Question

Express a decimal number ( ie 0.768) - process description please - as a rational number (96/125)?

Answer 1

The decimal representation of any number is a linear combination of powers of 10. In other words, given a number like 123.456, we can expand it as

`1\cdot10^2+2\cdot10^1+3\cdot10^0+4\cdot10^(-1)+5\cdot10^(-2)+6\cdot10^(-3)`

`10^(-n)=\frac1{10^n}` for any
`n`, so the above is the same as

`100+20+3+\frac4{10}+\frac5{100}+\frac6{1000}=(100000+20000+3000+400+50+6)/(1000)=(123456)/(1000)`

Similarly, we can write

`0.768=(768)/(1000)`

Now it's a question of reducing the fraction as much as possible. We have
`\mathrm{gcd}(768,1000)=8` so

`(768)/(1000)=(96\cdot8)/(125\cdot8)=(96)/(125)`