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 ($e,\pi$).

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3
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96 views

If $q^n$ is irrational for all $n>1$, then $q$ is irrational.

Theorem. Let $q \in \mathbb{R}$ an arbitrary given number. If $q^n$ is irrational for all $n>1$ integer, then $q$ is irrational. My Questions. What is a the name of this statement and what is the ...
-1
votes
1answer
63 views

How is circle closed?

I have this thought that circle in 'real' is not a closed figure. We all know that 'pi' is irrational.And integers are nodes in a 'monstrous' line of real numbers. Irrational numbers are ...
5
votes
4answers
160 views

Is $\ln\sqrt{2}$ irrational?

I know that the natural log of any positive algebraic number is transcendental, as a consequence of the Lindemann-Weierstrass theorem, but what about the natural log of the square root of two (which ...
1
vote
0answers
40 views

What is the limit of $k^2|\pi-n(k)/k |$, where $k$ minimizes $|k\pi -n|$?

Let $k\in \mathbb N$ and for any such n, let $k=k(n)$ minimizes the distance $|k\pi-n|\leq 2 \pi$. It is clear that, by fixing the value of $n$, it is possible to choose $k$ (and vice versa). ...
3
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1answer
68 views

Rational values of $\sin(\log(x))$

Apart from the trivial solution $\sin(\log(1))=0$, is $$\sin(\log(x))$$ ever rational if $x$ is rational?
0
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2answers
103 views

How do you multiply infinite quantities?

Out of curiosity I was watching this video from njwildberger on youtube: https://www.youtube.com/watch?v=4DNlEq0ZrTo Where he says that you can't define associativity between irrational numbers ...
2
votes
1answer
26 views

Existence of a sequence of integers $\lbrace a_k\rbrace_{k\geq 1}$ so that the first $k$ digits of $a_k\alpha$ are $0$ where $\alpha$ is irrational.

Let $\alpha$ be an irrational number. Is there a sequence $\lbrace a_k\rbrace_{k\geq 1}$ of integers so that the first $k$ digits of the fractional part of $a_k·\alpha$ are $0$? (in base $2$, for ...
4
votes
4answers
223 views

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

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 ...
0
votes
1answer
59 views

$1$ is not congruent because of Fermat's Last Theorem?

I would like someone to explain something I did not understand. I was reading a page called "nuking the mosquito" where they give very complex proofs for very simple results. The proof I want to talk ...
2
votes
3answers
121 views

Irrational number “test”?

Suppose we have a finite quantity $a$, which we would like to prove to be irrational, supposing that it is indeed irrational. Then, would it be enough to show that ...
2
votes
2answers
71 views

Are numbers like $\left ( -2 \right )^{\sqrt{2}}$ real or complex?

I know that numbers with rational power can be converted to radicals and based on the degree of the radical we can say that whether they are real or complex. But what about numbers like $\left ( -2 ...
0
votes
2answers
132 views

Lambert's Original Proof that $\pi$ is irrational.

I am trying to find Lambert's original proof that $\pi$ is irrational. Wikipedia has a little description but it is quite lacking. Can someone direct me to Lambert's original proof or post his proof ...
6
votes
7answers
775 views

A question regarding irrational lengths in reality

I have a square stone slab 1 metre by metre, by the Pythagorean identity the diagonal from one corner to another is given as $\sqrt 2$. However $\sqrt 2$ is an irrational number, could someone ...
0
votes
0answers
47 views

Archimedes' Apprxomation of Square Roots

Supposing a square root $\sqrt{X}$, let $x$ be the approximation of $\sqrt{X}$, then we get these 2 formulas to estimate $\sqrt{X}$: $x_{n+1}=\frac{x_n+\frac{X}{x_n}}{2}$ and ...
0
votes
2answers
52 views

Help With a proof (Irrational Number)

Prove the following statement by proving its contrapositive: if $r$ is irrational, then $r^\frac{1}{5}$ is irrational. Its contrapositive will be: If $r^\frac{1}{5}$ is not irrational, then $r$ is ...
4
votes
1answer
279 views

Proof that $\sqrt[m]{a} + \sqrt[n]{b}$ is irrational

Is there a way to prove that $\sqrt[m]{a} + \sqrt[n]{b}$ ($\sqrt[m]{a}$ and $\sqrt[n]{b}$ are irrational); $a, b, m, n \in \mathbb{N}$; $m, n \neq 2$; is irrational without using the theorem mentioned ...
2
votes
2answers
78 views

$x+y\sqrt{2}$ infimum ($x,y\in \mathbb{Z}$)

I've looked for help with this question but I have not found anything, I hope this is not a duplicate. Define the set $A=\{\mid x+y\sqrt{2}\mid \ : x,y\in \mathbb{Z}\ \mbox{and} \mid ...
0
votes
1answer
109 views

Show that $\sqrt{2}$ is irrational using integer root theorem

Show that $\sqrt{2}$ is irrational using integer root theorem. Let $P(x)=x^2-2$. Since $\sqrt{2}$ is a root of this polynomial, had it been a rational (suppose $\sqrt{2}=\frac{p}{q}$) no, by ...
0
votes
1answer
49 views

The minimum number of digits after the floating-point, which uniquely identify every irrational square root

Let the following: $B:$ a natural number larger than $1$ $S:$ a set of irrational numbers in the range $(0,1)$ represented in base $B$ $L:$ the minimal prefix length which uniquely identifies every ...
-3
votes
1answer
67 views

How do we prove sqrt2 is irrational? [duplicate]

What kind of number is sqrt2, rational or irrational? And what are the analytic and non analytic ways of proving it?
9
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2answers
1k views

Deciding whether a number is rational (2 examples)

1) Prove that number irrational $\sqrt{7-\sqrt{2}}$ I created a polynomial $x=\sqrt{7-\sqrt{2}}$ so $P(x)=x^4-14x^2+47$ and since $47$ is prime we check $P(x)$ for $ {1,-1,47,-47}$ and since all of ...
0
votes
0answers
39 views

Curve avoiding semi-rational points

A rational point is a point in $\mathbb{R}^d$ all of whose $d$ coordinates are rational. Let me define a semi-rational point as one that has at least one rational coordinate (but whose other ...
4
votes
2answers
134 views

The set $E= \{x\in [0,1]: \sum_{j=1}^\infty t^j|x−q_j|^{-r} <\infty\}$ does not contain all irrational numbers in $[0,1]$

Let $q_1,q_2,q_3,...$ be an enumeration of $\mathbb{Q}\cap[0,1]$ and let $r,t \in (0,1).$ Consider the set $$E= \{x\in [0,1]: \sum_{j=1}^\infty t^j|x−q_j|^{-r} <\infty\} $$ (a) Show that $E\neq ...
3
votes
2answers
604 views

The sum of two irrational square roots

This is very similar to this question, but I was wondering if there was a simpler proof. In particular, a proof that would prove that $\sqrt{x}+\sqrt{y}$ is an irrational number if both $\sqrt{x}$ ...
1
vote
2answers
84 views

Definition of Rational/ Irrational Numbers reguarding denominators

The definition of a Irrational number is "Irrational numbers don't include integers OR fractions. However, irrational numbers can have a decimal value that continues forever WITHOUT a pattern." So ...
8
votes
2answers
308 views

Dimension of R over Q without cardinality argument. [duplicate]

I am looking for the easiest (elementary) proof that $\mathbb R$ is infinite dimensional as a $\mathbb Q$-vector space, without using cardinality. It should be understandable at highschool level. So ...
3
votes
2answers
78 views

Are there any non-trivial counterexamples to the non-closure of the irrational numbers over addition?

It is trivial to show that the set of irrational numbers is not closed under addition. Just choose an irrational number $p$ and add it to its additive inverse $-p$ to get $0\in\mathbb{Q}$. However, I ...
7
votes
3answers
148 views

Using decimals of $\pi$ to store data

I read recently about an idea to, instead of storing actual data, converting the data to a string of digits and then store the index of where this pattern occurs in some number, for example $\pi$. The ...
0
votes
2answers
52 views

How to find irrational approximates

Say I have a rational number, $n$, that approximates an irrational number of the form: $$n \approx {a+\sqrt b \over c}$$ in terms of being irrational. What is a good way of finding the unknown ...
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vote
2answers
48 views

Does the absence of horizontal lines shows that there are no $n,m\in \mathbb{N}$ such that $n^2=2m^2$?

When I was learning about the proof of the irracionality of $\sqrt{2}$, I remember of trying to visualize it by ploting the graphs of $f(n)=n^2$ and $g(m)=2m^2$, but at the time I got confused and ...
0
votes
0answers
19 views

Using Descarte’s rule of signs to determine the number of positive roots.

Using the Descarte’s rule of signs to determine the number of positive roots. \begin{equation} f(q)=[(k_f+k_d+k_p*(1-q))(\lambda_b* \gamma ...
2
votes
2answers
114 views

Conjecture: if $a+b$ and $ab$ are rational, $a$ and $b$ are rational

I can't find a rigorous proof but I have a feeling it's true. Informal argument: Suppose $a+b$ and $ab$ are rational, $a$ and $b$ are irrational (since just one can't be irrational). Then $a$ and ...
2
votes
1answer
81 views

Show that $\arctan(n)$ is irrational for all $n \in \mathbb{N}$

Question : Show that $\arctan(n)$ is irrational for all $n \in \mathbb{N}$. Hint: My solution doesn't use continued fraction. I am interested in other possible proofs for this question.
9
votes
4answers
1k views

Is the fact that there are more irrational numbers than rational numbers useful?

Although it is known that the cardinality of the set of irrational numbers is greater than the cardinality of the set of rational numbers, is there any usefulness/applications of this fact outside of ...
1
vote
2answers
83 views

Prove that if $n \geq 2$, then $\sqrt[n]{n}$ is irrational. Hint, show that if $n \geq 2$, then $2^{n} > n$.

Prove that if $n \geq 2$, then $\sqrt[n]{n}$ is irrational. Hint, show that if $n \geq 2$, then $2^{n} > n$. So, my thought process was that I could show that $2^{n} > n$ using induction, but ...
2
votes
1answer
115 views

Does the limit of a sequence with floor function exist?

Question : Let $a_n=n\alpha-\lfloor n\alpha\rfloor\ (n=1,2,\cdots)$ where $\alpha$ is an irrational number. Then, does the limit $n\to\infty$ of $(a_n)^n$ exist? I know that ...
1
vote
1answer
100 views

How to prove that $\cos(n)$ is irrational?

We know that $\cos(1)$ is real and transcendental (1). Then by using the fact that for every $n \in \mathbb{N}$ there exists a polynomial $P_n$ of degree $n$ with integer coefficients such that ...
3
votes
1answer
87 views

Irrationality measure.

I would like someone to give me a definition of what irrationality measure is, I have stumbled over several definitions which may be equivalent but as I lack understanding I cant see this correlation. ...
14
votes
1answer
307 views

Can we prove that the solutions of $\int_0^y \sin(\sin(x)) dx =1$ are irrational?

Can we prove that the solutions of $$\int_0^y \sin(\sin(x)) dx =1$$ are irrational? Wolfram Alpha gives two approximate sets of solutions as $\{4.58+2\pi k|k\in\mathbb{Z}\}$ and $\{1.69+2\pi ...
4
votes
3answers
110 views

Real numbers that are not the roots of any polynomial equation with algebraic coefficients

An algebraic number is a number which is a root of some non-zero polynomial equation with rational coefficients. A transcendental number is a number which is not a root of any non-zero polynomial ...
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3answers
56 views

negative powers $(x^{-2} = 1/x^2)$

I need clarification for negative power of a number. I understand $x$ to the power of $2$ is equal to $x\cdot x$ But how $x$ to the power of $-2$ is equal to $\dfrac{1}{x^2}$ ?
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0answers
31 views

Integer algorithm

I have this equation: $$\ln_p y = x+\ln_k z$$ for $p, y, x, k, z \in \mathbb{N}$ Now consider that I have the values for $y$ and I can generate in anyway possible, the value for $x$. How would i ...
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0answers
48 views

“Building blocks” for computable functions

In an (otherwise very enlightening) answer to another question of mine the question came up What functions are allowed as building blocks for computable functions? I was astonished that there ...
4
votes
4answers
460 views

Why there are irrational numbers?

I do not quite get it. Why can't we represent all real numbers as a sum of rational numbers? Why do we need irrational numbers? For example, ...
2
votes
1answer
74 views

is $(\mathbb{Q} \times (\mathbb{R}\setminus\mathbb{Q}))\cup((\mathbb{R}\setminus\mathbb{Q})\times\mathbb{Q})$ connected? path connected?

let $$X=(\mathbb{Q} \times (\mathbb{R}\setminus\mathbb{Q}) ) \cup ((\mathbb{R}\setminus\mathbb{Q})\times\mathbb{Q}) $$ and let $$\tau=\tau (\text{euclid})$$ what are the connected components of ...
2
votes
1answer
40 views

Solving surds without compairing

Question: Let $a + \sqrt{2b} = 3 - 2\sqrt{2}$ .Find the value of $a - \sqrt{2b}$ What I did: I compared the whole numbers and the irrational numbers in both sides and calculated the answer $3 + ...
3
votes
2answers
158 views

A dense set on $[0,1)$

Let $x\in \mathbb{R}$ an irrational number. Define $X=\{nx-\lfloor nx\rfloor: n\in \mathbb{N}\}$. Prove that $X$ is dense on $[0,1)$. Can anyone give some hint to solve this problem? I tried ...
2
votes
1answer
33 views

Need help to simplify irrational equation

I have faced a problem simplifying this equation. . I tried to solve it this way: , but I just can't get the correct answer. This equation is from high school course, so it must have quite a simple ...
0
votes
3answers
47 views

A pretty much simple number theory problem

Let $x$ be an irrational number, and $n$ be a positive integer. Will there ever be a set of $(n,x)$ which satisfies $x(n-x) \in \mathbb{Z}$ ? If so, could you suggest those numbers? And, if not, ...
0
votes
1answer
42 views

Square root of an odd composite being irrational

Is there an odd composite number $n$ such that $\sqrt{n}$ is irrational?