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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.

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3answers
51 views

Irrationality of $(a_1+\sqrt{b_1})(a_2+\sqrt{b_2})$

Sorry, for a rather silly question. Suppose $a_1$, $b_1$, $a_2$, $b_2$ are integers, all different from zero, while $b_1$ and $b_2$ are co-prime positive integers, neither being a complete square. ...
0
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1answer
26 views

Sequences of Irrational numbers

Let $ \alpha $ is fixed irrational number. Let $ [-h,h] $ be an interval. Is it true that for any sequence of irrational numbers $ \{h_n\} $ converges to zero in $ [-h,h] \, $ both $ \, \alpha + h_n ...
3
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5answers
49 views

Find two irrational numbers $x,y$ such that $x+y$ and $xy$ are both rational.

I know how to satisfy one of the statements but never both together. $(a+b)*(a-b)=a^2-b^2$ so taking $a=\sqrt k_1$ and $b=\sqrt k_2$, $k_1,k_2\in\mathbb{Q}$ such that $a,b \notin\mathbb{Q}$ would ...
3
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1answer
59 views

Evaluating $\sum_{n=1}^\infty \frac{1}{(n^5)!} \approx 1$ and proving that is irrational.

Define $\delta = \sum_{n=1}^\infty \frac{1}{(n^5)!}$. Wolfram says it converges by the ratio test. Trying to prove that $\delta$ is irrational, begin defining $S_n$ as: \begin{align} S_n = (n^5)!\...
2
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2answers
50 views

Proof that $e$ is irrational. (Proof verification)

Here, $e = \sum_{k=0}^\infty 1/k!$. Define $S_n$ as: \begin{align} S_n = n!e - n!\sum_{k=0}^n \frac{1}{k!} \end{align} where $n!\sum_{k=0}^n \frac{1}{k!}$ is an integer. Write $e = 1/0!+1/1!+...+1/n!+...
0
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1answer
22 views

Simple continued fraction for irrational numbers.

I read it here that: "What you must have read is that a number with an infinite simple continued fraction expansion is irrational. A continued fraction is "simple" if all the partial numerators are ...
0
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2answers
75 views

$\sqrt{98} + 4^{\frac{1}{3}}$ is irrational [closed]

Problem is to prove that $\sqrt{98} + 4^{\frac{1}{3}}$ is irrational. I know that $\sqrt{98}$ and $4^{\frac{1}{3}}$ is irrational, but I don't know how to use it in this problem.
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0answers
61 views

Looking for a proof that $\pi$ is irrational using a series representation.

I'm searching for a proof that $\pi$ is irrational using a series representation for $\pi$. I've seen that Apery proved that $\zeta(2)$ is irrational by using the series \begin{align} \zeta(2) = \...
0
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0answers
34 views

Is the sum of the reciprocals of $lcm([1,\cdots ,n])$ irrational? [duplicate]

Denote $$f(n):=lcm([1,\cdots ,n])$$ Is the value of the sum $$\sum_{j=1}^\infty \frac{1}{f(j)}$$ irrational ?
1
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5answers
594 views

If irrational numbers are uncountable, then why did I find this? [closed]

I understand that irrational numbers are uncountable. I've seen the proof and it makes perfect sense. However, I came up with this (most likely false) proof that says that they're countable. ...
0
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1answer
19 views

Conditions for solutions to $p^a = q^b$

When (for what conditions on $p,q$) can we solve this equation for integers. I know that $p = q^b$ can only be solved when $p$ is an integer to the power of $b$. By analysing the prime factorisation ...
0
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0answers
32 views

Proof that $log_q(p)$ is (mostly) irrational for integers q,p

Does this proof make sense, especially the last part? Take $q>p$. We consider the equation $p^aq^b=p^{a'} q^{b'}$. Given that $a,b$ and $a',b'$ are distinct respectively. We solve, realising that $...
2
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1answer
72 views

Prove constructively that $\log_2 3$ is irrational.

The usual proof that $\log_2 3$ is irrational is by contradiction. For instance: Assume the negation: that $\log_2 3 = m/n$ for some integers $m$ and $n$. Then, by the property of logarithms, $2^{m/...
3
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2answers
45 views

Inequality deduced from relatively prime numbers.

If $a_n \text{ and }b_n$ are relatively prime for all $n$ and $$\frac{a_n}{b_n}=\frac{1}{n}+\frac{1}{n(n+1)}+\frac{1}{n(n+1)(n+2)}+\cdots$$ Deduce that $$b_n\geq b_{n+1}$$ CURRENT THOUGHTS I can ...
0
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0answers
23 views

Trying to extend distributive property of modulo operation to real numbers

Here Wikipedia states that modulo operation is distributive: $$a \cdot b\ mod\ n = (a\ mod\ n)\cdot (b\ mod\ n)\ mod\ n$$ Which is true for every natural number. Unfortunately it is not for rational ...
0
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1answer
14 views

How many Decimal Places are Needed For Accuracy to a Given Number of Significant Figures?

How many Decimal Places are Needed For Accuracy to a Given Number of Significant Figures? For example: $1234/3.141 = 392.8$, but $1234/3.14159$ changes the result to $392.7$, so how do I am using ...
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2answers
38 views

We can't exactly draw a line of length square root of 2 but can be constructed using Pythagoras theorem in an isosceles right angle triangle?

We can't exactly draw a line of length square root of 2 but in an isosceles right angle triangle of sides 1 unit each, the length of hypotenuse will be the square root of 2. Now does it mean we can ...
2
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2answers
50 views

Prove $\log_{4}6$ is irrational

Thanks for taking the time to verify my approach and as well as my answer. Background: B.S. in Business from a 4-year university taking CS courses online I would like some help with a basic proof ...
0
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1answer
36 views

Find natural number $0 < n < 30,000$ such that $\sqrt[3]{5n}+\sqrt{10n}$ is rational

I was thinking that I could try to make some sort of substitution to convert $\sqrt[3]{5n}+\sqrt{10n}$ into a polynomial with integer coefficients then use the Rational Roots Theorem to find a ...
0
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1answer
43 views

For $0 \le \theta \le \pi/2$, When are both $\theta/\pi$ and $\sqrt2\sin\theta$ rational?

For $0 \le \theta \le \pi/2$, when are both $\theta/\pi$ and $\sqrt2\sin\theta$ rational? I think $\theta=0, \pi/4$ is the only cases. This problem seems to be related to Niven's theorem, but I ...
0
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0answers
43 views

How to convert any decimal number in form of $\sqrt a+\sqrt b$?

Is there any method to directly convert any decimal number in form of $\sqrt a+\sqrt b$, given a and b are integers? Like $4.88=\sqrt5+\sqrt7$ (approximately)
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1answer
41 views

How to convert any decimal number in form of a√b [closed]

Is there any method to directly convert any decimal number in form of a√b, given a and b are integers? like 8.66=5√3(approx)
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0answers
351 views

Zeros square root of two

Why can a sequence of k zeros at the square root of 2 take place only at the k-th decimal place ( k-th digit after comma). And how can i proove/ show it? Thanks a lot :)
0
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4answers
48 views

Show $(3 + \sqrt{2})^{2/3}$ is irrational using RZT

I am asked to prove that $(3+\sqrt{2})^{2/3}$ is irrational via the rational zeroes theorem. This is what I have so far: $ x = (3+\sqrt{2})^{2/3} $ $ x^3 = (3+\sqrt{2})^{2} $ $ x^3 - 11 - 6\sqrt{2}...
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0answers
12 views

Can raising a number to an irrational power have infinite solutions?

$a^{\frac{1}{2}}$ is generally considered to be the positive square root of $a$, but it also makes sense (depending on context) to consider it to be multivalued, returning all square roots of $a$ ...
1
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2answers
78 views

Prove or disprove non-constructively there exist irrationals $a, b, c$ such that $a^{b^c}$ is rational.

Consider the interesting question: Do there exist irrationals $a$ and $b$ such that $a^b$ is a rational? Alternatively, prove or disprove that there exist irrationals $a$ and $b$ such that $a^b$ is ...
0
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3answers
74 views

Silly Question about $π$ [closed]

In our junior classes, we learnt that $π$ is an irrational number. Now, also we know about rational numbers. So, if I say that I have a thread of 44cm long and we may convert it into a circle. Then ...
14
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3answers
260 views

Sum of $\{n\sqrt{2}\}$

I would like to prove (rigorously, not intuitively) that $$\sum_{n=1}^N \{n\sqrt{2}\}=\frac{N}{2}+\mathcal{O}(\sqrt{N})$$ where $\{\}$ is the "fractional part" function. I understand intuitively why ...
0
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3answers
38 views

Arithmetic laws for rational numbers vs. real numbers

Are there any arithmetic laws that are always true for the set of rational numbers but not always true for the set of real numbers? This came up because I was doing various exercises in different ...
3
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0answers
78 views

Proving that $\zeta(2)$ is irrational.

We have the following integral: \begin{align} I_n=\int_0^1 P_n(x)\frac{\ln x}{1-x}dx = \frac{a_n\zeta(2)+b_n}{d_n^2} \end{align} $P_n(x)$ is a polynomial with integer coefficients and degree $n$. $a_n,...
4
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0answers
78 views

Prove a limit involving the ceiling function

I found a pattern that I want to prove: $$f(x) = 2^{\lceil \log_2(3^x)\rceil} - 3^x\quad \{x\in\mathbb{Z}^+\} $$ $$ \lim_{x\rightarrow\infty} f(x) = \infty $$ Discussion: $$ f(x) = 2^{\lceil \...
0
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1answer
42 views

Confusion on last step of Hardy's proof on the square root of “a rational number with imperfect square component”'s inability to be a rational number

Here is a copy of Hardy's proof: For suppose, if possible, that $p^2/q^2 = m/n$, $p$ having no factor in common with $q$, and $m$ no factor in common with $n$. Then $np^2 = mq^2$. Every factor of $...
3
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1answer
62 views

Trying to prove $e$'s irrationality

Knowing that $\lim\limits_{x\to\ 0}\ $$\frac{\sin(x)}{x}$$= 1$ , $\frac{1}{n+1}<n!r_n<\frac{1}{n}$, where $r_n=e- \sum _{ k=0 }^{ \ n}{ \frac { 1 }{k!}} $ By studying $\lim\limits_{n\to\infty}\ ...
1
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1answer
39 views

Prove of irrationality [closed]

how can I prove that $(1-\sqrt{2})^z $ is never rational for any integer $z$ different from $0$ ?
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0answers
31 views

Are there more real numbers than irrational numbers? [duplicate]

Wrapping my head around the mathematical definition of infinity and just curious here: Are there more real numbers than irrational numbers? It would intuitively seem so, but they are both just ...
1
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1answer
31 views

every positive rational number can be written as a finite sum of distinct numbers of the form 1/n where n is natural

every positive rational number can be written as a finite sum of distinct numbers of the form 1/n where n is natural I don't quite understand this question. Actually, what does distinct numbers of ...
1
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2answers
82 views

Did Babylonians know the Pythagorean theorem before his time?

On old tablets the Babylonians were able to work out the digits to the square root of two from the hypotenuse of a $45^\circ-45^\circ-90^\circ$ triangle. How could they have figured this out without ...
1
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1answer
23 views

Is a polygon with irrational internal angles possible?

I'm aware that the sum of internal angles of an $n$-sided polygon must be $180^\circ(n-2)$, but that doesn't say anything about the individual angles.
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2answers
56 views

Proving $e$ is irrational using a beukers like integral

We know the following, for some integers $a_n,b_n$ and $n$ where $n\geq0$: \begin{align} &I_n = \int_0^1 x^n(1-2x)^n e^x dx = a_ne+b_n \\ &|I_n|= \left\lvert \int_0^1 x^n(1-2x)^n e^x dx \...
6
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1answer
155 views

How to analyze $(-1)^{\left \lfloor n\theta \right \rfloor}$ (in which $\theta$ is an irrational number)?

Let $\theta = \frac{\sqrt{5}-1}{2}$. Define $a_{n}=(-1)^{\left \lfloor n\theta \right \rfloor}$. Please judge whether $S_{n}=\sum_{k=1}^{n} a_{k}$ is unbounded. I tried to relate $\left\lfloor n\...
0
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1answer
37 views

Sequence of fractions $c_n = a_n / b_n$ converges to irrational $x$, prove $a_n$ and $b_n$ diverges to infinity. [duplicate]

$a_n$ and $b_n$ are sequences of natural numbers, $\lim_{n->\infty}\frac{a_n}{b_n} = x$, where $x$ is irrational. Prove that $a_n$ and $b_n$ diverge to infinity. I've proved that if a sequence of ...
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votes
1answer
39 views

Proof for primes $\sqrt{p_1 p_2} \notin \mathbb{Q}$ [closed]

Let $p_1$, $p_2$ be two prime numbers such that they aren't equal. Prove: $\sqrt{p_1p_2} \notin Q $ I understand that there is a way to prove it using $m/n$ when $ \gcd {(m,n)}=1$ but I didn't ...
0
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1answer
25 views

Find real numbers satisfying some conditions of rational dependance

Does there exist $\xi_1,\ldots,\xi_4\in\mathbb R$ such that $$\forall (\alpha,\beta)\in\mathbb Q^2\setminus\{0,0\},\quad \begin{cases}\dim_{\mathbb Q}(\alpha \xi_1+\beta\xi_3,\,\alpha\xi_2+\beta\...
0
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2answers
113 views

Proof that the nth root of a rational s/t is irrational unless s and t are perfect nth powers

I'm trying to prove that $\sqrt[n]{\frac{s}{t}}$ is irrational unless both s and t are perfect nth powers. I have found plenty of proofs for nth root of an integer but cannot find anything for ...
0
votes
1answer
58 views

Does $lcm\{1,2,…,n\} = \prod_{p\leq n, p\in\mathbb{P}}p^{\lceil \frac{log(n)}{log(p)}\rceil}$?

I am trying to understand Apery's proof of the irrationality of $\zeta(3)$ from start to end, with this document. I apologise for having 2 questions in one, but both are relatively simple (I just need ...
0
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2answers
26 views

Maximum Period of Decimal Expansion

My question is similar to (but different from) the one here. I came across this sentence on Wikipedia: "The decimal expansion of a rational number always either terminates after a finite number of ...
9
votes
2answers
116 views

Prove that $\sqrt[3]{5} + \sqrt{2}$ is irrational

I tried with both squaring and cubing the statement, it got messy, here's my latest attempt: Assume for the sake of contradiction: $\sqrt[3]{5} + \sqrt{2}$ is rational $\sqrt[3]{5} + \sqrt{2}$ = $\...
5
votes
1answer
145 views

Show that $\lim_{k\to\infty}\lim_{j\to\infty}\cos^{2j}k!\pi x=0$ [duplicate]

The Dirichlet function is defined as the indicator function of rational numbers. I have also seen this function described by: $$f(x)=\lim_{k\to\infty}\lim_{j\to\infty}\cos^{2j}k!\pi x$$ How does this ...
2
votes
1answer
47 views

Interior and closure of $\mathbb{Q}\cap (0,1)$.

Determine the interior and closure of the set $A=\mathbb{Q}\cap (0,1)$. My approach: First note that the set of rationals $\mathbb{Q}$ and irrational $\mathbb{I}=\mathbb{R}\setminus\mathbb{Q}$ are ...
1
vote
2answers
36 views

proving transcendental numbers are irrational

I don't understand how every transcendental number is irrational, is there a way to prove that? I know it just means it's a non-algebraic number, but how does that correlate to irrationality?