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

$1\cdot10^2+2\cdot10^1+3\cdot10^0+4\cdot10^{-1}+5\cdot10^{-2}+6\cdot10^{-3}$

$10^{-n}=\dfrac1{10^n}$ for any $n$, so the above is the same as

$100+20+3+\dfrac4{10}+\dfrac5{100}+\dfrac6{1000}=\dfrac{100000+20000+3000+400+50+6}{1000}=\dfrac{123456}{1000}$

Similarly, we can write

$0.768=\dfrac{768}{1000}$

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

$\dfrac{768}{1000}=\dfrac{96\cdot8}{125\cdot8}=\dfrac{96}{125}$