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Questions tagged [irrational-numbers]

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.

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Find an integer coefficient polynomial for this number [duplicate]

At my latest exam was the following problem: Prove that $\sqrt{2}+\sqrt[3]{2}$ is irrational. The solution is to find a polynomial with integer coefficients, a root of which is $\sqrt{2}+\sqrt[3]{...
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4answers
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Request help on proof of irrationality of $\sqrt {8}$.

Prove that $\sqrt{8}$ is an irrational number. As know that $2\lt \sqrt{8} \lt 3$, so if assume (to attempt proof by contradiction) that $\sqrt{8} = \frac pq$, where $p,q$ are coprime integers; then $...
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1answer
25 views

Trying to understand the irrationality criterion with examples.

If the irrationality criterion below is wrong feel free to edit this post. The irrationality criterion, as far I understood is that if exists sequences $p_n,q_n (n=1,2,3,...)$ where $p_n,q_n$ are ...
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2answers
49 views

How can this binomial expansion result in two different approximations of root 2?

I have been working on a problem on approximating $\sqrt{2}$ using the first three terms of the following binomial expansion, and a substitution of $x = -\frac{1}{10}$ : $$(4 - 5x)^.5 = 2 - \frac{5x}...
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1answer
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Finite Sequence in an Infinite Non-Cyclic Sequence

My friend claims, that the digits in the decimal representation of pi contains every finite sequence of digits. For example my phone number will occur eventually. He claims that this is because there ...
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2answers
39 views

How to show $\sqrt{p}$ is irrational [duplicate]

How to show that $\sqrt{p}$ is an irrational number, given that $p$ is a prime number? $\sqrt{p}=\frac{a}{b}$ $p=\frac{a^2}{b^2}$
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1answer
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Alternation of Rationals and Irrationals?

I'm in a lunch group at work of recreational math geeks and we came up with a question which we need help to resolve. I apologize in advance, if my explanation is not perfectly rigorous. Given these ...
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Proving a number is irrational based off of the fact that another is irrational [duplicate]

Here's the question: Use the fact that √6 is irrational to prove that √2 + √3 is irrational. What would be the best method to use here? How would I lay it out using contradiction? Thanks.
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1answer
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How to prove the irrationality of a number generated by the “$6n \pm 1$ property” of primes?

Assuming that $i > 0$ and $p_1 = 5$, let $p_i$ denote an $i$-th prime. Then we can assume that the value of $b_i$ is $0$ if $p_i = 6n-1$ and the value of $b_i$ is $1$ if $p_i = 6n+1$ (where $n$ ...
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Is my proof of $\pi$’s irrationality correct?

$1.$ Viète's formula for $\pi$ is easy to derive without trigonometry or calculus. Starting with a square inscribed in a circle of unit diameter, the number of sides of the inscribed polygon is ...
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1answer
56 views

Irrationality of square root of 2

I was trying to study Understanding Analysis by Stephen Abbott and I am stuck at the very second page. I feel stupid please help me. It says this: Given two line segments $ab$ and $cd$ it would ...
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How $|a - b| < \epsilon$ implies $|\{a\} - \{b\}| < \epsilon$?

Inside the proof of Theorem 6.3. in the book Irrational Numbers by I Niven, it supposes the following : Let $\zeta$ be an irrational and $\frac{h_i}{k_i}$ its i-th convergent (of $\zeta$ as continued ...
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1answer
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Flea on infinite chessboard jumping with irrational vector eventually changes square color

Question from Engel's Problem Solving Strategies: An infinite chessboard consists of $1 \times 1$ squares. A flea starts on a white square and makes jumps by $\alpha$ to the right and $\beta$ ...
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Why are $sin(x)$ and $cos(x)$ irrational for (non-zero) rational values of $x$? [duplicate]

$sin(x)$ is the ratio between two sides of a triangle. One or more sides must be irrational for $sin(x)$ to be irrational. Why does a rational value of $x$ guarantee this?
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If $R:S^{1}\rightarrow S^{1}$ is a irrational rotation, $\{R^{n}([x])\}$ is dense in $S^{1}$ for all points. [duplicate]

Let $\alpha$ a irrational number, and $R:S^{1}\rightarrow S^{1}$ the irrational rotation, i.e., $[x]\rightarrow[x+\alpha]$. I need to prove that, for all $[x]\in S^{1}$, the set $\{R^{n}([x])\}$ is ...
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1answer
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Prove that: If $a+b$ and $b$ are irrational, then $a+kb$ is irrational.

How can I prove that, If $a+b$ is irrational and $b$ is irrational, then $a+2b$ irrational; where $a,b>0$ More generally, if $a+b$ and $b$ are irrational, then $a+kb$ is irrational; where $...
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5answers
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Using the fact that $\sqrt{n}$ is an irrational number whenever $n$ is not a perfect square, show $\sqrt{3} + \sqrt{7} + \sqrt{21}$ is irrational.

Question: Using the fact that $\sqrt{n}$ is an irrational number whenever $n$ is not a perfect square, show $\sqrt{3} + \sqrt{7} + \sqrt{21}$ is irrational. Following from the question, I tried: ...
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Can this product formula for $\pi$ be used to prove the irrationality of $\pi$?

A less known product formula for $\pi$, due to Sondow, is the following: $$ \frac{\pi}{2}= \left(\frac{2}{1}\right)^{1/2} \left(\frac{2^2}{1\cdot3}\right)^{1/4} \left(\frac{2^3\cdot4}{1\cdot3^3}\...
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1answer
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What are the greater uses of classifying numbers in advanced mathematics [closed]

In high school throughout college, we have been thought what is rational, integers etc. But how does is fit to the greater scheme of things. Trigonometry, Calculus, Mensuration have very obvious ...
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1answer
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Correlation between two sequences of irrational numbers

Let us consider the sequence $x(n+1) = \{b+x(n)\}$ with $x(0) = 0$. Here the brackets represent the fractional part function. Thus $x(n)= \{nb\}$ is related to Beatty sequences. If $b$ is irrational, ...
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1answer
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How do we know that a non recurring number will not repeat after many digits?

We are told that there are rational numbers that either terminate or repeat and irrationals that neither terminate nor repeat. But how are we so sure that a non terminating non recurring number will ...
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2answers
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Is $\frac{\pi}{e}$ irrational? Can we make nonconstant polynomial functions that $\frac{f(\pi)}{g(e)}$ = rational number?

If: $f(\pi) = a_n *\pi^n + a_{n-1} *\pi^{n-1}... a_1 *\pi$ and $g(e) = b_n *e^n + b_{n-1} *e^{n-1}... b_1 *e$ $f(\pi) <> 0$, and $g(e) <> 0$ where $a_n$'s and $b_n$'s are whole or ...
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1answer
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Is it true that irrational roots of rational polynomial comes in pairs? How to prove this? If not give an example please.

I have no information about this. In some places it is written true in some places it is written false. I am sure that if polynomial is quadratic then it is true. But what if not a quadratic ...
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1answer
62 views

Prove that $\log_5 7$ is irrational

Using contradiction, suppose $\log_5 7$ is rational. I found $7^q=5^p$ and I have to explain why this is a contradiction. I don't know how to explain it. This is for homework and my teacher is very ...
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1answer
47 views

How to show an irrational equation have no solution?

There are different cases, one of them is when I have a negative $\sqrt{x}$ so I know there is no solution because $\sqrt{x}$ can't be negative. However there is another case where I just can't ...
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2answers
70 views

Proving $1+\sqrt2+\sqrt3$ is irrational [duplicate]

How can I prove that $1+\sqrt2+\sqrt3$ is an irrational number, without proving first $\sqrt2$ and $\sqrt3$ are irrational numbers? Please give some hints or suggestion to proceed with this proof. ...
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Are there any irrational/transcendental numbers for which the distribution of decimal digits is not uniform?

I conjecture that for irrational numbers, there is generally no pattern in the appearance of digits when you write out the decimal expansion to an arbitrary number of terms. So, all digits must be ...
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Proof verification: Prove $\sqrt{n}$ is irrational.

Problem Let $n$ be a positive integer and not a perfect square. Prove $\sqrt{n}$ is irrational. Proof Consider proving by contradiction. If $\sqrt{n}$ is rational, then there exist two coprime ...
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1answer
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Is it always possible match two non-equal closed form Algebraic Irrational Numbers (real or complex) so the product is a rational number?

(This is a question from an engineer and extremely naïve mathematician when it comes to the topic of irrational and transcendental numbers and the precise distinction between them.) I begin with two ...
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Trying to prove that $\pi$ is irrational using Legendre Polynomials.

Unfortunately, numerical data sugggest that is not possible to show that $\pi$ is irrational with the polynomials below. I've to search for another polynomial... But I've little faith that such ...
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5answers
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Why does this iterative way of solving an equation work?

I was solving some semiconductor physics problem and in order to get the temperature I got this nasty equation: $$ T = \dfrac{7020}{\dfrac{3}{2}\ln(T)+12}.$$ It seems that I can solve this kind of ...
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1answer
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Hopping to infinity along a string of digits

Let $s$ be an infinite string of decimal digits, for example: \begin{array}{cccccccccc} s = 3 & 1 & 4 & 1 & 5 & 9 & 2 & 6 & 5 & 3 & \cdots \end{array} Consider ...
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3answers
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How to represent reciprocal of an irrational number on number line e.g $1/\sqrt{7}$? [closed]

If we try to rationalize this reciprocal of irrational number like for $1/\sqrt{7}$ then we get $\sqrt{7}/7$ which poses a new problem of how to divide an irrational number into $7$ parts. Dividing an ...
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1answer
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Is this number Liouville?

If $b\in\mathbb{Z}_{\geqslant2}$, is the number given by the following sum: $M=\displaystyle\sum_{n\in\mathbb{Z}_{\geqslant0}}b^{-10^{n}}$ a Liouville number? It has lots of zeros, for sure, and by ...
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3answers
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Does all proofs of irrationality are the same way: “Find an integer between $0$ and $1$”?

Excluding square/cubic etc roots of numbers, I've read proofs about the irrationality of $e,\pi,\ln 2,\zeta(2)$ and $\zeta(3)$. In all of them is: assume $x=p/q$, where $x$ is the number trying to ...
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3answers
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Suppose that $x$ and $y$ are irrational, but $x + y$ is rational. Prove that $x -y$ is irrational.

i was wondering if someone could check my proof $Q= \{a/b , c,d : a,c ∈ \mathbb Z , b,d ∈ N>0\}$ $a/b =x+y$ $a/b -y=x$ proof by contradiction. Let $x-y$ is rational $c/d = x-y$ sub $a/b -y = ...
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Proving the negation of a conditional using proof by contradiction

CONTEXT: Question made up by uni maths lecturer Prove the following statement using a proof by contradiction: For all nonzero rational numbers $x$, if $y$ is irrational then $\frac{x}{y}-3$ is ...
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A geometry theory without irrational numbers?

Is there any theory or theorem of geometry -- whether used in practice or not -- which denies or forbids the use of irrational numbers? If not, were there any notable attempts at it? Disclaimer: I ...
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Rainbow numbers: Can mapping digits to different bases produce different varieties of irrationality?

This is a follow-up to the question, "Irrationality of 0.123456789101112 … and similar numbers." There I took some decimal number, in one case Champernowne's constant, $$ n_{10} = 0....
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Irrationality of 0.123456789101112 … and similar numbers

Consider four numbers in $(0,1)$: $n_1$ in base $10$ is formed by listing the decimal digits $1,2,3,4,\ldots$; $b_1$ in binary is formed by $0$ and $1$ for each even and odd digit of $n_1$: $$ n_1 = 0....
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1answer
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Leibniz convergence test to compute the limit

Can we compute the limit of a series using Leibniz test? This is the problem I am struggling with: " Let $(a_{n})_{n \ge 1}$ be a sequence of natural numbers, $a_{n} \ge 2$ , Let $b_{n} = 1 - \frac{...
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0answers
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Question about periodicity in Fibonacci numbers

This is related to Pisano periods, that is, the periods of the Fibonacci numbers modulo $k=2, 3, \cdots$. I am studying the sequence $x(n+1)=\{b x(n)\}$ (here the brackets represent the fractional ...
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1answer
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digamma function inverse and special value

What the inverse of the digamma function?, and how can I write the x for $$ ψ(x)=1$$ and $$1<x$$ $[x ≈ 3.20317146837693106929448152]$ as irrasional number [not a new one a familiar old one]
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1answer
56 views

De-rationalisation of a surd expression $\sqrt p - \sqrt {pq} + q$

Consider two dissimilar surds $\sqrt p$ and $\sqrt q$. Then the problem asks to find rational numbers $a,b,c$ and $d$ such that for $x=\sqrt p + \sqrt q$ we can write, $$ \sqrt p - \sqrt {pq} + q = \...
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1answer
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How to check rationality, irrationality and algebraicity in this case?

This question was asked in a test and the use of calculator was not allowed. Choose the correct one: (A) $\log_e x$ can be defined as a real-valued function of $x$ for all $x\in R$ (B) $\...
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2answers
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Is infinite sequence of irrational numbers digits mathematically observable?

I have a little question. In fact, is too short. Is infinite sequence of irrational numbers digits mathematically observable? I would like to explain it by example because the question seems ...
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3answers
58 views

Prove that root of number is rational.

Consider $ x_1, x_2, ..., x_n \in \mathbb{R}$. We have to prove that each $\sqrt x $ is rational if the sum of $\sqrt x_1 + \ldots + \sqrt x_n $ is rational. I think that I could prove it using ...
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3answers
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Cauchy sequences of rationals with limit irrational: natural, or geometric examples

As we know, real numbers are constructed by filling up gaps between rationals by the limits of all Cauchy sequences of rationals. Q. What are examples of sequence of rationals such that its easy ...
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1answer
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Is ok to differentiate and integrate when trying to prove that a number $x=p/q$ is irrational?

For example consider $x=e.$ So $0<e=p/q$ where $p,q$ are both integer and positive. I'm trying to show that there is no integer $p,q$ that satisfies the equation $e=p/q$. If we differentiate $e=p/q$...
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0answers
58 views

Nth-digit of an irrational number

Is there a way to directly find the Nth digit of the fractional part of an irrational number. For example how to 1000th digit of $\{\pi\}$?