Section B.2 Subtraction
Whenever we end up with “less” or “fewer” of some amount, we can use subtraction.
\begin{equation*}
\text{\pear \pear \pear \pear \pear \pear} - \text{\pear \pear} = \text{\pear \pear \pear \pear}
\end{equation*}
As with addition, if the values are small we can find the difference of two numbers by counting. Unlike addition, the order of subtraction matters. It's possible to take away \(2\) ants from \(6\text{,}\) but not possible to take away \(6\) ants from only \(2\text{.}\) Though there are situations where subtracting a larger number from a smaller number makes sense, the order of subtraction still matters. (Subtraction is not a commutative operation.)
For larger values, there are subtraction algorithms that work well for computing differences by hand.
Process B.2.1.
\begin{equation*}
\begin{array}{rrrr} \amp 6 \amp 2 \amp 8 \\ - \amp 5 \amp 5 \amp 0 \\ \hline{} \amp \amp \phantom{7} \amp \phantom{8} \end{array}
\end{equation*}
\begin{equation*}
\begin{array}{rrrr} \amp 6 \amp 2 \amp 8 \\ - \amp 5 \amp 5 \amp 0 \\ \hline{} \amp \amp \phantom{7} \amp \highlight{8} \end{array}
\end{equation*}
\begin{equation*}
\begin{array}{rrrr} \amp \highlight{\cancelto{5}{6}}\amp \highlight{12} \amp 8 \\ - \amp 5 \amp 5 \amp 0 \\ \hline{} \amp \amp \phantom{7} \amp 8 \end{array}
\end{equation*}
\begin{equation*}
\begin{array}{rrrr} \amp 5 \amp 12 \amp 8 \\ - \amp 5 \amp 5 \amp 0 \\ \hline{} \amp \amp \highlight{7} \amp 8 \end{array}
\end{equation*}
Process B.2.2.
\begin{equation*}
\begin{array}{rrrr} \amp 6 \amp 2 \amp 8 \\ - \amp 5 \amp 5 \amp 0 \\ \hline{} \amp \phantom{1} \amp \phantom{2} \amp \phantom{8} \\ \amp \amp \phantom{5} \amp \phantom{0} \\ %\hline{} \amp \amp \phantom{7} \amp \phantom{8} \end{array}
\end{equation*}
\begin{equation*}
\begin{array}{rrrr} \amp 6 \amp 2 \amp 8 \\ - \amp 5 \amp \phantom{5} \amp \phantom{0} \\ \hline{} \amp \highlight{1} \amp \highlight{2} \amp \highlight{8} \\ \amp \amp \phantom{5} \amp \phantom{0} \\ %\hline{} \amp \amp \phantom{7} \amp \phantom{8} \end{array}
\end{equation*}
\begin{equation*}
\begin{array}{rrrr} \amp 6 \amp 2 \amp 8 \\ - \amp 5 \amp \phantom{5} \amp \phantom{0} \\ \hline{} \amp 1 \amp 2 \amp 8 \\ - \amp \amp 5 \amp \phantom{0} \\ \hline{} \amp \amp \highlight{7} \amp \highlight{8} \end{array}
\end{equation*}
Do you see how these algorithms work? (Again, the first one should be familiar.) Time for more practice doing arithmetic by hand.
Checkpoint B.2.3.
Perform each subtraction by hand, then check your results.
