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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irrational, proving or finding counterexapmle [duplicate]

Prove or find a counterexample: For all real numbers x and y it holds that x + y is irrational if, and only if, both x and y are irrational. Can anyone give me a hint just about how to start cause i ...
6
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0answers
75 views

Does the Cantor set contain any irrational algebraic numbers?

I've been trying to characterise the irrationals in the Cantor set $\mathcal{C}$ and this is proving to be surprisingly difficult. In particular I am trying to investigate whether $\mathcal{C}$ ...
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5answers
119 views

Prove or find a counterexample: For all real numbers x and y it holds that x + y is irrational if, and only if, both x and y are irrational.

The following is a Homework Question that I've been working on and I would like some feedback on my answer: Prove or find a counterexample: For all real numbers $x$ and $y$ it holds that $x + y$ is ...
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4answers
2k views

Can every irrational number be written in terms of finitely many rational numbers?

Consider the irrational number $\sqrt{2}$. It can be written in terms, i.e., in a closed form expression, of two rational numbers as $2^{\frac{1}{2}}$. Does it hold in general that every irrational ...
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1answer
66 views

Are there numbers that if proven rational (or irrational) will have important consequences to mathematics?

We see all the time conjectures and proofs that specific (real) numbers are (more often than not) irrational. I'm wondering that apart from the mathematical curiosity motivating such proof attempts, ...
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2answers
136 views

Let a, b, c, d be rational numbers… [closed]

Let $a, b, c, d$ be rational numbers, where $\sqrt{b}$ and $\sqrt{d}$ exist and are irrational. If $a + \sqrt{b} = c + \sqrt{d}$, prove that $a=c$ and $b=d$.
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1answer
132 views

Is $\log 2\pi$ rational?

Is it known whether $\log 2\pi$ is rational (where the base of the logarithm is $e$)? Or algebraic?
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1answer
66 views

Confusing rational numbers

Question: If $$x = \frac{4\sqrt{2}}{\sqrt{2}+1}$$ Then find value of, $$\frac{1}{\sqrt{2}}*(\frac{x+2}{x-2}+\frac{x+2\sqrt{2}}{x - 2\sqrt{2}})$$ My approach: I rationalized the value of $x$ to ...
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1answer
50 views

Logic verification: $x^3$ is irrational, then $x$ is also irrational

Prove, by contraposition, if $x^3$ is irrational, then $x$ is also irrational. Just a verification do I need to show that given $x$ is rational $x^3$ is also rational? Suppose $x \in \mathbb{Q}$ ...
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0answers
132 views

Are there integers $a, b$ s.th. $\pi^a = e^b$?

Is $\log \pi $ a rational number? That is, are there non-zero integers $a, b$ s.th. $\pi^a = e^b$ ?
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1answer
77 views

A general method to efficiently calculate the floor of an element of $\mathbb{Q}[\sqrt{2}]$

I had to decide whether to post this question here on the Mathematics Stack Exchange or on Stack Overflow, but I decided that the question was essentially a mathematical one despite being inspired by ...
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7answers
282 views

Proving that $\sqrt{2}+\sqrt{3}$ is irrational [duplicate]

This is from Spivak. Prove that $\sqrt{2}+\sqrt{3}$ is irrational. So far, I have this: If $\sqrt{2}+\sqrt{3}$ is rational, then it can be written as $\frac{p}{q}$ with integral $p, q$ and in ...
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0answers
53 views

Products of irrational numbers field of mathematics?

Recently a friend posed the question "can the product of two irrational numbers be rational?" We the trivial answers like for example $\sqrt{2}\sqrt{8} = 4$. I have become somewhat obsessed with the ...
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4answers
94 views

why the occurrence of 4,5,6 and 9 in pi differs?

i´m playing around with pi, i have this document with the first 5million decimal numbers after comma. http://www.aip.de/~wasi/PI/Pibel/pibel_5mio.pdf and i build a script that i put in for example ...
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4answers
64 views

Cancelling out square roots gives 2?

Question: If $$N = \frac{\sqrt{\sqrt{5}+2}+\sqrt{\sqrt{5}-2}}{\sqrt{\sqrt{5}+1}}$$Find N (This is a subset of a larger question) My approach: After rationalizing the denominator, by ...
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0answers
40 views

Combination of integers and irrationals numbers

Show that if $j$ is a positive irrational number, then for all $\varepsilon>0$ there exist integers $h$ and $k$ such that $0<h+kj<\varepsilon$. I need this result in a problem, but I don't ...
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0answers
60 views

Irrational numbers and series

Let $$f(x) = \prod_{n = 0}^\infty \left(1 + \frac{x}{2^n}\right)$$ According to an exercise in a packet of problems in elementary number theory, this function and all its derivatives are irrational ...
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3answers
510 views

Proving that e is irrational

Prove that that e is irrational. Recall e $ = \exp(1)$ so assume $\mathrm{e}$ is rational , then $$\sum\limits_{k=0}^\infty \frac{1}{k!} = \frac{a}{b}\quad \text{for some positive integers}\,\,\, ...
5
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1answer
124 views

If $(n_k)$ is strictly increasing and $\lim_{n \to \infty} n_k^{1/2^k} = \infty$ show that $\sum_{k=1}^{\infty} 1/n_k$ is irrational

Prove that for a strictly increasing natural sequence $(n_k) $ satisfying $\lim_{n \to \infty} n_k^{1/2^k}=\infty$, $\sum_{k=1}^{\infty} 1/n_k$ is irrational. This is another problem "problems in ...
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1answer
53 views

Proving that a series converges to an irrational number.

I am taking honors Calculus II and have been doing reasonably well in the course until the current problem set which is due tomorrow. One exercise that is really giving me trouble is this: Prove ...
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2answers
77 views

Proving either $x^2$ or $x^3$ is irrational if $x$ is irrational

I had a test today in discrete mathematics and I am dubious whether or not my proof is correct. Suppose $x$ is an irrational number. Prove that either $x^2$ or $x^3$ is irrational. My Answer: ...
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3answers
396 views

Polynomials with Integer Coefficients and irrational roots

Is there a polynomial with integer coefficients which has √2 +√7  as a root?
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0answers
119 views

Irrational numbers to the power of other irrational numbers: A beautiful proof question

The following theorem has a very beautiful proof. Theorem: There exist two irrational numbers $x$ and $y$ such that $x^y$ is rational. Proof: If $\sqrt{2}^{\sqrt{2}}$ is rational then we ...
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3answers
471 views

Finding set of non recurring non terminating decimals

I need to find a set of two Integers P and Q such that ...
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1answer
79 views

Rational and trascendental numbers: $\pi$, $e$ and $\pi+e$ [duplicate]

The numbers $\pi,e$ are trascendentals, but if consider: $\pi+e$ then is rational, trascendental? Thanks
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2answers
134 views

Prove that if $a$ is irrational then $\sqrt a$ is irrational

Just hints but solution thx. Any hints for me? I simply suppose that $a = \dfrac mn$ then $\sqrt a = \sqrt{\dfrac mn}$ But this does not make sense ..
7
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3answers
455 views

Sequences of Rationals and Irrationals

Let $(x_n)$ be a sequence that converges to the irrational number $x$. Must it be the case that $x_1, x_2, \dots$ are all irrational? Let $(y_n)$ be a sequences that converges to the rational number ...
5
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4answers
147 views

Cardinality of Irrational Numbers

I know and I have proved more than once that the set of irrational numbers ($\mathbb{I}$) is uncountable, but now I'm given to solve this problem: Show that $|\mathbb{I}|=|\mathbb{R}|$, How can I ...
0
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1answer
31 views

Filling up space with irrational fractional parts [duplicate]

While trying to generalise a mechanics exercise with a friend, we came up with this question, in an attempt to understand wether sine curves with irrational period defined inside an annulus will end ...
3
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1answer
74 views

Enough Dedekind cuts to define all irrationals?

Assuming that there are uncountably infinitely many irrationals between any two consecutive rationals, how can the Dedekind cuts (defined on the countably infinite rationals) define all the ...
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1answer
68 views

Difference between density and measure

In terms of definition, I know the difference between the two. However, the set of rationals $\mathbb{Q}$ has measure zero but is dense in $\mathbb{R}$. Whenever I envision this, I see a set of ...
10
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1answer
171 views

Does $\lfloor(4+\sqrt{11})^{n}\rfloor \pmod {100}$ repeat every $20$ cycles of $n$?

I recently came across a post on SO, asking to calculate the least two decimal digits of the integer part of $(4+\sqrt{11})^{n}$, for any integer $n \geq 2$ (see here). The author presented a Java ...
0
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1answer
75 views

How to understand Apostol's proof of the irrationality of $\sqrt{n}$ if $n$ is not a perfect square?

Recently I am reading the textbook of Apostol, Mathematical Analysis, Second Edition. On page 7, there is a theorem 1.10: If $n$ is a positive integer with is not a perfect square, then $\sqrt{n}$ is ...
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3answers
85 views

Prove that there is no rational number solution for an equation.

Prove that there is no rational number solution to the equation $x^2-3x+1=0$. (Note, we do not assume that we know all the solutions of $x^2-3x+1=0$ are given by quadratic formula)
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5answers
293 views

Sum of two periodic functions is periodic?

I have following paragraph taken from the Stanford's study material. Question: Is the sum of two periodic functions periodic? Answer: I guess the answer is no if you are Mathematician, yes ...
2
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1answer
58 views

What Will Happen Without Decimal Expansion?

After a discussion on the complexity of decimal expansion (such as $0.\bar{9}=1$), some of my students (middle school) decided to throw away the decimal expansion of some numbers! Namely, the numbers ...
2
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3answers
35 views

A good site documenting approximations of irrationals

I'm thinking of Sloane here but I believe that only takes sequences/series into account. Basically I've derived an interesting, appealing formula for e and want to know if it's already been ...
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0answers
199 views

Irrationality of $\displaystyle\sum_{p\in\mathbb{P}} \frac{1}{2^{p}}$

Let $\mathbb{P}$ be the set of prime numbers, and consider $m=\displaystyle\sum_{p\in\mathbb{P}} \frac{1}{2^{p}}$. Is $m$ irrational? In the following paper, the author recalls several sufficient ...
5
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3answers
868 views

Is a cube root of a prime number rational?

The question is: if $P$ is prime is $P^{1/3}$ rational? I have been able to prove that if $P$ is prime then the square root of $P$ isn't rational (by contradiction) how would I go about the cube ...
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1answer
60 views

How do you prove $\sqrt{n}$ is an integer or it is irrational? [duplicate]

I have tried this problem five times but I keep getting stuck. I keep following the proof for $\sqrt{2}$. I know that I have to prove that the set is nonempty. Which I do by induction. $2^1 > 1$ ...
8
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2answers
196 views

Why are $e$ and $\pi$ so common as results of seemingly unrelated fields?

I'm sure this gets asked all the time but I swear I googled with no useful result. What I'm looking for is a reasonably intuitive answer. Those two constants have some pretty interesting properties. ...
0
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1answer
51 views

representation of rational field

I want to know how is represented general form of rational field, for example definition of ${\mathbb Q}(\sqrt{2})$ is represented as $p+q \sqrt{2}$, where $p$ and $q$ are rational numbers, for ...
5
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3answers
348 views

Prove that 2.101001000100001… is an irrational number.

My try: This number is non-terminating and non-repeating, so this is an irrational number. But how do I prove it more formally in a more mathematically rigorous way?
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2answers
152 views

Proof of irrationality of $\dfrac{\sqrt{8}}{\sqrt{7}}$

We have to prove that $\dfrac{\sqrt{8}}{\sqrt{7}}$ is irrational(try not to use the Rational Root Theorem) At first,we prove that the expression is not an integer. ...
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5answers
232 views

What is the name of $0.\overline{0}1$

Short question: What is the name of the number closest but not equal to zero? Long question: Some programmers were discussing about the smallest number close to zero, which is ...
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1answer
80 views

Calculation of irrational numbers [closed]

I am looking for the fastest parallelized algorithm to calculate 1 billion decimal places of ANY irrational number, be it e, pi or any other. May main aim is to obtain 1 billion of non repeating ...
6
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0answers
130 views

Does the number $2.3\,5\,7\,11\,13\ldots$ exist and, if so, is it rational or irrational &/or transcendental? [duplicate]

Does there exist a number which contains in its digits all of the prime numbers listed in order: $$2.3\,5\,7\,11\,13\ldots\ldots$$ if so, will it be rational or irrational &/or transcendental?
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4answers
476 views

Understanding non-solvable algebraic numbers

Background We know from Galois theory that the zeros of a polynomial with rational coefficients whose Galois group is solvable can be expressed in a formula that involves rational powers of the ...
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4answers
141 views

Help me to Prove that log2 3 is irrational. [closed]

seemingly simple homework assignment, help? Was never the best with logarithms, how would I go about proving? Sorry the question read IRrational!
3
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1answer
103 views

$f$ differentiable, $f(x)$ rational if $x$ rational; $f(x)$ irrational if $x$ irrational. Is $f$ a linear function?

Let $f$ be an everywhere differentiable function whose domain consists of all real numbers. Assume that $f(x)$ is rational for rational $x$ and irrational for irrational $x$. Can we conclude that $f$ ...