+2446 I happen to remember what this decimal is represented in a fraction form, but here is the method one can use to convert \(0.\overline{142857}\) into a fraction.
This first step is very simple; just set it equal to a variable. I'll use the normal x as my variable for this example. Therefore, \(0.\overline{142857}=x\)
The next goal is to get the repeating portion into the whole numbers part. It is probably easier showing by example than by explaining in words:
| \(0.\overline{142857}=x\) | Multiply by 1000000 on both sides. |
| \(142857.\overline{142857}=10000000x\) | As you can see, the repeating section is now in whole numbers. Now, subtract both equations from each other. |
| \(142857=999999x\) | Now, divide by 999999 on both sides. |
| \(x=\frac{142857}{999999}\div\frac{142857}{142857}\) | It is probably hard to realize here, but the GCF of the numerator and denominator is 142857. |
| \(x=\frac{1}{7}\) | |
Look at that! \(0.\overline{142857}=\frac{1}{7}\). That's quite a nice fraction.
