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Section B.3 Multiplication

There's more than one way to think about multiplication. For counting, multiplication helps us compute the number of cats in the table below.

\cat \cat \cat \cat
\cat \cat \cat \cat
\cat \cat \cat \cat

If we notice three rows with four cats each, we have

\begin{equation*} \overbrace{4 + 4 + 4}^{3\text{ times}} = 3 \times 4 = 12 \text{ cats}\text{,} \end{equation*}

but if we notice four columns with three cats each, that's

\begin{equation*} \overbrace{3 + 3 + 3 + 3}^{4 \text{times}} = 4 \times 3 = 12 \text{ cats}\text{.} \end{equation*}

Either way, multiplication can serve as shorthand for repeated addition. And it's clear that the order of the factors (the numbers we multiply together) has no effect on the value of the product (the result of multiplication). Multiplication, like addition, is a commutative operation.

Multiplication isn't always a shortcut for repeated addition. We also use multiplication for scaling. If a map's scale is “\(1\) inch to \(2\) miles,” a distance of \(4\) inches on the map would translate into \(4\cdot2=8\) miles for the actual distance.

Of course, when we multiply larger values by hand, we use an algorithm. Again, there are many algorithms that will work. Here are two examples:

Symbolically, there is more than one way to indicate multiplication. All of the following symbolic expressions mean the same thing:

\begin{equation*} 0.6 \times 3 \quad (0.6)(3) \quad 0.6(3) \quad (0.6)3 \quad 0.6 * 3 \quad 0.6 \cdot 3 \end{equation*}

In algebra, the times symbol, as in \(0.6\times3\text{,}\) has fallen out of favor because it looks like the letter \(x\text{,}\) which we use to represent an unknown amount. Most mathematicians write two quantities right next to each other (maybe wrapping one or both in parentheses) or they use the dot, as in \(0.6\cdot3\text{,}\) to indicate multiplication. If you're typing, use parentheses or the asterisk, *, for multiplication, and both people and computers will understand you.

Checkpoint B.3.3.

Perform each multiplication by hand, then check your results.

The product of any number and \(0\) is always \(0\text{.}\) For example: \(2.2\cdot0=0\text{.}\)