Numbers not expressible as a ratio of two integers. Examples: $\sqrt{2},\phi,e,\pi,\zeta(3)$. Some of them are algebraic ($\sqrt{2},\phi$) and some transcendental ($e,\pi$).

learn more… | top users | synonyms

1
vote
0answers
116 views

Asymptotic behavior of $\sum_{j=1}^n \cos^p(\pi u j)$ for large $n$ and $p$?

Consider the sum $$S=\sum_{j=1}^n \cos^p(\pi u j),$$ where $n$ and $p$ are positive integers and $u$ is irrational. Let's say $p$ is even. I'm interested in the asymptotic behavior of this for $n$ ...
3
votes
2answers
92 views

How do continuity, distance and irrationals arise from discreteness?

Consider a square as rendered on a computer screen: its width and height are $N$ pixels each, and its area is $N^2$ pixels. Its diagonal, when measured in pixels, is also $N$ pixels long. If you ...
58
votes
4answers
6k views

Can an irrational number raised to an irrational power be rational?

Can an irrational number raised to an irrational power be rational? If it can be rational, how can one prove it?
3
votes
5answers
3k views

How can one prove that the cube root of 9 is irrational?

Of course, if you plug the cube root of 9 into a calculator, you get an endless stream of digits. However, how does one prove this on paper?
20
votes
4answers
1k views

Uncountable set of irrational numbers closed under addition and multiplication?

Is such a thing even possible? There's not much to say really. Obviously if there was a set it would be full of transcendental numbers. This led me to think of a function generating transcendental ...
37
votes
6answers
3k views

$\sin 1^\circ$ is irrational but how do I prove it in a slick way? And $\tan(1^\circ)$ is …

In the book 101 problems in Trigonometry, Prof. Titu Andreescu and Prof. Feng asks for the proof the fact that $\cos 1^\circ$ is irrational and he proves it. The proof proceeds by contradiction and ...
1
vote
3answers
486 views

Dedekind's method for irrational number

I am now reading the definition of irrational number, which we can describe by the following terms: suppose that we have divided all rational numbers into two classes, a lower class and an upper ...
2
votes
3answers
681 views

What is the ratio of rational to irrational real numbers?

There exists an infinite amount of rational and irrational numbers. But is there more irrational numbers than rational? And if so can a ratio of one to the other be calculated?
6
votes
1answer
3k views

Sum of irrational numbers

Well, in this question it is said that $\sqrt[100]{\sqrt3 + \sqrt2} + \sqrt[100]{\sqrt3 - \sqrt2}$, and the owner asks for "alternative proofs" which do not use rational root theorem. I wrote an ...
23
votes
9answers
5k views

Prove $2^{1/3}$ is irrational.

Please correct any mistakes in this proof and, if you're feeling inclined, please provide a better one where "better" is defined by whatever criteria you prefer. Assume $2^{1/2}$ is irrational. ...
5
votes
0answers
321 views

Is ${^5\pi}$ an integer? [duplicate]

Possible Duplicate: How to show $e^{e^{e^{79}}}$ is not an integer Is ${^5\pi}$ an integer? It is "obviously" not, right? But can we prove it? Here ${^5\pi}$ means the result of tetration ...
23
votes
2answers
824 views

Is there any real number except 1 which is equal to its own irrationality measure?

Is there any real number except $1$ which is equal to its own irrationality measure? If so, then what is the cardinality of the set of all such numbers? Is the set dense on any interval? Is it ...
18
votes
3answers
940 views

Prove $x = \sqrt[100]{\sqrt{3} + \sqrt{2}} + \sqrt[100]{\sqrt{3} - \sqrt{2}}$ is irrational

Prove $x = \sqrt[100]{\sqrt{3} + \sqrt{2}} + \sqrt[100]{\sqrt{3} - \sqrt{2}}$ is irrational. I can prove that $x$ is irrational by showing that it's a root of a polynomial with integer coefficients ...
0
votes
1answer
247 views

When are $\theta$ and $\sin\theta^\circ$ both rational? [duplicate]

Possible Duplicate: Sine values being rational I'm guessing that if I look in Ivan Niven's elementary book on irrational numbers, I'll find the answer to this quickly, but I'm posting it ...
7
votes
4answers
2k views

When is $\sin(x)$ rational?

Obviously, there are some points (like $\pi,30$) but I am unsure if there are more. How can it be proved that there are no more points, or what those points will be? EDIT: I largely meant to ask ...
10
votes
1answer
179 views

Is there a dense subset of $\mathbb{R}^2$ with all distances being incommensurable?

Is there a set $S$ of points on the real plane $\mathbb{R}^2$ such that: there is a point belonging to $S$ in any neighborhood of every point of $\mathbb{R}^2$ (so, $S$ is dense) and ratio of any ...
11
votes
1answer
136 views

Irrationality of Two Series

Show that if the integers $1<b_1<b_2<\cdots$ increase so rapidly that$$\frac{1}{b_{k+1}}+\frac{1}{b_{k+2}}+\cdots<\frac{1}{b_{k}-1}-\frac{1}{b_{k}},\quad k\geq 1,$$ then the number ...
1
vote
1answer
163 views

Does anybody know the formula for a quasicrystal structure?

I am an architecture student researching into quasicrystals with the hope of applying it to form a complex truss system. I was wondering if anyone new of a formula for the structure? thanks in ...
5
votes
4answers
308 views

Existence of irrationals in arbitrary intervals

I was studying for my analysis mid-term paper and was going over the properties of real numbers. I was wondering how to prove the following statement: (Not a textbook problem, it just popped into my ...
14
votes
2answers
2k views

ArcTan(2) a rational multiple of $\pi$?

Consider a $2 \times 1$ rectangle split by a diagonal. Then the two angles at a corner are ArcTan(2) and ArcTan(1/2), which are about $63.4^\circ$ and $26.6^\circ$. Of course the sum of these angles ...
1
vote
3answers
289 views

How to prove $e$ isn't a $\frac {a}{b}$. Not irrationality with other ways or about transcendental, only about fractions

I would like a proof that $e$ isn't a fraction $\frac{a}{b}$, for $a,b \in Z$ and $mdc(a,b)=1$. Just a observation =) I'd like a proof with fractions, not about $e$ irrationality or if $e$ is ...
1
vote
0answers
248 views

What is the exact definition of a rational power?

I was taught in school that $$x^{a/b} = \sqrt[b]{x^a}$$ however, wolfram says this is not always true: $\sqrt[3]{x^2} \ne x^{2/3}$ ...
21
votes
3answers
2k views

Proving that $m+n\sqrt{2}$ is dense in R

I am having trouble proving the statement: Let $S = \{m + n\sqrt 2 : m, n \in\mathbb Z\}$. Prove for every $\epsilon > 0$, The intersection of $S$ and $(0, \epsilon)$ is nonempty.
0
votes
1answer
114 views

non-perfect square of number [duplicate]

Possible Duplicate: $a^{1/2}$ is either an integer or an irrational number I would like to know the better proof for the following one. question: non perfect square of any integer is an ...
0
votes
1answer
231 views

How to prove $\sqrt3$ is irrational? [duplicate]

How to prove $\sqrt3$ is irrational using Fermat's infinite descent method? Like says in Carl Benjamim Boyer's book. Isnt the same prove to $\sqrt2$, in Boyer's book says something like this. ...
11
votes
3answers
432 views

Closed form for a pair of continued fractions

What is $1+\cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{4+\cdots}}}$ ? What is $1+\cfrac{2}{1+\cfrac{3}{1+\cdots}}$ ? It does bear some resemblance to the continued fraction for $e$, which is ...
4
votes
0answers
146 views

Is the maximal temperature of the curlicue fractal acheived by $e\times\gamma$?

The Curlicue Fractal is defined as follows: Choose an irrational number $s$ and a horizontal unit segment with angle $\phi_0 = 0$. Define $\theta_{n+1} = \theta_{n} + 2 \pi s \pmod{2 \pi}$, with ...
4
votes
1answer
381 views

rational angles with sines expressible with radicals

An angle x is rational when measured in degrees. sin(x) is can be written using radicals. What are the conditions on x? If nested square roots are allowed? What I know so far: If sin(x) can be ...
2
votes
7answers
2k views

How to prove that $\sqrt 3$ is an irrational number? [duplicate]

Possible Duplicate: $a^{1/2}$ is either an integer or an irrational number I know how to prove $\sqrt 2$ is an irrational number. Who can tell me that why $\sqrt 3$ is a an irrational ...
3
votes
3answers
616 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 ...
5
votes
1answer
398 views

Is it ever really Pi Time?

Walking with my son at 3:14pm the other day, I mentioned to him, "Hey, it's Pi Time". My son knows 35 digits of $\pi$ (don't ask), and knows that it's transcendental. He replied, "is it exactly ...
18
votes
6answers
13k views

Proving Irrationality

How is it possible to prove a number is irrational? First part of that question: How it possible to know that a number will go on infinitely? Second part: How is it possible to know that no ...
3
votes
1answer
369 views

How to express an irrational as a continued fraction in computer with high precision?

Background I'm writing a C++ library for continued fraction using MPIR (Multiple Precision Integers and Rationals) library http://www.mpir.org/ due to the limitation of built-in ...
0
votes
3answers
186 views

What is $\{y\in\mathbb Q\mid y=\cos(x),\quad x\in[0,2\pi]\cap\mathbb Q\}?$

Given a range of the rational numbers, $x$, between $0$ and $2\pi$\, what is the set of rational numbers $ y = \cos(x) $? I was inspired by the stackoverflow question Can $\cos(a)$ ever equal $0$ in ...
13
votes
0answers
389 views

Irrationality of e

I have a question about the irrationality of $e$: In proving the irrationality of $e$, one can prove the irrationality of $e^{-1}$ by using the series $$e^x = 1+x+\frac{x^2}{2!} + \cdots + ...
23
votes
3answers
5k views

Can you raise a number to an irrational exponent?

The way that I was taught it in 8th grade algebra, a number raised to a fractional exponent, i.e. $a^\frac x y$ is equivalent to the denominatorth root of the number raised to the numerator, i.e. ...
4
votes
2answers
224 views

Extrapolating properties of rational numbers to irrational/transcendental numbers

I've had this idea in my head for a while, but I've never told anybody because... well, I really don't know. I just never thought that it might even be remotely correct, but here goes. Here is just an ...
6
votes
2answers
395 views

For any irrational number such as pi, would any sequence of length n appear in its decimal places?

If pi is an irrational number that goes on infinitely forever, does it mean that I can get any sequence of numbers of any length, and somewhere in the decimals of Pi, this sequence will exist. Eg. ...
4
votes
4answers
269 views

Relationship between degrees of continued fractions

I'm trying to compute the values of differing degrees of continued fractions like $\sqrt 2$, e and other similar fractions. My theory was to take the reduced fraction at an arbitrary depth and the ...
3
votes
1answer
415 views

Can we have a mirror image of an irrational decimal?

Is it possible to have a number that extends to the left of the decimal point in mirror image of an irrational number? Such as <...95141.30000...>, to write pi as a mirror image.
7
votes
6answers
8k views

Prove that the product of a rational and irrational number is irrational

Could you please confirm if this proof is correct? Theorem: If $q \neq 0$ is rational and $y$ is irrational, then $qy$ is irrational. Proof: Proof by contradiction, we assume that $qy$ is rational. ...
8
votes
1answer
291 views

What bizarrities lurk within the decimal expansion of an irrational number?

i.e. can we write $\pi = 3.14159\dots X\dots$ where $X$ consists of (say) $10^{100}$ consecutive zeroes? [Originally asked on reddit without response :-( ]
14
votes
2answers
557 views

How come such different methods result in the same number, $e$?

I guess the proof of the identity $$ \sum_{n = 0}^{\infty} \frac{1}{n!} \equiv \lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x $$ explains the connection between such different calculations. How ...
3
votes
3answers
768 views

If $n$ is any positive integer, prove that $\sqrt{4n-2}$ is irrational

If $n$ is any positive integer, prove that $\sqrt{4n-2}$ is irrational. I've tried proving by contradiction but I'm stuck, here is my work so far: Suppose that $\sqrt{4n-2}$ is rational. Then we ...
1
vote
2answers
121 views

-$\frac{2\sqrt{2}-6}{7}$ = $\frac{6-2\sqrt{2}}{7}$ correct?

When asked to rationalize the denominator for $\frac{2}{\sqrt{2}+3}$, I came up with $\frac{6-2\sqrt{2}}{7}$ but my algebra book gives -$\frac{2\sqrt{2}-6}{7}$ as the answer. I think we're both ...
5
votes
4answers
423 views

why does $\sqrt2 = \frac{2}{\sqrt2}$?

I noticed just now that $\sqrt2 = \frac{2}{\sqrt2}$ I'm suprised because isn't this like saying $x = \frac{2}{x}$?
20
votes
1answer
2k views

Is there a proof that $\pi \times e$ is irrational?

A little reading suggests: It is known that either $\pi + e$ or $\pi \times e$ is transcendental (or possibly both), but no proof is known that one of those two numbers in particular is ...
8
votes
2answers
1k views

Is the area of a circle ever an integer?

Is the area of a circle ever an integer? I was trying to answer someone else's question on yahoo answers today and I got thumbs down from people on my answer and have come here to get a thorough ...
3
votes
1answer
239 views

How can I determine a number is irrational?

I have a hypothesis about regular polygons, but in order to prove or disprove it I need a way to determine whether an expression is rational. Once I boil down my expression the only part that could be ...
25
votes
2answers
860 views

Is this number transcendental?

My son was busily memorizing digits of $\pi$ when he asked if any power of $\pi$ was an integer. I told him: $\pi$ is transcendental, so no non-zero integer power can be an integer. After tiring of ...