Section B.1 Addition
When we consider counting, addition simply means “more.” Starting with three apples, then getting two more gives:
\begin{equation*}
\text{\apple \apple \apple} + \text{\apple \apple} = \text{\apple \apple \apple \apple \apple}
\end{equation*}
It doesn't matter if we add two apples to three apples or three apples to two apples, either way we end up with five apples. Order isn't important for addition.
\begin{equation*}
3+2=2+3
\end{equation*}
They both equal \(5\text{.}\) When the order of an operation doesn't matter, mathematicians say the operation is commutative.
For small whole numbers (\(0\text{,}\) \(1\text{,}\) \(2\text{,}\) \(3\text{,}\) \(4\text{,}\) etc.), we can add by counting. For larger whole numbers, we use an algorithm if we need to add by hand. Over the centuries, people have developed many algorithms that work. Adding \(697\) apples to \(223\) apples, here are two possible algorithms:
Process B.1.1.
\begin{equation*}
\begin{array}{rrrr} \amp\phantom{_1} \amp\phantom{_1} \amp \\ \amp 6 \amp 9 \amp 7 \\ + \amp 2 \amp 2 \amp 3 \\ \hline{} \amp \phantom{9 }\amp \phantom{2 }\amp \phantom{0} \end{array}
\end{equation*}
\begin{equation*}
\begin{array}{rrrr} \amp\phantom{_1} \amp\highlight{_1} \amp \\ \amp 6 \amp 9 \amp 7 \\ + \amp 2 \amp 2 \amp 3 \\ \hline{} \amp \phantom{9 }\amp \phantom{2 }\amp \highlight{0} \end{array}
\end{equation*}
\begin{equation*}
\begin{array}{rrrr} \amp\highlight{_1} \amp_1 \amp \\ \amp 6 \amp 9 \amp 7 \\ + \amp 2 \amp 2 \amp 3 \\ \hline{} \amp \phantom{9 }\amp \highlight{2} \amp 0 \end{array}
\end{equation*}
\begin{equation*}
\begin{array}{rrrr} \amp_1 \amp_1 \amp \\ \amp 6 \amp 9 \amp 7 \\ + \amp 2 \amp 2 \amp 3 \\ \hline{} \amp \highlight{9} \amp 2 \amp 0 \end{array}
\end{equation*}
Process B.1.2.
\begin{equation*}
\begin{array}{rrrr} \amp 6 \amp 9 \amp 7 \\ + \amp 2 \amp 2 \amp 3 \\ \hline{} \amp \phantom{8} \amp \phantom{0} \amp \phantom{0} \\ \amp \phantom{1} \amp \phantom{1} \amp \phantom{0} \\ \amp \phantom{1} \amp \phantom{0} \\ %\hline{} \amp \phantom{9} \amp \phantom{2} \amp \phantom{0} \end{array}
\end{equation*}
\begin{equation*}
\begin{array}{rrrr} \amp 6 \amp 9 \amp 7 \\ + \amp 2 \amp 2 \amp 3 \\ \hline{} \amp \highlight{8} \amp \highlight{0} \amp \highlight{0} \\ \amp \phantom{1} \amp \phantom{1} \amp \phantom{0} \\ \amp \phantom{1} \amp \phantom{0} \\ %\hline{} \amp \phantom{9} \amp \phantom{2} \amp \phantom{0} \end{array}
\end{equation*}
\begin{equation*}
\begin{array}{rrrr} \amp 6 \amp 9 \amp 7 \\ + \amp 2 \amp 2 \amp 3 \\ \hline{} \amp 8 \amp 0 \amp 0 \\ \amp \highlight{1} \amp \highlight{1} \amp \highlight{0} \\ \amp \amp \phantom{1} \amp \phantom{0} \\ %\hline{} \amp \phantom{9} \amp \phantom{2} \amp \phantom{0} \end{array}
\end{equation*}
\begin{equation*}
\begin{array}{rrrr} \amp 6 \amp 9 \amp 7 \\ + \amp 2 \amp 2 \amp 3 \\ \hline{} \amp 8 \amp 0 \amp 0 \\ \amp 1 \amp 1 \amp 0 \\ \amp \amp \highlight{1} \amp \highlight{0} \\ \hline{} \amp \highlight{9} \amp \highlight{2} \amp \highlight{0} \end{array}
\end{equation*}
Do you see how both algorithms work? (The first one should be familiar, the second might require some thought.) Whether we add by hand or use technology (calculator, computer, cell phone, etc.) we have \(697 + 223 = 920\text{.}\)
In mathematics, we call each number being added together a term, and the result of addition is called a sum. In your algebra course, you may be required to perform calculations by hand on exams. Here are a few practice problems.
Checkpoint B.1.3.
Perform each addition by hand, then check your results.
