Prove alebracically that 0.5 recurring is 5/9

Let's first assign X the value of the repeating decimal. So X = 0.55555555 Let's multiply both sides by 10 10X = 5.55555555 Now let's subtract the 1st equality from the 2nd. 10X = 5.55555555 - (X = 0.55555555) = 9X = 5 And finally, divide both sides by 9. X = 5/9 You can use the above technique for ANY repeating decimal by multiplying by 10^n power where n is the number of digits in the repeating cycle. For instance, 1/7 = 0.142857142857142857142857142857142857 So the repeating portion is 142857 which is 6 digits long. So you'd multiply by 10^6 or 1000000.