# On Recurring decimals

$\large{0\red{.}\overline{x_1x_2x_3...x_n}=\dfrac{x_1x_2x_3...x_n}{10^n-1}}$

Proof of the above statement

Let $l=0\red{.}\overline{x_1x_2x_3...x_n}$ $10^nl=x_1x_2x_3...x_n\red{.}\overline{x_1x_2x_3...x_n}$ $\Rightarrow 10^nl-l={x_1x_2x_3...x_n}$ $(10^n-1)l=x_1x_2x_3...x_n$ ${l=\dfrac{x_1x_2x_3...x_n}{10^n-1}}$ $\boxed{0\red{.}\overline{x_1x_2x_3...x_n}=\dfrac{x_1x_2x_3...x_n}{10^n-1}}$

$\large{a_1a_2a_3...a_p\red{.}b_1b_2b_3...b_q\overline{x_1x_2x_3...x_n}=\dfrac{1}{10^q}(10^q\times a_1a_2a_3...a_p+b_1b_2b_3...b_q+\dfrac{x_1x_2x_3...x_n}{10^n-1})}$

Proof of the above statement

For any number $a_1a_2a_3...a_p\red{.}b_1b_2b_3...b_q\overline{x_1x_2x_3...x_n}$ $a_1a_2a_3...a_p\red{.}b_1b_2b_3...b_q\overline{x_1x_2x_3...x_n}=a_1a_2a_3...a_p+0\red{.}b_1b_2b_3...b_q\overline{x_1x_2x_3...x_n}$ $=\dfrac{1}{10^q}(10^q\times a_1a_2a_3...a_p+b_1b_2b_3...b_q\red{.}\overline{x_1x_2x_3...x_n})$ $=\dfrac{1}{10^q}(10^q\times a_1a_2a_3...a_p+b_1b_2b_3...b_q+0\red{.}\overline{x_1x_2x_3...x_n})$ $=\boxed{\dfrac{1}{10^q}(10^q\times a_1a_2a_3...a_p+b_1b_2b_3...b_q+\dfrac{x_1x_2x_3...x_n}{10^n-1})}$

Note :

• $x_1x_2$ act as number with digits $x_1,x_2$ for example if $x_1=5$ and $x_2=8\Rightarrow x_1x_2=58$ dont confuse ($x_1x_2\cancel{=}x_1\times x_2$), same for $x_1x_2x_3$ and $x_1x_2x_3...x_{n-1}x_n$

• $0\red{.}\overline{a}=0\red{.}aaaaa...$

Note by Zakir Husain
6 months, 2 weeks ago

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Sort by:

- 6 months, 2 weeks ago

Well.. I am impressed.

- 6 months, 2 weeks ago

✨ brilliant +1

- 6 months, 2 weeks ago

By the way, I have an interesting #Geometry problem!
Given points $A,B,C,D$, find the square $\square PQRS$ with A on PQ, B on QR, C on RS, D on SP.
I figured out the first part, where we can construct circles with diameters AB, BC, CD, DA respectively, so if a point W is on arc AB, $\angle AWB=90^\circ.$

- 6 months, 2 weeks ago

I tried the problem and got an algorithm to construct a rectangle $PQRS$ with points $A,B,C$ and $D$ on sides $PQ,QR,RS,SP$ respectively. Also there will be infinitely many such rectangles for given points $A,B,C,D$

- 6 months, 2 weeks ago

What about a square?

- 6 months, 2 weeks ago

I will try it also! and will inform you as I get any results.

- 6 months, 2 weeks ago

Let’s start a discussion! That might help :)

- 6 months, 2 weeks ago

square is also a rectangle..

- 6 months, 2 weeks ago

But a rectangle isn’t a square, so I hope to find an algorithm to construct a square (I know it is possible but I don’t know a specific way to do it except for brute-force :P) :)

- 6 months, 2 weeks ago