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.

learn more… | top users | synonyms

2
votes
2answers
101 views

Can the exponentiation of an integer by a rational be a non-integer rational?

Consider a strictly positive integer $n\in\mathbb{N^*}$ and a rational $r=\frac{p}{q}\in\mathbb{Q}$. My question is the following: what is the nature of $n^r$? My first guess is that $n^r$ is an ...
1
vote
1answer
173 views

Subsets of real line without accumulation points; also, accumulation points of irrationals

I had to answer three questions about accumulation points. I think my work is correct, but I'd appreciate if someone would look over them for me. (I'm not sure if I read question 2 correctly.) In my ...
0
votes
0answers
2k views

The Conjugate Roots Theorem for Irrational Roots

The Conjugate Roots Theorem for Irrational Roots states that for a polynomial $f(x)$ with integer coefficients, if a root of the equation $f(x) = 0$ is expressed as $a+\alpha$, where $a\in\mathbb{Q}$ ...
1
vote
0answers
483 views

Third degree polynomial with integer coefficient and three irrational roots

There are some polynomial with the above characteristic, and real roots of such polynomials cannot be found using rational number theorem and irrational conjugate theorem. The example of such function ...
-1
votes
1answer
171 views

ZFC and irrational numbers [duplicate]

I understand how integers and rationals are expressed/derived in ZFC. But what about the irrational numbers? Can they also be expressed? If not, are there other axiomatic set theories able to express ...
0
votes
3answers
101 views

The irrationality of the square root of 2 [duplicate]

Is there a proof to the irrationality of the square root of 2 besides using the argument that a rational number is expressed to be p/q?
-2
votes
1answer
360 views

how $\pi$ is irrational if it is a ratio [duplicate]

How can $\pi$ be an irrational number if it is a ratio of the circumference over the diameter? Thanks!
0
votes
3answers
252 views

Can there exist a function with discontinuity at Cantor's Set union Z?

I know there can't exist function with discontinuities only at irrational points,since cantor set is also uncountable like irrational numbers,I thought that the answer is no. Also if yes can you give ...
0
votes
3answers
307 views

Proving that $\sqrt{3} +\sqrt{7}$ is rational/irrational [duplicate]

I took $\sqrt{3}+\sqrt{7}$ and squared it. This resulted in a new value of $10+2\sqrt{21}$. Now, we can say that $10$ is rational because we can divide it with $1$ and as for $2\sqrt{21}$, we divide ...
2
votes
7answers
1k views

Why is a repeating decimal a rational number?

$$\frac{1}{3}=.33\bar{3}$$ is a rational number, but the $3$ keeps on repeating indefinitely (infinitely?). How is this a ratio if it shows this continuous pattern instead of being a finite ...
3
votes
1answer
396 views

When to rationalize numerator and/or denominator? [duplicate]

Sometimes, we have to rationalize either the numerator or the denominator, and sometimes we can still work the problem without rationalizing. So, in some cases, rationalizing can be done, although it ...
3
votes
0answers
218 views

Minkowski's question mark function iterations

The Minkowski's question mark function (we use the sign $?$ to note this function) was designed in 1904 by Minkowski. It can be defined as an increasing bijection between $\mathbb Q$ and the set of ...
2
votes
1answer
76 views

If $(F_n)$ is increasing and $\lim_{n\to\infty}\frac{F_1\dotsb F_n}{F_{n+1}}=0$ then $\sum\limits_{n=1}^\infty\frac1{F_n}$ is irrational [closed]

Let $F_n$ be integers, and $F_1<F_2<\cdots<F_n<\cdots$. Suppose that $$\lim_{n\to\infty}\frac{F_1F_2\cdots F_{n-1}}{F_n}=0.$$ Prove then $$\sum_{n=1}^\infty \frac{1}{F_n}$$ is convergence, ...
30
votes
10answers
4k views

Rational + irrational = always irrational?

I had a little back and forth with my logic professor earlier today about proving a number is irrational. I proposed that 1 + an irrational number is always irrational, thus if I could prove that 1 + ...
4
votes
4answers
1k views

The n-th root of a prime number is irrational

If $p$ is a prime number, how can I prove by contradiction that this equation $x^{n}=p$ doesn't admit solutions in $\mathbb {Q}$ where $n\ge2$
4
votes
6answers
2k views

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

How do you prove that $\sqrt{2} + \sqrt{5}$ is irrational? I tried to prove it by contradiction and got this equation: $a^2/b^2 = \sqrt{40}$.
0
votes
2answers
51 views

algebraically determining if a number is irrational or not

Is it possible to use an algebraic formula, equation, concept, or principle to determine with perfect accuracy (or high precision, if not perfect) whether or not a number is rational? An example ...
4
votes
1answer
216 views

Proof that ${\pi}$ can(not) be expressed as a root or as a root in combination with a fraction

I was doing some math for a programming project of myself and ran into decimal numbers and how to define them without losing precision while calculating an expression, so I tried writing them down as ...
8
votes
5answers
261 views

Different ways to prove $\sqrt p$ irrational for $p$ prime.

I know this fact can be proved by contradiction(reductio ad absurdum) but please give proofs by different methods.
6
votes
6answers
6k 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 ...
2
votes
9answers
1k views

Why does $(3\sqrt3)^2 = 27$?

How does $(3\sqrt3)^2 = 27$? I've tried to solve this using binomial expansion and using the FOIL method from which I obtain $9 + 3\sqrt3 +3\sqrt3 + 3$. it has been a while since I've done this kind ...
1
vote
1answer
75 views

Question on a subset $S$ of $[0,1]\times[0,1]$ where for each $(x,y)\in S$ at least one of $x$ and $y$ is irrational

If $S$ is a subset of $[0,1]\times[0,1]$ such that one point of the ordered pair is rational and the other is irrational or both are irrationals. Then which of the following is true? a) $S$ is closed ...
4
votes
3answers
109 views

Number of irrational roots of the equation $(x-1)(x-2)(3x-2)(3x+1)=21$?

The number of irrational roots of the equation $(x-1)(x-2)(3x-2)(3x+1)=21$ is (A)0 (B)2 (C)3 (d)4 Actually im a 10 class student i don't know any of it,but my elder brother(IIT Coaching) cannot ...
4
votes
1answer
107 views

$f_1, f_2 : \mathbb{R} \rightarrow \mathbb{R}$ nonconstant, continuous, with period $1, \sqrt{2}$, respectively, then $f_1 + f_2$ is not periodic

I've been working on this problem for several hours, but I keep getting stuck. Suppose $f_1, f_2 : \mathbb{R} \rightarrow \mathbb{R}$ periodic with period $1, \sqrt{2}$, respectively, and that each of ...
1
vote
1answer
58 views

Prove $k!(e-s_k)$ is irrational.

Given that $\frac{p}{q} = e = 1 + \frac{1}{1!} + \frac{1}{2!} + ... + \frac{1}{k!} + \frac{e^{z}}{(k+1)!}$ for some $z$ in $[0,1]$ (using Taylor's theorem), and that $s_k = 1 + \frac{1}{1!} + ...
2
votes
3answers
96 views

If $a$ and $b$ are odd integers, then $\sqrt{a^2+b^2}$ is irrational

If $a,b\in\mathbb{N}$ are odd then demonstrate: $$ {\sqrt{a^2 + b^2}} \not\in \mathbb{Q}$$ I try to guess that $$ {\sqrt{a^2 + b^2}} \in\mathbb{Q}.$$ Then i write $$ {\sqrt{a^2 + b^2}= m/n}.$$ ...
2
votes
1answer
70 views

If $a$ and $b$ are rational, then $a + b{\sqrt{2}} \ne {\sqrt{3}} $

If $a,b\in\mathbb{Q}$ then demonstrate:$$a + b{\sqrt{2}} \ne {\sqrt{3}} $$ I raised and squared the equation but it didn't work.
2
votes
0answers
54 views

Apery's constant

I read that it is unknown if $\zeta (3)$ is algebraic but it is known to be irrational. Has anyone proved anything of the form $\zeta (3)$ is not a root of a polynomial of degree $12345$ with integer ...
2
votes
0answers
44 views

Experimental calculation and $\mathbb{Q}$

I have been reading this article and have a question about the first line of the second paragraph on the first page. It says: The basis for this suggestion is the simple fact that all experimental ...
2
votes
3answers
640 views

Set of irrationals between two reals is uncountable

I know that between any two reals, there is an irrational number. See: Proving that there exists an irrational number in between any given real numbers Now let a, b $\in$ $R$ such that a < b. And ...
1
vote
2answers
865 views

Check if this proof about real numbers with an irrational product is correct.

Can anyone confirm if my proof is correct, please? Claim:- “If $x$ and $y$ are real numbers and their product is irrational, then either $x$ or $y$ must be irrational.” Proof:- Assume that both ...
1
vote
3answers
879 views

How to write this in mathematical notation?

I have the following claim: “If $x$ and $y$ are real numbers and their product is irrational, then either $x$ or $y$ must be irrational.” I'm supposed to write this in mathematical notation. It's ...
-1
votes
1answer
303 views

Proof of $\pi$+$e$ irrational

The wikipedia tells that it is not known that $\pi+e$ is irrational? Immediately after reading this my mind came with this proof- Let $x =\sqrt{\pi^2}+\sqrt{e^2}$ be rational, then $ \quad ...
5
votes
2answers
247 views

Fraction raised to integer power

if I have $(p/q)^n$ where $p,q,n$ are integers and $p/q$ is a... I don't know what you call it. Not a whole number, but something like 15/7 where you can't reduce it any more and it's non-integer. Can ...
2
votes
3answers
5k views

Proving/Disproving Product of two irrational number is irrational

I saw this question where I had to prove/disprove that: Ques. Product of two irrational number is irrational. I tried 'Proof by Contraposition'. Product of two irrational number is irrational. p ...
4
votes
2answers
469 views

If $x$ and $y$ are rational then is $x^y$ also rational?

I can think of the counter example $x = 2$ and $y = 1/2$ but how would a proof to disprove this look like?
-1
votes
1answer
285 views

Is this a rational or irrational number? [closed]

It is given that $$z=\sqrt\frac{\sqrt{3x+1}}{\sqrt{3x-1}}$$ How does one find whether $z$ is a rational or irrational number?
1
vote
1answer
747 views

a square root of an irrational number

I wonder if a square root of an irrational number is always irrational? I would tend to think that yes, but I can´t think of any justification. Also there are cases which are rather hard to decide ...
5
votes
3answers
3k views

sum of irrational numbers - are there nontrivial examples?

I know that the sum of irrational numbers does not have to be irrational. For example $\sqrt2+\left(-\sqrt2\right)$ is equal to $0$. But what I am wondering is there any example where the sum of two ...
0
votes
1answer
325 views

Spiral of Theodorus - Discussion

The fact that $\sqrt2$ is not rational goes back to Theodorus of Cyrene from the school of Pythagoras, and is discussed in Plato's dialog "Theaetetus". Of course, $\sqrt n$ is not rational for any ...
1
vote
1answer
299 views

integral factors of an irrational number

If the radicand of a square root is a non-square (making the root an irrational), and if the non-square is either a prime number, or a composite number that does not have a square divisor (other than ...
20
votes
6answers
3k views

Prove that $\sqrt 2 +\sqrt 3$ is irrational. [duplicate]

Please prove that $\sqrt 2 + \sqrt 3$ is irrational. One of the proofs I've seen goes: If $\sqrt 2 +\sqrt 3$ is rational, then consider $(\sqrt 3 +\sqrt 2)(\sqrt 3 -\sqrt 2)=1$, which implies ...
2
votes
3answers
173 views

Do irrational numbers have equivalence classes the way rational numbers do?

Rational numbers are defined as equivalence classes of ordered pairs (less formally, "fractions") of integers, where $m_1n_2=m_2n_1$. This equivalence relation justifies the common practice of ...
5
votes
1answer
129 views

What is the sum of $4\sqrt{28}$ and $3\sqrt{7}$ ?

As far as I can simplify it - $$4\sqrt{7*4} + 3\sqrt{7} = 8\sqrt{7} + 3 \sqrt{7} = \sqrt{7} * 11$$ However , The options for the correct answer are - A) $ 8/3$ B) $ 16/3$ C) $ 18/3$ D) $24/3$ I ...
0
votes
1answer
215 views

Simplify : $\frac{\sqrt{6}}{\sqrt{2} + \sqrt{3}} + \frac{3\sqrt{2}}{\sqrt{6 + \sqrt{3}}} - \frac{4\sqrt{3}}{\sqrt{6 + \sqrt{2}}}$

My exams are approaching fast and this question was in one of the sample papers . I have to simplify $$\frac{\sqrt{6}}{\sqrt{2} + \sqrt{3}} + \frac{3\sqrt{2}}{\sqrt{6 + \sqrt{3}}} - ...
2
votes
2answers
436 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 : ...
5
votes
4answers
235 views

Can an irrational always be found by multiplying irrationals?

I was thinking about the function $\ f(a,b) = a/b $ where $a$ and $b$ where both irrational. It quickly stood out to me that the codomain of that function would include every rational number. But, ...
3
votes
2answers
855 views

How to evaluate $\sqrt{5+2\sqrt{6}}$ + $\sqrt{8-2\sqrt{15}}$ ?

My exams are approaching fast and I found this question in one of the unsolved sample papers. I tried squaring the whole term but couldn't work out the answer . I am a ninth grader so please try to ...
3
votes
2answers
805 views

Pi might contain all finite sets, can it also contain infinite sets?

In a previous, and quite popular, question it was discussed about whether or not $\pi$ contains all finite number combinations. Let us assume for a moment that $\pi$ does in fact contain all finite ...
12
votes
1answer
259 views

Multiplying by an irrational number in combinatorial problems

Everybody knows that the number of derangements of a set of size $n$ is the nearest integer to $n!/e$. It is also widely known that the $(n+1)$th Fibonacci number $F_{n+1}$ is the nearest integer to ...