Lesson Seven: Multiplication and Addition

Now we know how to combine two numbers with multiplication, and we know how to combine two numbers with addition, but how do multiplication and addition interact?

Multiplication First:

It is an unfortunate truth that multiplication and addition do not get along as well with each other as addition gets along with addition and multiplication gets along with multiplication. Since both are associative 2+3+4 is unambiguously 9, and 2*3*4 is unambiguously 24. However 2+3*4 takes different values depending on if you interpret it as (2+3)*4, which is 20, or 2+(3*4), which is 14. So, to resolve this ambiguity, we decree that multiplication occurs before addition. This is a common convention upon which we agree, rather than a mathematical truth which we discover, but we should stick with it nonetheless. So, things within parenthesis happen before things outside of them and multiplication happens before addition, these are conventions by which we must agree to live.

Distribution

The other interesting way in which addition and multiplication interact is called distribution. Consider 2(3+4), which is 2*7 or 14. This is the same value as 2*3+2*4 which, since we know to multiply before we add, is 6+8 or still 14. This is not a coincidence, when we switch from doing addition first to doing multiplication first, we must make sure that we multiply both values in the sum by 2. Consider if you had two bank accounts, into one you deposited a certain amount of money each month to save up for a vacation, and into the other you deposited a different amount of money to save for a rainy day. If, after three months, you wished to know how much money was in the accounts total, you could find out in two different ways. You could first figure out how much you deposited each month, by adding the amounts going into each account together, then multiply this monthly deposit size by 3, this way you add first. You could also figure out how much money was in each account individually after three months, by multiplying the deposit to each account by 3 separately, then add these account totals together to obtain the overall total, this way you multiply first. Notice that in the second case, the amount of money you put into the first account must be multiplied by 3 and the amount you put into the second account must be multiplied by 3, we say that the 3 is distributed to each of them.

A Note on Order

When it is necessary or useful to indicate that something is happening before something else, I shall enclose the things to be done first in parentheses. For example, I might have broken down 3+4 into (1+1+1)+(1+1+1+1), indicating that you can recover 3+4 by adding the first three 1's and the last four 1's together before you finally add their sums. Of course, since each time you add a one you simply move to the next number, no matter what order you add them, seven ones must add up to 7. This is why we may write 1+1+1+1+1+1+1 unambiguously, it does not matter how you add them. Since you can do addition in any order, this means that 1+(2+3)=(1+2)+3. It doesn't matter whether the 2 is grouped, or associated, with the 1 or the 3, the sum is still 6. Because addition works the same no matter how the numbers are associated, we say that addition is associative.