Amira Serrano

2022-10-13

What is the formula for the decimal representation of $\frac{a}{b}$ where $b$ is not coprime to 10?

If we want to turn a proper fraction $\frac{a}{b}$ into a decimal, then when $b$ is coprime to 10 we just need to rewrite $\frac{a}{b}$ in the form

$\frac{m}{{10}^{\varphi (b)}-1}$

where $\varphi $ is Euler's totient function. And then the repeatend is $m$ and the period of repetition is $\varphi (b)$.

But my question is, what do we do when $b$ is not coprime to 10? In that case we would need to rewrite $\frac{a}{b}$ in the form

$\frac{k+\frac{l}{m}}{{10}^{n}}$

where $k<{10}^{n}$, $\frac{l}{m}$ is a proper fraction, and $m$ is coprime to 10. Then $k$ would be the non-repeating part, $n$ would be the length of the nonrepeating part, and we can convert $l/m$ into a decimal using the procedure given in the beginning of my post.

But is there a formula for the four numbers $k$, $l$, $m$, and $n$ in terms of $a$ and $b$, short of doing long division to convert the fraction into a decimal?

If we want to turn a proper fraction $\frac{a}{b}$ into a decimal, then when $b$ is coprime to 10 we just need to rewrite $\frac{a}{b}$ in the form

$\frac{m}{{10}^{\varphi (b)}-1}$

where $\varphi $ is Euler's totient function. And then the repeatend is $m$ and the period of repetition is $\varphi (b)$.

But my question is, what do we do when $b$ is not coprime to 10? In that case we would need to rewrite $\frac{a}{b}$ in the form

$\frac{k+\frac{l}{m}}{{10}^{n}}$

where $k<{10}^{n}$, $\frac{l}{m}$ is a proper fraction, and $m$ is coprime to 10. Then $k$ would be the non-repeating part, $n$ would be the length of the nonrepeating part, and we can convert $l/m$ into a decimal using the procedure given in the beginning of my post.

But is there a formula for the four numbers $k$, $l$, $m$, and $n$ in terms of $a$ and $b$, short of doing long division to convert the fraction into a decimal?

Kaylee Evans

Beginner2022-10-14Added 20 answers

$\frac{\sqrt{2x+3y}+\sqrt{2x-3y}}{\sqrt{2x+3y}-\sqrt{2x-3y}}=p$

Apply componendo and dividendo,

$\frac{\sqrt{2x+3y}+\sqrt{2x-3y}+\sqrt{2x+3y}-\sqrt{2x-3y}}{\sqrt{2x+3y}+\sqrt{2x-3y}-\sqrt{2x+3y}+\sqrt{2x-3y}}=\frac{p+1}{p-1}$

$\frac{\sqrt{2x+3y}}{\sqrt{2x-3y}}=\frac{p+1}{p-1}$

Put x=y

$\frac{\sqrt{5y}}{\sqrt{-y}}=\frac{p+1}{p-1}$

Squaring both sides,

$5.(p-1{)}^{2}=-1(p+1{)}^{2}$

Sorry I have little mistake here, $5{p}^{2}-10p+5=-{p}^{2}-2p-2$

$5{p}^{2}-10p+5=-{p}^{2}-2p-1$

$6{p}^{2}-8p+6=0$

$3{p}^{2}-4p+3=0$

$D={b}^{2}-4ac$

= $(-4{)}^{2}-\mathrm{4.3.3}$ = 16 - 36 = - 20

$p=\frac{-b\pm \sqrt{D}}{2a}$

= $\frac{-(-4)\pm \sqrt{-20}}{2.3}$

= $\frac{4\pm \sqrt{20{i}^{2}}}{6}$

= $\frac{4\pm 2i\sqrt{5}}{6}$

p = $\frac{2+i\sqrt{5}}{3},\frac{2-i\sqrt{5}}{3}$

Edit-

In above formula D is discriminat. It is used when we can't factorise equation using factorisation method.

Apply componendo and dividendo,

$\frac{\sqrt{2x+3y}+\sqrt{2x-3y}+\sqrt{2x+3y}-\sqrt{2x-3y}}{\sqrt{2x+3y}+\sqrt{2x-3y}-\sqrt{2x+3y}+\sqrt{2x-3y}}=\frac{p+1}{p-1}$

$\frac{\sqrt{2x+3y}}{\sqrt{2x-3y}}=\frac{p+1}{p-1}$

Put x=y

$\frac{\sqrt{5y}}{\sqrt{-y}}=\frac{p+1}{p-1}$

Squaring both sides,

$5.(p-1{)}^{2}=-1(p+1{)}^{2}$

Sorry I have little mistake here, $5{p}^{2}-10p+5=-{p}^{2}-2p-2$

$5{p}^{2}-10p+5=-{p}^{2}-2p-1$

$6{p}^{2}-8p+6=0$

$3{p}^{2}-4p+3=0$

$D={b}^{2}-4ac$

= $(-4{)}^{2}-\mathrm{4.3.3}$ = 16 - 36 = - 20

$p=\frac{-b\pm \sqrt{D}}{2a}$

= $\frac{-(-4)\pm \sqrt{-20}}{2.3}$

= $\frac{4\pm \sqrt{20{i}^{2}}}{6}$

= $\frac{4\pm 2i\sqrt{5}}{6}$

p = $\frac{2+i\sqrt{5}}{3},\frac{2-i\sqrt{5}}{3}$

Edit-

In above formula D is discriminat. It is used when we can't factorise equation using factorisation method.

Which expression has both 8 and n as factors???

One number is 2 more than 3 times another. Their sum is 22. Find the numbers

8, 14

5, 17

2, 20

4, 18

10, 12Perform the indicated operation and simplify the result. Leave your answer in factored form

$\left[\frac{(4x-8)}{(-3x)}\right].\left[\frac{12}{(12-6x)}\right]$ An ordered pair set is referred to as a ___?

Please, can u convert 3.16 (6 repeating) to fraction.

Write an algebraic expression for the statement '6 less than the quotient of x divided by 3 equals 2'.

A) $6-\frac{x}{3}=2$

B) $\frac{x}{3}-6=2$

C) 3x−6=2

D) $\frac{3}{x}-6=2$Find: $2.48\xf74$.

Multiplication $999\times 999$ equals.

Solve: (128÷32)÷(−4)=

A) -1

B) 2

C) -4

D) -3What is $0.78888.....$ converted into a fraction? $\left(0.7\overline{8}\right)$

The mixed fraction representation of 7/3 is...

How to write the algebraic expression given: the quotient of 5 plus d and 12 minus w?

Express 200+30+5+4100+71000 as a decimal number and find its hundredths digit.

A)235.47,7

B)235.047,4

C)235.47,4

D)234.057,7Find four equivalent fractions of the given fraction:$\frac{6}{12}$

How to find the greatest common factor of $80{x}^{3},30y{x}^{2}$?