Wondrous Arithmetic : Division

Remember having to divide numbers by hand ? If you have a rational number you can move from fractional form (a/b) to decimal form and back. Going from fractional to decimal form is done by performing the long division method. Going the other way is done using a ‘trick’. Any rational number written in decimal form will either have a finite number of decimal places or an infinite number of decimal places with some repeating sequence. In case of a finite number of decimal places it is simply a matter of removing the decimal separator by multiplying by the correct power of 10 and then dividing this number by the same power of 10.

For example, 16,75 has 2 decimal places. We multiply by 10² to remove the decimal separator and divide by 10² giving the fractional form ¹⁶⁷⁵⁄₁₀₀.

In case of an infinite number of decimal places this simple method will not work. Take as example 14,2363636… We call this number x. Then we multiply x with a power of 10 so that the decimal is placed immediately after the first instance of the repeating sequence (36 in this example). Here we need to multiply x by 1000 to get 14236,363636… Then we multiply x with a power of 10 so that the decimal is placed immediately in front of the first repeating sequence, here 10 to get 142,363636… We now have 1000x = 14236,363636 and 10x = 142,363636. Or 1000x – 10x = 14236,363636 – 142,363636 which equals 990x = 14094 or x = ¹⁴⁰⁹⁴⁄₉₉₀ = ⁷⁸³⁄₅₅