Just wondering what is the best way of converting a non ending repeating decimal number into a fraction.
Ex:
Convert 0.0315 into a fraction (315 is repeating. Sorry don't know how to dot numbers in latex!)
Cheers
BIOS
Just wondering what is the best way of converting a non ending repeating decimal number into a fraction.
Ex:
Convert 0.0315 into a fraction (315 is repeating. Sorry don't know how to dot numbers in latex!)
Cheers
BIOS
Hello, BIOS!
$\displaystyle \text{Convert }0.0\overline{315}\text{ into a fraction.}$
I'll show you what skeeter did . . .
Let $\displaystyle x \,=\,0.0315315\hots$
$\displaystyle \begin{array}{cccccc}\text{Multiply by 10,000:} & 10,\!000x &=& 315.315315\hdots \\
\text{Multiply by 10:} & \quad\;\; 10x &=& \;\;\;0.315315\hdots \\
\text{Subtract:} & \;\; 9990x &=& 315\qquad\qquad\;\; \end{array}$
Hence: .$\displaystyle x \;=\;\dfrac{315}{9990} \;=\;\dfrac{7}{222}$
It works because it works - there's not really any other way to explain it. You need to find which multiple of your original number has the same repeating digits, then subtract to eliminate the repeating digits. Division gives you the fraction...
Anyway, Plato's method is using infinite geometric series...
$\displaystyle \displaystyle 0.0\overline{315} = 0.0315315315315\dots$
$\displaystyle \displaystyle = \frac{315}{10\,000} + \frac{315}{10\,000\,000} + \frac{315}{10\,000\,000} + \dots$
$\displaystyle \displaystyle = \frac{315}{10}\left(\frac{1}{1000} + \frac{1}{1\,000\,000} + \frac{1}{1\,000\,000\,000} + \dots\right)$
$\displaystyle \displaystyle = \frac{315}{10}\left[\left(\frac{1}{1000}\right)^1 + \left(\frac{1}{1000}\right)^2 + \left(\frac{1}{1000}\right)^3 + \dots\right]$.
The stuff in brackets is a convergent infinite geometric series with $\displaystyle \displaystyle a = \frac{1}{1000}$ and $\displaystyle \displaystyle r = \frac{1}{1000}$. So its sum is $\displaystyle \displaystyle \frac{a}{1 - r} = \frac{\frac{1}{1000}}{1 - \frac{1}{1000}} = \frac{\frac{1}{1000}}{\frac{999}{1000}} = \frac{1}{999}$.
So $\displaystyle \displaystyle \frac{315}{10}\left[\left(\frac{1}{1000}\right)^1 + \left(\frac{1}{1000}\right)^2 + \left(\frac{1}{1000}\right)^3 + \dots\right]=\frac{315}{10}\cdot \frac{1}{999}$
$\displaystyle \displaystyle = \frac{7}{222}$.