We were grouping numbers in a different way, not changing the answer. When we cut 9 in half, we dropped the remainder because 9 is an odd number. Because our product has decreased by 8, we have to add 8 back in again at the end. We can think of the addition as restoring 1 group of 8, for the remainder of 1 that we dropped earlier. In a different problem, we might have to restore several different groups of numbers.
Binary numbers are numbers written in base two instead of base ten.
Russian peasant multiplication is actually a quick way to convert two numbers to binary form, multiply them together, and convert back to our number system. The connection is not surprising, because binary numbers use base two, and Russian peasant multiplication depends on multiplying and dividing by two. Halving You can convert a number to binary form by repeatedly dividing by two and keeping track of the remainders. Let's try cutting twelve in half again, the same way. This time, we'll write everything in base two.