd or even?

My sister just asked this question to me: "Is an irrational number odd or even?" I told her that decimals are not odd or even and that does imply that not recurring and non repeating decimals will ...
2
votes
3answers
84 views

Is a prime to the power of a fraction always irrational?

Let $p$ be a prime number and let $x$ be a faction, i.e. $x \in \mathbb{Q} - \mathbb{N}$. It seems to be the case that $p^x$ is always irrational. How do I prove this?
-4
votes
4answers
42 views

Suppose $a$ is an irrational positive real number. Prove that if $x$ and $y$ are integers and $x\neq y$, then $(x + a)/(y + a)$ is irrational. [closed]

I need your assistance. Please provide a proof of the following proposition: Proposition: Suppose $a$ is an irrational positive real number. Prove that if $x$ and $y$ are integers and $x\neq y$, then ...
8
votes
0answers
64 views

Is there any known application for normal numbers?

Background: I am writing a master thesis on the complexity of the expansions of algebraic numbers in some complex basis $\beta$ with $|\beta| > 1$. This is a very small step towards proving the ...
-1
votes
2answers
43 views

Does an irrational number $C$ exist such that $C \cdot \sqrt 2 \in \Bbb{Q}$?

Does an irrational number $C$ exist such that $C \cdot \sqrt 2 \in \Bbb{Q}$, where $\sqrt2 \not\mid C$? I just thought of this, I'm trying to find answers that aren't of the form $C=a\sqrt2, ...
-1
votes
4answers
1k views

Prove that $\log_2 3$ is irrational [closed]

Prove that $\log_2 3$ is irrational Seemingly simple homework assignment. Was never the best with logarithms, how would I go about proving?
0
votes
1answer
20 views

Incommensurable units as ratios

I am having a bit of trouble understanding the concept of an incommensurable unit. From what I have gathered so far, it is simply a magnitude that cannot be expressed as the ratio of two natural ...
4
votes
1answer
25 views

Several values of irrational exponentiation

When talking about a number to a rational exponent, there are as many answers as the denominator of the exponent. Like the question: Is $9^{1/2}$ equal to $3$ or $-3$. However when we have an ...
11
votes
1answer
237 views

Integer parts of multiples of irrationals

Let $\alpha>0$ and define $$S(\alpha)=\big\{\lfloor n \alpha \rfloor: n\in\Bbb Z^+ \big\}.$$ Here $\lfloor x\rfloor$ is the integer part of $x$ and $\mathbb Z^+$ the set of positive integers. ...
-4
votes
2answers
33 views

Can we prove $\sqrt6$ is irrational by the method of contradiction? [closed]

Can we prove that $\sqrt 6$ is irrational number using the meathod of contradiction?
2
votes
1answer
32 views

Approximating non-rational roots by a rational roots for a quadratic equation

Let $a,b,c$ be integers and suppose the equation $f(x)=ax^2+bx+c=0$ has an irrational root $r$. Let $u=\frac p q$ be any rational number such that $|u-r|<1$. Prove that $\frac 1 {q^2} \leq |f(u)| ...
6
votes
0answers
95 views

The sum $\sum_{n=1}^\infty \min_{k\le n}\{\alpha k\}$ for irrational $\alpha$

Let $\alpha$ be an irrational number. For every $n$ let $z_n$ be the integer closest to the number $\alpha n$. Then we can define $$A(\alpha):= \sum_{n=1}^\infty |\alpha n - z_n|.$$ We can also ...
5
votes
4answers
626 views

Number raised to power of irrational number

What is the consequence of raising a number to the power of irrational number? Ex: $2^\pi , 5^\sqrt2$ Does this mathematically makes sense? (Are there any problems in physics world where we ...
2
votes
2answers
360 views

Proving that any rational number can be represented as the sum of the each cube of three rational numbers

I found the following question in a book: Prove that any integer can be represented as the sum of the each cube of five integers. The answer : ...
9
votes
2answers
136 views

when index is irrational number with inequality

Let $x>0$, show that $$x^{\sqrt{3}}+x^{\frac{\sqrt{3}}{2}}+1\ge 3\left(\dfrac{1+x}{2}\right)^{\sqrt{3}}$$ we consider $$f(x)=2^{\sqrt{3}}(x^{\sqrt{3}}+x^{\dfrac{\sqrt{3}}{2}}+1)- ...
3
votes
1answer
43 views

Problem understanding this specific proof that $\sqrt{2}$ is irrational.

The proof (taken from http://www.themathpage.com/aPreCalc/rational-irrational-numbers.htm#proof): "To prove that there is no rational number whose square is 2, suppose there were. Then we could ...
17
votes
7answers
7k views

What rational numbers have rational square roots?

All rational numbers have the fraction form $$\frac a b,$$ where a and b are integers($b\neq0$). My question is: for what $a$ and $b$ does the fraction have rational square root? The simple answer ...
6
votes
5answers
507 views

For an irrational number $a$ the fractional part of $na$ for $n\in\mathbb N$ is dense in $[0,1]$ [duplicate]

How to prove that the $\{$ fractional part of $n\alpha\mid n \in \mathbb{N}$ $\}$ is dense in $[0,1]$ for an irrational number $\alpha$. NOTICE that $n$ is in $\mathbb{N}$ Also notice that this is ...
1
vote
2answers
32 views

Find the relationship between $n$ and $m$ (both natural numbers) such that $m^{1/n}$ is a rational number.

I know how to show that specific numbers such as $2^{1/2}, 2^{1/3}, 3^{1/2}, etc.,$ are irrational, but what about the general form $m^{1/n}$?
2
votes
1answer
65 views

Is $12^{1/3}$ irrational?

Is $12^{1/3}$ irrational? Give a proof that justifies your answer So far I have: Suppose $12^{1/3}$ is rational.This means there exists integers a and b such that $12^{1/3} = \frac{a}{b}$ where ...
19
votes
10answers
3k views

Critiques on proof showing $\sqrt{12}$ is irrational.

My only exposure to proofs was in a math logic class I took in University. I was wondering if my attempt at proving that $\sqrt{12}$ is irrational is OK. $$\Big(\frac{m}{n}\Big)^2 = 12$$ ...
7
votes
4answers
131 views

Irrational Numbers : Show that $0.1248163264…$ is irrational

I was working through some basic Number Theory Problems in Rosen and came across the following problem : Show that the real number $0.1248163264...$ represented in ...
2
votes
1answer
44 views

Rational Irrational Numbers

I know that a rational number can always be expressed as a fraction, but can't we also say that it is a number that follows a definite pattern? Like one-third for example; it is never ending as a ...
0
votes
0answers
36 views

By induction, show that $\sqrt{p}\notin\mathbb{Q}(\sqrt{p_1},\sqrt{p_2},\cdots,\sqrt{p_k})$ [duplicate]

By induction, show that $\sqrt{p}\notin\mathbb{Q}(\sqrt{p_1},\sqrt{p_2},\cdots,\sqrt{p_k})$ where $p_1,p_2,\cdots ,p_k,p$ are distinct primes. My try: For $k=1, ...
9
votes
1answer
83 views

If all the numbers $(1^\alpha,\,2^\alpha,\,3^\alpha,\,\dotsc)$ are integer, then $\alpha$ is an integer.

A theorem of Siegel asserts that If $\beta>0$ and $2^\beta,\,3^\beta,\,5^\beta$ are integers, then $\beta$ is an integer. The following result is a beautiful consequence of this theorem ...
5
votes
4answers
429 views

William Lowell Putnam Integral Problem [closed]

Prove That $$ \frac{22}{7}-\pi = \int_{0}^{1}\frac{x^{4}\left(1 - x\right)^{4}}{1 + x^{2}}\,{\rm d}x $$
3
votes
4answers
656 views

Is the constant $e$ infinitely long?

The number $e = 2.718281828...$ is the base of the natural logarithm. Its decimal representation is infinitely long. Why does this mathematical constant contain an infinite number? What is the reason ...
1
vote
5answers
46 views

Irrational number multiplied by its fractional part becomes rational (SOLVED)

Here's a Korean middle school midterm problem I've been struggling for quite some time now. "$X$ is an irrational number such that $X>0$, and $Y$ is fractional part of $X$. If $$X^2+Y^2=27$$, find ...
-3
votes
2answers
74 views

Difficult Complex Number Proof. Given $|w| =1$ or $|v|=1$ [closed]

Let $z, w$ be distinct complex numbers. Show that if $|z| = 1$ or $|w| = 1$, then $$\left|\frac{w-z}{1-\overline{w}z}\right| = 1$$ Hint: Note that $|a|^2 = a\overline a$ I have been ...
8
votes
3answers
475 views

“Length” of rationals in an interval

For $x \in \mathbb{R}$, define $r(x)$ as follows: $$ r(x)= \begin{cases} 1 &\text{if $x$ is rational},\\ 0 &\text{if $x$ is irrational}. \end{cases} $$ Q. What is $\int_0^1 r(x) dx$ ? I ...
3
votes
1answer
481 views

Non-integer bases and irrationality

I read somewhere: When it comes to properties like prime, irrational, rational, divisible by 2, etc., nothing changes when you change base. But I'm not sure about the rational/irrational one. ...
0
votes
4answers
63 views

Definition of irrational number

What is a formal definition of a irrational number? Usually, we say that it is a number that it is not rational. Is it enough?
3
votes
2answers
60 views

Closeness of $n! \ x$ to integers for irrational $x$

This question came up in the comments to another question. Is there an irrational number $x$ such that, for sufficiently large $n$, the product $$ n! \ x $$ is arbitrarily close to an integer? More ...
3
votes
2answers
44 views

When do we know for sure that we have the correct digits of an irrational number?

This comes from a programming assignment I was given using MATLAB. The objective was to calculate the difference between $\pi/4$ and the Leibniz series for computing $\pi/4$ with $n = 200$. This ...
4
votes
4answers
139 views

How to prove that the value of $e$ is irrational without using the number $e$ itself [closed]

Recently I have tried to prove that the value of $e$ is irrational without using the number $e$ itself. I have seen that the number $e$ can be expressed as $$\lim_{n\to\infty}(1 + 1/n)^n;$$ however, ...
1
vote
8answers
168 views

Rational number that approximates $\sqrt{3}$

Questions: Show that is is theoretically possible to find a rational number that approximates $\sqrt{3}$ with an error less than $0.001$. Explain how you would go about determining a ...
2
votes
0answers
28 views

Continued fraction for $[1,2,3,4,5,6,\dots]$ [duplicate]

Any continued fraction that does not terminate or repeat can't be rational or a quadratic irrational. It is not hard to write something that does not fit these two categories. Can we still get a ...
0
votes
1answer
74 views

Show that $4^\frac{1}{3}$ is an algebraic number?

How do you show that $4^\frac{1}{3}$ is an algebraic number? I don't understand the question nor how to begin on describing the proof to show what the question is asking.
1
vote
0answers
52 views

Natural bijection between $\mathbb{N}$ and algebraic numbers?

Q. Is there a canonical, explicit bijection between the natural numbers $\mathbb{N}$ and the algebraic numbers? The earlier MSE question, "Bijection for algebraic numbers," does not quite ...
3
votes
2answers
88 views

Integer Part and Arithmetic Progressions

For $A=\{\lfloor n \alpha \rfloor: n\in\Bbb Z \}$, where $\alpha$ irrational, $\alpha \gt 2$, we aim to show the following: There exists $m$ elements contained in $A$ that form an arithmetic ...
7
votes
11answers
4k views

Are there any irrational numbers that have a difference of a rational number?

Are there any irrational numbers that have a difference of a rational number? For example, if you take $\pi - e$, it looks like it will be irrational ($0.423310\ldots$) - however, are there any ...
6
votes
6answers
4k views

what's the difference between a rational number and an irrational number?

I tried to understand the difference between rational numbers and irrational numbers. I understand what is a rational number (a number that can be expressed as the ratio of two numbers p/q). what ...
0
votes
1answer
101 views

On a theorem of Kronecker! [closed]

Let $\alpha$ be an irrational number and $\beta$ be an arbitrary real number, Prove that there are infinitely many pair of integers $(x,y)$ with $x\in\mathbb{N}$ such that: ...