1.9 Introduction and Key Insights

Our "picture of arithmetic" is almost finished. We have our real numbers and operations for combining them ($+$, $-$, $\times$, $\div$), and we have learned about some properties these operations enjoy. But our last puzzle piece asks: How do these operations interact with one another?

Addition and subtraction interact in understandable ways: these are inverse operations. The same is true for multiplication and division. But how do addition and multiplication interact?

Our answer is encoded in the distributive law for numbers. This property tells us how to distribute a multiplication across a sum (we write the sum in parentheses). For example, we can use the distributive property to rewrite $2\times (3+5)$ as $(2\times3)+(2\times5)$. The answer is the same, and writing it this way makes it easier to simplify large calculations and figure out the answer without having to write it down (mental math).

Here is an abstract statement of the distributive law:

$a\times(b+c)=(a\times b)+(a\times c)$

As its name suggests, we are distributing the multiplied number $a$ to each number in the sum.