# 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.

1,689 questions
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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. ...
3answers
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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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