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
votes
3answers
457 views

Sum of all real number for any interval.

We know that sum of natural numbers over any interval always exists. For example sum from 0 to 10 of all natural numbers is $$S=\sum_{n=0}^{10}{n}=\frac{0+10}{2}\times{10}=55$$ But what about real ...
3
votes
2answers
60 views

Does $E^2 \; ( E \approx 1.2640847\ldots)$ equal $D \approx 1.5979102\ldots$?

Does $E^2=D$? Where $E$ is a constant used in the closed form of the Sylvester Sequence (see: Closed form formula and asymptotics) and $D$ is a constant for the closed formula of the sequence A007018 ...
4
votes
2answers
37 views

probability that a number defined by a random process is irrational

What if we write $0$. and then throw a coin and depending on the result continue the number with 1 or $0$ and continue this process indefinitely. It is clear that the result of this procedure is a ...
0
votes
1answer
37 views

recipe for infinitely many irrational numbers - or is it?

What if we write 0. and then throw a coin and depending on the result continue the number with 1 or 0 and continue this process indefinitely. It seems like a recipe for producing irrational numbers. ...
9
votes
2answers
151 views

$\arctan$ of a square root as a rational multiple of $\pi$

I know that if $x$ is a rational multiple of $\pi$, then $\tan(x)$ is algebraic. Is there a fairly simple way to express $x$ as $\pi\frac{m}{n}$, if $\tan(x)$ is given as a square root of a rational? ...
1
vote
0answers
37 views

Proving irrationality i p,s and k are primes number?

Can you prove that $\frac{k^{\frac13}-p^{\frac13}}{s^{\frac13}-p^{\frac23}}$ is irrational if p, k and s are different prime numbers. I am certain it is but i dont know how to prove it.
3
votes
2answers
72 views

Existence of five real numbers satisfying a given condition.

Let $a_1,\dots,a_5$ be five distinct non-zero real numbers. Suppose that for $i\neq j$ either $a_i+a_j$ or $a_ia_j$ or both are rational numbers, does it implies that $a_i^2$ are rational numbers for ...
2
votes
2answers
76 views

Does $\pi \ | \ 2 \pi$

Does $\pi$ divide $2 \pi?$ Clearly $\frac{2 \pi}{\pi}=2$ and 2 is an integer, so it would seem to make sense to say that $\pi \ | \ 2 \pi$. Does it make sense to write, for example, $$\pi \ | \ x ...
0
votes
1answer
118 views

Logic: Prove Log(base 9) 15 is irrational

Im having trouble with the following proof... Ill post what I have completed so far.. Prove $\log_915$ is irrational. Ill attempt by contradiction assuming $\log_915$ is rational. So, $\log_915 = ...
0
votes
1answer
85 views

The shape of a graph of a function with $n$th-roots?

Not just these type of functions: $$\sqrt[3]{x}=x^{1/3} \;\;\;\text{and} \;\;\; \sqrt[8]{x}=x^{1/8}$$ But also more complicated expressions, like expressions that have $n$th roots inside of ...
2
votes
1answer
182 views

Can you get any irrational number using square roots?

Given an irrational number, is it possible to represent it using only rational numbers and square roots(or any root, if that makes a difference)? That is, can you define the irrational numbers in ...
3
votes
1answer
38 views

Is the fraction of the irrational exponentiations of two coprime integers by a rational an irrational?

Consider two strictly positive integer coprimes $n, m\in\mathbb{N^*}$ and a rational $r=\frac{p}{q}\in\mathbb{Q}$. Consider furthermore that the three number statifies the following condition: ...
2
votes
2answers
92 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 ...
0
votes
0answers
468 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
210 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
117 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
67 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?
-3
votes
1answer
135 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
126 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
193 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 ...
1
vote
7answers
288 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 ...
1
vote
1answer
104 views

When to rationalize numerator and/or denominator?

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 ...
2
votes
0answers
74 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
58 views

A series of reciprocal of integers [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, ...
26
votes
9answers
3k 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 + ...
1
vote
4answers
610 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
991 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
42 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
153 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 ...
7
votes
5answers
224 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.
5
votes
6answers
1k 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
988 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
64 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 ...
3
votes
3answers
78 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
95 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 ...
0
votes
2answers
722 views

Can we ever get an irrational number by dividing two rational numbers?

If we try to divide any two random arbitrarily long rational numbers like 103850.2387209375029375092730958297836958623986868349693868398659825528365... and ...
1
vote
0answers
44 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
41 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
461 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
676 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
757 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 ...
-2
votes
1answer
238 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 ...
4
votes
2answers
144 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 ...
4
votes
2answers
260 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
180 views

Is this a rational or irrational number?

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
185 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 ...
4
votes
3answers
703 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
201 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
99 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
2k 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 ...