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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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 ...
2
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2answers
76 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
384 views

Polynomials with Integer Coefficients and irrational roots

Is there a polynomial with integer coefficients which has √2 +√7  as a root?
2
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0answers
114 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
353 views

Finding set of non recurring non terminating decimals

I need to find a set of two Integers P and Q such that ...
0
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1answer
75 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
132 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 ..
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3answers
452 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 ...
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4answers
135 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 ...
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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
71 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
66 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
167 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 ...
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1answer
72 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
83 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
266 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
56 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
184 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
801 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
58 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
191 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
50 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
341 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?
2
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2answers
147 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
127 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
469 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
138 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
votes
1answer
101 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 $f$ ...
3
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1answer
147 views

How is this a proof of the irrationality of $\sqrt2$

Proof. Suppose for the sake of contradiction that $\sqrt2$ is rational, and choose the least integer, $q \gt 0$, such that $(\sqrt2 − 1)q$ is a non negative integer. Let $q':=(\sqrt2 − 1)q$. Clearly ...
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1answer
63 views

Will negative bases with irrational exponents get a real or imaginary number?

Here are a few examples: $$(-1)^{\sqrt{2}},(-2)^{\pi},(-3)^{e}$$ From what I've learned, negative bases must have denominators of the exponent odd. Normally if we do $(-2)^{0.258}$ it would be the ...
6
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4answers
533 views

Teaching irrational numbers?

I'm interested in teaching the irrational numbers to high-school students, and I need your ideas on how to do in an 'optimal' and innovative way. And my question is: What should the teacher know ...
6
votes
1answer
167 views

Predicting digits in $\pi$

Is it possible to predict next digit in $\pi$ using $N$ previous digits, so on and so forth? Or is this impossible because it's irrational? Basic assumption is that the person doesn't know a ...
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3answers
90 views

$x^2-p=0$, with $p$ prime, have irrational roots.

Unaware that $\sqrt{p}$ is irrational, prove that as $x^2-p=0$ have irrational root for $p$ prime? How would you use the criterion of Eisenstein?
3
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2answers
109 views

Polynomial in $\mathbb{Q}[x]$ with root $\sqrt[3]{2}+\sqrt[3]{3}$

What is a polynomial $P(x)\in \mathbb{Q}[x]$ with root $\sqrt[3]{2}+\sqrt[3]{3}$? I write $x=\sqrt[3]{2}+\sqrt[3]{3}$, so $(x-\sqrt[3]{2})^3=3$, but the expansion of the left side contains two cube ...
4
votes
3answers
215 views

Prove that $\sqrt[3]{5-\sqrt{2}}$ is not a rational number

My attempt: Consider the polynomial $ (x^3-5)^2 - 2 = x^6 -10x^3 + 23 = 0 $. By the rational zeros theorem, we can conclude that $ \pm 1$ and $ \pm 23 $ are the only possible rational solutions*. ...
4
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2answers
242 views

Proof e is irrational — Choice of Sign of m/n = e and Uncanny Step

Assume $e$ is rational. Then, there exist coprime integers $m$ and $n$, and we can choose $n$ to be positive, such that: $\displaystyle \frac m n = e = \sum_{i \mathop = 0}^\infty \frac 1 {i!}$ from ...
19
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2answers
269 views

Is there an elementary proof that $\sum_{n=1}^\infty {1\over n^s\{n\pi\}}<\infty$ for some $s>0$?

Edit: David Speyer's answer made me realize a couple of things and I would like to clarify. Sorry if the length of this is getting out of hand. First, it is now clear that no estimate can be obtained ...
0
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0answers
52 views

Solutions of $3a^2-2b^2=1$ [duplicate]

The question is to find the integral solutions of $$3a^2-2b^2=1$$ In the solution it is given that: The solutions of $3a^2-2b^2=1$ can be obtained from ...
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5answers
354 views

Direct proof: $\sqrt{13}$ is irrational

Show that $\sqrt{13}$ is an irrational number. How to direct proof that number is irrational number. So what is the first step.....
2
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3answers
187 views

Proof that there are infinitely many irrational numbers!

I want to prove that there are infinitely many irrational numbers! How can I do that? I don't know where or how to start so any hint is appreciated. Thanks! :)
3
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1answer
79 views

Irrational number?

Is the solution of the equation $$x + \arctan(x) = \pi$$ irrational ? The equation of $x + \arctan(x) = 1$ must be transcendental because for any nonzero algebraic $x$, $arctan(x)$ is ...
12
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1answer
351 views

Prove that this number is irrational

The number $a=0.12457...$ is defined as follows: The digit on the $n$-th place after the dot is the first digit left to the dot of the number $n\sqrt2$. For example, for $n=1$ we have ...
8
votes
1answer
75 views

Number made from ending digits of primes

Consider the number 0.23571379391713739171393971379371799173739113791379391173917133713717793 ... The number is formed by the ending digits of the prime numbers. Is it known whether this number is ...
12
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4answers
175 views

$\sqrt{2}\notin\mathbb{Q}$ but …

Ok, it's easy to prove that prime roots are irrational (i.e. $ \sqrt{p} \not\in \mathbb{Q}, \text{ if } p \in \mathbb{P} $) Consider $ \sqrt{2} $. We can quickly prove that $ \sqrt{2} \not\in ...
9
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1answer
188 views

Integer parts of multiples of irrationals

Let $\alpha>0$ and define $S(\alpha)=\{\lfloor n \alpha \rfloor: n\in\Bbb Z^+ \}$. (Here $\lfloor x\rfloor$ is the integer part of $x$ and $\mathbb Z^+$ the set of positive integers.) Question. Is ...
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2answers
230 views

Do irrational numbers really exist?

Isn't it possible that an irrational number is in reality the quotient of two infinitely long integers that even if there were repeating sections in it, it would take infinite digits to find the first ...
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2answers
125 views

Draw an irrational number on the number line (without pythagoras sentence)

Let's say im a guy for ancient greece and I only have a string and a pencil. And I want to draw a line, the width of the line is the square root of 6. And I only know how to draw a line in the width ...