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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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2answers
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Integer solutions of $ X+Y+Z=X\cdot Y\cdot Z $ [closed]

An integer solution of above equation is $(X,Y,Z)=(1,2,3)$. But I am wondering: are there other natural solutions? And what about rational or irrational solutions, where $X,Y,Z$ are different ...
0
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1answer
46 views

Prove ${e}^{x}$, $x\in\mathbb{Z}$ is irrational [duplicate]

I was thinking of random problems in my head when this came to mind. But I have absolutely no idea how to solve it, so I can't show my attempts. Please help me with this. Prove $\boldsymbol{e^x}$, $\...
1
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1answer
43 views

When does a polynomial with integer coefficients have only irrational roots?

Given a polynomial $P$ with integer coefficients, is there any simple criterion (other than explicitly calculating the roots) with which I can check whether its roots are irrational?
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2answers
21 views

Are the roots of polynomials “with almost integer” roots irrational?

Ok, this may be not the most clear title, but my question is straightforward. Say we choose $n$ integers $\{ z_1,\dots,z_n \}$ and we construct the polynomial $$ P(x) = \prod_{i=1}^n (x-z_i). $$ Are ...
0
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1answer
57 views

Show that $\sqrt{abc}$ is irrational if $a, b, c$, and $\sqrt{a} + \sqrt{b} + \sqrt{c}$ are irrational. [closed]

Assume $a$,$b$,$c$ are the irrational numbers, and $\sqrt a + \sqrt b + \sqrt c$ is irrational number. Show that $\sqrt{abc}$ is irrational number. Please help me this problem, thank you for watching!
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2answers
52 views

If $x$ and $y$ are irrational, then $x^y$ is irrational

I thought it was true, however my textbook claims it to be false. I need a counter example but I can't really think of one.
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1answer
66 views

If $x^2$ and $x^3$ are rational, does it imply that $x$ is rational? [closed]

It is given that $x^2$ is rational and $x^3$ is rational. Is $x$ rational for all cases satisfying these conditions or is there are case where $x$ won't be rational? If so, then what other condition(...
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0answers
16 views

How does Siegel's Lemma and properties of auxiliary functions helps in transcendental/irrationality proofs?

I know this Lemma appears more in trascendental number theory, but I saw a proof of the irrationality of $e^\pi$ it's a proof by contradiction, finding a integer between $0$ and $1$, but to that it's ...
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1answer
46 views

Prove that $\sqrt[3]p+\sqrt[3]{p^5}$ is irrational if $p$ is prime [closed]

Prove that $\sqrt[3]p+\sqrt[3]{p^5}$ is irrational when $p$ is a prime. First I suppose $x=\sqrt[3]p+\sqrt[3]{p^5}$. Cubing gives $$x^3=p+p^5+p^2x$$ And then what properties of prime, and how to test ...
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4answers
107 views

Proving $1 +\sqrt[3]{5-\sqrt2}$ is rational via the rational roots theorem [closed]

Find a polynomial that has $x = 1 +\sqrt[3]{5-\sqrt2}$ as a root, then use rational roots theorem to show that $x$ must be rational. I came up with the polynomial $100,000,000,000x^2 = 640393215809$ ...
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1answer
39 views

Proving piecewise function is not continuous [closed]

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be given by $f(x) = \left\{ \begin{array}{ll} 2x & \quad x \text{ is rational} \\ -2x & \quad x \text{ is irrational }...
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0answers
54 views

Can we find the exact sum of series $\sum_{n=0}^\infty \frac{1}{(n!)^n}$ [closed]

Can we find the exact sum of series $\sum_{n=0}^\infty \frac{1}{(n!)^n}$? We know thaf the sum is $e$ without that power 'n' in the denominator.
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2answers
56 views

Find all non-negative real numbers $a_1$ $\le$ $a_2$ $\le$ $\ldots$ $\le$ $a_n$

Find all non-negative real numbers $a_1$ $\le$ $a_2$ $\le$ $\ldots$ $\le$ $a_n$ satisfying $\sum_{i=0}^n a_i = 12$ , $\sum_{i=0}^n (a_i)^2 = 18 $, and $\sum_{i=0}^n (a_i)^3 = 27 $ I have a ...
0
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1answer
34 views

Two different rational numbers to the power irrational both rational

I'm trying to generalize the question Is there such an $x$ that both $2^{\frac{x}{3}}$ and $3^{\frac{x}{2}}$ are simultaneously rational?. Do there exist $a,b \in \mathbb{Q}^+ \setminus \{1\}$ and $x ...
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2answers
36 views

Examples of Irrational Algebraic Functions

In my textbook, it says that an irrational algebraic function is a function in which the independent variables appear under a radical sign or in a power with a rational number for its exponent. I ...
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0answers
28 views

Quasi-periodic sequence

Let $f(\theta)$ be some $2\pi$-periodic function which takes the values $f(\theta) \in \{1,-1\}$. Further let $Q$ be some number which is rationally independent of $2\pi$ (More specifically take $Q/(2\...
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1answer
49 views

How many digits do I need to determine if the product of a whole number an irrational number is odd or even? [closed]

So, say you have a really huge whole number like $5^{2000}$ and an irrational number like $\sqrt(5)$. If you were two multiply the two would you get an even or odd number after rounding to the ...
0
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0answers
42 views

Is it possible to manipulate Niven's proof to prove the Irrationality of $\sqrt{2}$.

I am looking for the generalization (if any) of Niven's proof (https://kconrad.math.uconn.edu/blurbs/analysis/irrational.pdf) of irrationality of $\pi$. Is it possible to manipulate his proof to prove ...
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1answer
21 views

Need Help with Complex Equation and finding all Zs

I'm preparing for an exam and I'm facing some troubles with complex numbers, so any help would be much much appreciated! I've been re-reading the chapter like 10 times by now and I just can't figure ...
0
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0answers
33 views

What does $\omega/\mathbb{Q}$ mean?

I have been reading a paper titled Applied Koopmanism several times but cannot find what does this mean (given on page 21): If $\lambda = e^{i2\pi \omega}$ is such that $\omega/\mathbb{Q}$ I am ...
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1answer
29 views

On the cardinality of rationals vs irrationals

I understand that there are infinitely many more irrational numbers than there are rationals - there are many ways in which one can intuitively understand this. However, consider the following: LEMMA:...
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2answers
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Is $ (3+\sqrt{2})^{2/3} $ an irrational number?

I am supposed to find out whether $ (3+\sqrt{2})^{2/3} $ is an irrational number and prove it, but I have no idea how to begin. Thanks
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2answers
81 views

Proof Verification - Elementary proof that $\sqrt3$ is irrational

Sorry for the dumb question; something about this proof seems off and I was wondering what (if anything) is wrong with it. Assume $p$, $q$ are integers. We prove by contradiction. $\sqrt3 = p/q$ $(...
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1answer
32 views

Finding the rationalizing factor of rational numbers with denominator 1

I have a question which I could not solve after hours of research. It goes like this: Find the rationalizing factor of $$\sqrt[3]{16} - \sqrt[3]{4} + 1$$ I can rationalize the denominator but can’...
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3answers
23 views

Some numbers represented by symbols

I am trying to find some numbers that are represented by symbols, such as π, e, i, φ. I couldn't find more. Can you guys help me? (English is not my main language and it is for school project.)
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1answer
52 views

$\sum_{i=1}^n {(a_i\sqrt{b_i})} \ne 0$

In a surd $a\sqrt{b}$   ($b \in \mathbb{Z^+}$)   the value of $b$ can assumed to be a square-free integer ($b = p_1p_2\dots p_k$, where $p_i$ are distinct primes), since otherwise a ...
7
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3answers
148 views

What mathematical consequences might there be if Euler Mascheroni constant is rational?

So far as I know, no one has proved the irrationality of Euler Mascheroni constant. There are discussions about the difficulty of proving the irrationality of this constant. Since we cannot prove ...
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1answer
42 views

Expressing A Strange Irrational Number As An Infinite Sum

I was just thinking about math when a very simple thought occurred to me. Is there any way to express $0.39278124372921876561...$ as an infinite sum. If you could not see the pattern in the number, ...
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1answer
39 views

Well-Ordering Irrationality

Let $D$ be a positive integer and the let the square root of $D$ be a real number. Assuming that the square root of D is not an integer (i.e. $D$ is not a perfect square), use Well-Ordering to prove ...
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2answers
42 views

Sum of two irrational numbers being rational or irrational

I am currently doing a project on irrational and transcendental numbers and part of this project requires me to look at sums and products of irrational numbers. I am aware that the sum of 2 ...
13
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8answers
5k views

Why is Euler's number $2.71828$ and not anything else? [closed]

Why is Euler's number $\mathtt 2.71828$ and not for example $\mathtt 3.7589$? I know that $e$ is the base of natural logarithms. I know about areas on hyperbola ...
0
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1answer
51 views

Abstract Algebra Square Roots Are Irrational

For part (a), I begin by trying to prove $S$ is empty implies the square root of $D$ is irrational. If we take the contrapositive of this implication, this is equivalent to proving that if the square ...
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3answers
1k views

Proving irrationality of $\sqrt[3]{3}+\sqrt[3]{9}$ [duplicate]

I need to prove $$\sqrt[3]{3}+\sqrt[3]{9}$$ is irrational, I assumed $$\sqrt[3]{3}+\sqrt[3]{9} = \frac{m}{n}$$ I cubed both sides and got $$\sqrt[3]{3}+\sqrt[3]{9} = \frac{m^3-12n^2}{9n^3}$$ I tried ...
2
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5answers
197 views

Proving $\sqrt3 + \sqrt[3]{2}$ to be irrational

In a test I tried to solve recently I came across the following question: Prove $$\sqrt3 + \sqrt[3]{2}$$ is irrational I tried proving it by saying it is equal to some rational number $$\sqrt3 + \...
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0answers
64 views

Sums and Products of Algebraic and Transcendental numbers

I am doing a project about irrational and transcendental numbers and part of this project involves looking at sums and products of various combinations of rational, irrational, algebraic and ...
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0answers
31 views

Irrational packing of Euclidean spaces (with no gaps).

This appears to be a new question on MSE. The only post on here after a search using the string irrational "packing" does not mention (explicitly) what I have in ...
0
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1answer
19 views

simple question : sequential criterion for continuity

Let $A$ be a subset of $\mathbb{R}$, and let $f:A\to\mathbb{R}$ be a function. Now, let $\{r_{n}\}$ be any rational sequence in $A$, and let $\{s_{n}\}$ be any irrational sequence in $A$. Suppose ...
3
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1answer
87 views

Hint Needed: Proving $\sqrt{2}$ is irrational using induction

I came across this question in the book Challenge and Thrill of Pre-College Mathematics: Prove that $\sqrt{2}$ is irrational using induction. Apart from the fact that this is hardly the usual ...
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1answer
54 views

Why do we consider $\pi$ as a irrational number?

Why do we consider $\pi$ as a irrational number? Why is that? We all know that $\pi$ is the solution of circumference / diameter of a circle and there could be infinite amount of circles which can ...
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1answer
32 views

How can I prove that no limit exists for this function over this particular interval?

I was given the following function: $$ f(x) = \begin{cases} \frac{1}{q} &\text{if }x = \frac{p}{q} \text{ is rational, reduced to lowest terms} \\ 0 &\text{if }x \text{ is irrational}\end{...
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1answer
49 views

Proof that the nth root of a rational s/t is irrational unless s and t are perfect nth powers with gcd(s,t)=1

A few months ago I asked for help with the above question (linked here Proof that the nth root of a rational s/t is irrational unless s and t are perfect nth powers) and I need a bit more help. My ...
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3answers
62 views

Can the product of two rational numbers be an irrational number? (Kindly see the example in description)

I checked in many sources and I saw "Multiplication is closed under Rational Numbers Q". But consider $$ a = \frac{1}{7} ; \;\;\; b = \frac{22}{1} ;$$ both a, b are individually rational (either ...
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1answer
47 views

Prove $a_n = \lfloor ni \rfloor$ for some irrational $i$ has no pattern

I have a sequence $a_n = \lfloor ni \rfloor$ for an irrational $i$ and I want to prove that there is no 'pattern' to these terms. In particular, I have $1 < i <2$ and I want to prove that $\...
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2answers
142 views

Prove or disprove that there is a rational number $x$ and an irrational number $y$ such that $x^y$ is irrational.

In solving the following problem: Prove or disprove that there is a rational number $x$ and an irrational number $y$ such that $x^y$ is irrational. I let $x=2$ and $y = \sqrt 2$, so that $x^y = 2^\...
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0answers
30 views

Can nth root of 2 be a rational number for any natural number n > 1?

We know square and cube root of 2 are irrational numbers, but is it possible for any a rational number to be multiplied by itself finite times and it results in 2.
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1answer
29 views

How can an convergent series of rational numbers result in a irrational number?

In mathematics, nearly all significant irrational numbers can be expressed as a sum of an infinite convergent series, but according to law of addition of rational numbers, adding any to rational ...
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1answer
58 views

Is this sufficient to prove $\sqrt{n}$ is irrational if $n$ is not a perfect square

If $n$ is a natural number then $n$ is a unique product of primes to integer powers If $n$ is a perfect square then its prime factors will all be to even powers hence when taking the square root the ...
2
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3answers
69 views

Proving $\sqrt{2013^{2016}+2014^{2016}}$ is irrational

Prove $\sqrt{2013^{2016}+2014^{2016}}$ is irrational. I've looked at the proofs for proving $\sqrt{\mathstrut 2}$, $\sqrt{\mathstrut 3}$, and$\sqrt{\mathstrut 15}$ irrational. Using proof by ...
2
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3answers
77 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. ...
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1answer
34 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 ...