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Section 20.4 Closing Ideas

We opened this section with the idea that multiplication and division were related by thinking about how taking 10 objects and splitting it into two equal groups looks the same as taking half of a group of 10 objects. But we didn't elaborate on the nature of that relationship.

Multiplication and division are known as inverse operations. Basically, it means that one operation undoes the other. We've actually already seen this idea, but without those words. When we were solving equations, we would run into situations that look like the following:

\begin{equation*} \begin{aligned} \frac{x}{3} \amp = 4 \\ x \amp = 12 \amp \eqnspacer \amp \text{Multiply both sides by $3$} \end{aligned} \end{equation*}

The act of dividing both sides by is undoing the multiplication of by on the left side of the equation. In fact, we can do the same thing when solving equations that involve fractions.

\begin{equation*} \begin{aligned} \frac{x}{3} \amp = 4 \\ x \amp = 12 \amp \eqnspacer \amp \text{Multiply both sides by $3$} \end{aligned} \end{equation*}

The idea that multiplication is the inverse of division and that division is multiplication by the reciprocal has another important parallel. We run into the exact same situation with addition and subtraction. When solving equations, you have to subtract to undo addition and you have to add to undo subtraction.

\begin{equation*} \begin{aligned} x + 4 \amp = 9 \\ x \amp = 5 \amp \eqnspacer \amp \text{Subtract $4$ from both sides} \end{aligned} \end{equation*}
\begin{equation*} \begin{aligned} x + 4 \amp = 9 \\ x \amp = 5 \amp \eqnspacer \amp \text{Subtract $4$ from both sides} \end{aligned} \end{equation*}

It is often said that subtraction is addition of the opposite. This phrase is very close to our division phrase. We will put these phrases next to each other to see the comparison.

\begin{equation*} \begin{array}{ccccc} \text{Division} \amp \text{is} \amp \text{multiplication} \amp \text{by} \amp \text{the reciprocal.} \\ \text{Subtraction} \amp \text{is} \amp \text{addition} \amp \text{of} \amp \text{the opposite.} \end{array} \end{equation*}

This shows us that addition and multiplication are the fundamental operations on numbers. In some ways, this may help to explain why addition is easier than subtraction and why multiplication is easier than division. Some operations are just more basic and more fundamental than others. These ideas are also at the core of an area of mathematics known as field theory, which is built on many of the ideas that we've already encountered. So it turns out that high level mathematics has its roots in things that we teach to all students.