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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1
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4answers
387 views

Suppose that x and y are irrational, but x + y is rational. Prove that x - y is irrational. [on hold]

I can understand how it works in my head, I don't know how to prove it though.
0
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0answers
52 views

Approximation of irrationals by rationals!

Let $\alpha$ be an irrational number and $\beta$ be an arbitrary real number, Prove that there are infinitely many pair of integers $(x,y)$ with $x\in\mathbb{N}$ such that: ...
0
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1answer
27 views

constructing continuous function with range of rational numbers

Can we construct funtion $f$ non-constant and continuous on $\mathbb{R}$, $f$:$\mathbb{R}\rightarrow\mathbb{Q}$. Is that right "By intermediate value theorem, there exist irrational number between ...
6
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0answers
66 views

Does π start with two identical decimal sequences?

Let d$(x,y)$ be the sequence of decimals of π from the x:th one to the y:th one. My question: is there a number $n$ such that d$(1,n)$ = d$(n+1, 2n)$? I.e., does π start with two identical decimal ...
1
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0answers
21 views

Proof Verification of Sum of Irrational Numbers

The chosen answer here claims that for rational $u$ and $v$, $u^{\frac{1}n} + v^{\frac{1}n}$ is rational iff $u^{\frac{1}n}$ and $v^{\frac{1}n}$ are both rational. However, the link to a proof seems ...
3
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2answers
142 views

How can there exist a Liouville number?

I just found out about them today so forgive me if I'm just missing something obvious. As far as I'm aware, the following is true $(\forall a\in\mathbb R\geq0)(((\forall\epsilon\in\mathbb ...
3
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1answer
51 views

An infinite sum based on the mod-parity of Euler's totient function

Let $\bmod( m,k )$ be the remainder when $m$ is divided by $k$: $0,1,\ldots,m{-}1$. Let $\phi(n)$ be Euler's totient function: the number of relatively prime numbers smaller than $n$. So for ...
19
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6answers
1k views

How to prove that $\sqrt[3] 2 + \sqrt[3] 4$ is irrational?

So while doing all sorts of proving and disproving statements regarding irrational numbers, I ran into this one and it quite stumped me: Prove that $\sqrt[3]{2} + \sqrt[3]{4}$ is irrational. I tried ...
7
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3answers
1k views

How do I find the value of this weird expression?

How can I find the value of the expression $\sqrt{2}^{\sqrt{2}^{\sqrt{2}^...}} $? I wrote a computer program to calculate the value, and the result comes out to be 2 (more precisely 1.999997). Can ...
4
votes
2answers
79 views

Find conditions on positive integers so that $\sqrt{a}+\sqrt{b}+\sqrt{c}$ is irrational

Find conditions on positive integers $a, b, c$ so that $\sqrt{a}+\sqrt{b}+\sqrt{c}$ is irrational. My solution: if $ab$ is not the square of an integer, then the expression is irrational. I find it ...
0
votes
2answers
101 views

Prove $3 - 2 ^ {1/7}$ is Irrational

How to prove that $3 - 2 ^ {1/7}$ is irrational? If I do $$\frac p q = 3 - 2 ^ {1/7}$$ $$2 ^ {1/7} = 3 - \frac p q $$ Hint needed Should I multiply by $7$ times??
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3answers
76 views

Proof of $6 - \sqrt{2}$ is Irrational by Contradiction

What is a Proof by Contradiction, and how to prove by contradiction that $6 - \sqrt{2}$ is an irrational number?
2
votes
6answers
90 views

What type of number is this $\frac{\sqrt2}{2}$?

$$\frac{\sqrt{2}}{2}$$ In this polynomial, an irrational number is divided by a rational number. However this is not a general case but can any one tell me that when we divide an irrational number or ...
0
votes
1answer
30 views

Prove that there is an irrational number a such that $2\lt a \lt 3$. Prove also that there is an irrational number $b$ such that $2 \lt b^2 \lt 3$

I tried to prove a is irrational by subtracting two all three sides so I get 0 is less than a-2 less than 1. From there I proved that a-2 is irrational by saying a-2 is equal to rational which ...
-3
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2answers
38 views

How do we know a number isn't recurring? [closed]

How can we say for certain that a number never repeats it's digits such that it might be recurring?
1
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2answers
92 views

Solution of the equation $x^x=2$

Let $x$ be the solution of the equation $x^x=2$. Is $x$ irrational? How to prove this?
7
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5answers
171 views

Prove that $\sqrt{3}+ \sqrt{5}+ \sqrt{7}$ is irrational

How can i prove that $\sqrt{3}+ \sqrt{5}+ \sqrt{7}$ is irrational? I know that $\sqrt{3}, \sqrt{5}$ and $\sqrt{7}$ are all irrational and that $\sqrt{3}+\sqrt{5}$, $\sqrt{3}+\sqrt{7}$, ...
0
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0answers
46 views

Irrational numbers to irrational powers being rational?

So some of you may be familiar with the proof that some irrational numbers to irrational powers are rational, that is: if $A = \sqrt2^\sqrt{2}$ then it follows that $A^\sqrt{2} = 2$. So, I've found a ...
-1
votes
1answer
76 views

Irrational diagonal length problem.

Premise 1: All straight lines have the value of length equal to the numerical value of the end point, provided the starting point of the line is assigned the numerical value zero. Premise 2: No ...
0
votes
3answers
57 views

Prove $\sqrt[5]{672}$ is irrational

How would you prove $\sqrt[5]{672}$ is irrational? I was trying proof by contradiction starting by saying: Suppose $\sqrt[5]{672}$ is rational ...
-2
votes
1answer
61 views

Do irrational numbers exist? [closed]

Assuming the question is asked in the real number system: If an irrational number, such as Pi is infinitely long, then do parts of this number have to repeat? and if so, does it then become a ...
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votes
3answers
77 views

How to simply this fraction with irrational denominators? [closed]

How to simplify? $\frac{1}{1+\sqrt{3}} + \frac{1}{\sqrt{3}+\sqrt{5}} + \frac{1}{\sqrt{5}+\sqrt{7}} \frac{1}{\sqrt{7}+3}$
2
votes
0answers
30 views

Prove that $E_0$ is transcendental

Consider the non-negative natural numbers: $0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19\dots$ Encode the primes as $1$, the rest as $0$. $E = 0,0,1,1,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,1\dots$ ...
2
votes
2answers
44 views

Seeing the plane as a four (or more) dimensional vector space on $\mathbb Q$

As I was trying to answer a question about the enumeration of circuits one can build with a set of miniature train track elements, I realized that all plane positions that could be reached had ...
4
votes
4answers
173 views

How to compute a lot of digits of $\sqrt{2}$ manually and quickly?

After having read the answers to calculating $\pi$ manually, I realised that the two fast methods (Ramanujan and Gauss–Legendre) used $\sqrt{2}$. So, I wondered how to calculate $\sqrt{2}$ manually in ...
1
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2answers
32 views

Proof that 2^0.5 will not touch the 1.5 mark on the number line when we try to mark it exactly??

I know this would be kind of silly, but then this has been troubling me for the past few days. All of us know 2^(1/2) is irrational. Let us try to mark this on the number line "exactly", as in trying ...
-1
votes
3answers
155 views

What is the 3rd most famous non-algebraic real constant? [closed]

If we think of the non-integer, non-algebraic numbers (or rather, not known to be algebraic numbers) used in mathematics, certainly the most famous/useful is $\pi$—appropriate to mention on the ...
3
votes
0answers
40 views

When does the following construction generate a transcendental number?

Given $n\in[0,1]$ with base-b expansion $0.n_1n_2n_3\dots$, define $\Delta_b(n)$ to be the number with the following base-b expansion: $\huge{ 0.\underbrace{n_1}_{1^{st}\text{ ...
2
votes
0answers
63 views

What is the name of this irrational math constant and is there a compact way to write it? 0.10110111011110…

I think this number is a transcendental number and I've tried looking online to see who first made it, I'm not sure if it's a Liouville Number or if there is a more common or better name for it. Does ...
2
votes
2answers
47 views

Determine whether a fraction will produce a rational number with infinite digits after the decimal

This may be a naive question but I would like to know whether we can determine if a fraction (say $1/3$) will produce a rational number with an infinite number of digits after the decimal when ...
2
votes
1answer
38 views

Algorithm for eliminating irrationality in denominator

Good day. Suppose $a$ is rational number, $p$ is positive integer and $a^{1/p}$ is irrational. If we want to eliminate irrationality in the denominator of the fraction $\frac{1}{a^{1/p}}$, then there ...
1
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0answers
67 views

If (a+b)/2 is rational can we say that a,b are rational? Prove

The question is if it's given that $$ {a+b\over 2} \in \Bbb Q $$ prove or disprove $a,b \in \Bbb Q$. Since it is to disprove, i tried the following method by using examples Take $$a = 1 + \sqrt{2} ...
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0answers
34 views

Conceptual question [duplicate]

If pi is a ratio of circumference to diameter than why it is an irrational number?
4
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5answers
180 views

why is PI considered irrational if it can be expressed as ratio of circumference to diameter? [duplicate]

Pi = C / D (circumference / diameter) . I have read that if circumference can be expressed as an integer then diameter cannot and vice-versa, so that the ratio can never be expressed as a/b where both ...
2
votes
3answers
35 views

Prove the following based on number theory

$\log_5(2) \in \mathbb{R}\setminus \mathbb{Q}$ (irrational numbers). I know there is a question out there already for this but my problem is that I need to prove this using the fundamental ...
0
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1answer
32 views

Rational Number Density in a Square

It is well known that rational numbers are distributed on the number line everywhere compactly. If we consider a 'square' a parallelogram to be precise, formed by natural numbers p and q, i.e. ...
2
votes
5answers
57 views

Is there a way, in general, to tell whether the nth root of a integer is rational?

Is there a way, in general, to tell whether the $n^{th}$ root of a integer is rational? More explicitly, is it possible to elegantly determine whether the result of $k^{1/n}$ is rational for $k,n \in ...
1
vote
1answer
27 views

Methods for Improving Convergence of a sequence of Partial Sums

I have the following sum: $$\zeta(3)+\frac1{4}=\sum_{k=0}^{\infty}\frac{2k^2+7k+7}{(k+1)^3(k+2)(k+3)}$$ Are there any methods that I can use to speed up the convergence of the sequence generated by ...
8
votes
4answers
484 views

Simplification of an expression involving nested square roots.

I was trying to simplify $\sqrt{14} - \sqrt{16 - 4 \sqrt{7}}$. Numerical evaluation suggested that the answer is $\sqrt{2}$ and it checked out when I substituted $\sqrt{2}$ in the equation $x= ...
1
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5answers
92 views

Is $\frac{\sqrt7}{\sqrt[3]15}$ rational or irrational.

Is $\frac{\sqrt7}{\sqrt[3]15}$ rational or irrational? Prove it. I am having a hard time with this question. So far what I did was say, assume it's rational, then ...
-1
votes
1answer
88 views

Show that $\sqrt{2} + \sqrt{3} +\sqrt{5}$ is an irrational number. [closed]

Show that $\sqrt{2} + \sqrt{3} +\sqrt{5}$ is an irrational number.
42
votes
0answers
865 views

Is $ 0.112123123412345123456\dots $ algebraic or transcendental?

Since the decimal expansion $ 0.112123123412345123456\dots $ is non-terminating and non-repeating, clearly $ 0.112123123412345123456\dots $ is an irrational number. Can it be shown whether it is ...
3
votes
1answer
46 views

Need a help to show $x<1$

Now I am proving the number $$ \sum_{k=1}^{\infty}\frac{9}{10^{\frac{k(k+1)}{2}}}=0.90900900090... $$ is irrational. Here I use a similar method to the proof of e is irrational by Joseph Fourier. My ...
29
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8answers
6k views

How are first digits of $\pi$ found?

Since Pi or $\pi$ is an irrational number, its digits do not repeat. And there is no way to actually find out the digits of $\pi$ ($\frac{22}{7}$ is just a rough estimate but it's not accurate). I am ...
3
votes
2answers
54 views

Roots of $z^r=1,r\notin\mathbb{Q}$

If $a,b\in\mathbb{Z}$, and $\frac a b$ is in lowest terms, then $$z^{\frac a b}=1\\\implies z=\exp\left(\frac{2\pi in b}{a}\right)\forall n\in\mathbb{Z}$$ This means that $z$ has exactly $a$ distinct ...
3
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1answer
56 views

Is there a number $x\neq0$ whose products with $\pi$ and with $e$ are both rational?

Does there exist a number $x\neq0$, such that $[x\cdot\pi\in\mathbb{Q}]\wedge[x\cdot{e}\in\mathbb{Q}]$? I thought this question would be easy to answer, but it turns out otherwise. Obviously ...
6
votes
1answer
101 views

Representation of irrationals as $\sum_{n\ge 2}\frac{x_n}{n!}$

Prove that every $x\in(0,1)\setminus\mathbb{Q}$ has a unique representation as $x = \sum_{n\ge 2}\frac{x_n}{n!}$, where $x_n\in\mathbb{Z}_n = \{0,1,2,\ldots,n-1\}$. Probably this is well known, I'd ...
0
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0answers
9 views

Trascendental numbers permutation

Let $x_n$ be the infinite sequence of decimal digits of a fixed irrational/trascendental number. Can I obtain any other irrational/trascendental number's sequence of decimal digits through a ...
5
votes
3answers
171 views

Proof that the square root of 5 is not a natural number

I am a student and I want to know if this proof is correct. Please help me. Thanks in advance! Proof. If square root of 5 is a natural number, it should be even or odd. If it would be even, we could ...
1
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2answers
52 views

Why must the decimal representation of a rational number in any base always either terminate or repeat?

Wikipedia makes the following statement about rational numbers. The decimal expansion of a rational number always either terminates after a finite number of digits or begins to repeat the same ...