 ### Subtraction algorithm

Subtraction is straightforward if each digit in the number to be subtracted is smaller than those in the number we are subtracting from.
For example:

 9 8 7 6 − 5 4 3 2 4 4 4 4

When this is not the case, there are two often-used algorithms.

As discussed earlier, adding the same number to both numbers in a subtraction does not change the difference between them. This is the basis for the equal addition algorithm for subtraction. In this case we add ten to each number. Some people may know this as the borrow and pay back method for subtraction.

 $$3\$$ $$^1$$4 − 1$$_1$$ 6 $$1\$$ 8

For example, to calculate 34 – 16 we set the numbers one under the other. We add ten ones to 34 and one ten to 16. The algorithm works because, arithmetically, we are actually subtracting 26 from 44. The difference between 44 and 26 is the same as the difference between 34 and 16.

#### Decomposition

The decomposition algorithm is also known as the trading algorithm for subtraction. The names decomposition and trading come from the place-value representation of the numbers, as modelled by Dienes blocks (MAB). The subtraction of one from the tens column and addition of ten to the ones column can be viewed as the 'long' block being traded for, or decomposed into, ten 'ones'.

We calculate 34 − 16 by writing 34 as 20 + 14 instead of 30 + 4.

\begin{align} 34-16&=(20+14)-(10+6)\\\\&=(20-10)+(14-6)\\\\&=10+8 \end{align}

 $$^2 3\hspace{-3mm}\color{red}/$$ $$^1$$4 − 1 6 1 8

Algorithmically, we indicate the new regrouping as a conversion of one ten into ten ones.   