How to find out any digit of any irrational number?

Patatiniuh
2022-07-07
Answered

How to find out any digit of any irrational number?

You can still ask an expert for help

persstemc1

Answered 2022-07-08
Author has **18** answers

Let $\alpha $ be an irrational number. As long as there exists an algorithm the can decide whether $\alpha >q$ or $\alpha <q$ for any given rational $q$, you can obtain arbitrarily good rational approximations for $\alpha $. Especially, you can find upper and lower bounds good enough to uniquely determine any desired number of decimals.

For $\alpha =\sqrt{2}$, the decision algorithm is quit simple: If $q=\frac{n}{m}$ with$n\in \mathbb{Z},m\in \mathbb{N}$, then $\alpha <q\phantom{\rule{thickmathspace}{0ex}}\u27fa\phantom{\rule{thickmathspace}{0ex}}n>0\wedge {n}^{2}>2{m}^{2}$.

For $\alpha =\sqrt{2}$, the decision algorithm is quit simple: If $q=\frac{n}{m}$ with$n\in \mathbb{Z},m\in \mathbb{N}$, then $\alpha <q\phantom{\rule{thickmathspace}{0ex}}\u27fa\phantom{\rule{thickmathspace}{0ex}}n>0\wedge {n}^{2}>2{m}^{2}$.

Kolten Conrad

Answered 2022-07-09
Author has **1** answers

In general, no.

Suppose that for every irrational number $r$ there were an algorithm that takes a natural $n$ as input and returns the $n$-th digit of $r$. The possible algorithms are countable, all the irrationals are not, hence it is not possible to have such algorithms for every irrational.

However, such algorithms do exist for the so-called computable number.

Suppose that for every irrational number $r$ there were an algorithm that takes a natural $n$ as input and returns the $n$-th digit of $r$. The possible algorithms are countable, all the irrationals are not, hence it is not possible to have such algorithms for every irrational.

However, such algorithms do exist for the so-called computable number.

asked 2022-04-12

How can we get Nth digit of any irrational number in base $2$ using Taylor expansion?

asked 2021-01-10

Find the value of the following expression:

$\frac{\mathrm{sin}\left(x\right)}{\mathrm{cos}(-x)}+\frac{\mathrm{sin}(-x)}{\mathrm{cos}\left(x\right)}$

asked 2021-02-08

In which set(s) of numbers would you find the number $\sqrt{80}$

- irrational number

- whole number

- rational number

- integer

- real number

- natural number

- irrational number

- whole number

- rational number

- integer

- real number

- natural number

asked 2022-07-13

For instance, let the function:

$f=\prod _{n=1}^{\mathrm{\infty}}\sqrt[b]{a},$

for $b\in \mathbb{N}$, $b>1$ and $a$ irrational such that $f$ converges to the real number $k$. We immediately see that all partial products are irrational. Can we then also say that $k$ is irrational?

$f=\prod _{n=1}^{\mathrm{\infty}}\sqrt[b]{a},$

for $b\in \mathbb{N}$, $b>1$ and $a$ irrational such that $f$ converges to the real number $k$. We immediately see that all partial products are irrational. Can we then also say that $k$ is irrational?

asked 2022-07-08

Is this proposition $\sqrt{2}$ + $\sqrt{3}$ is an irrational number." true or false?

asked 2020-11-22

How to find a rational number halfway between any two rational numbers given infraction form , add the two numbers together and divide their sum by 2. Find a rational number halfway between the two fractions in each pair.

$\frac{1}{4}$ and $\frac{3}{4}$

asked 2021-02-02

Cindy separated her fruit flies into equal groups. She estimates that there are 2¹⁰ fruit flies in each of 2² jars. How many fruit flies does Cindy have in all?