Lesson Nine: In Which We Separate

We have seen that multiplication provides us a powerful tool for adding a number to itself many times. However, suppose we mix things together that we ought have left alone, how can we get back to our original quantity? Asked another way, how do we undo multiplication?

Suppose 40 students arrive in each of 5 buses to their school in the morning. At this point the school has 40(5) = 200 students in it. However, one of them burns the building down, and now all the students need to be sent home, how many should go on each bus so that there are the same number of students on each bus? Because we counted them in the morning, we already know the answer is 40 students, but we would like a way to reach that answer without knowing it before hand.

Reciprocals:

Remember when we needed to undo addition we introduced new numbers, the negatives, which undid addition. This worked because n+(-n) = 0, and adding zero to a number does not change that number. We do a similar thing to undo multiplication.

Since multiplying by 1 does not change a number, to undo multiplying by n, we need another number which, when multiplied by n, becomes one. Then these two numbers will cancel each other out. We call this new number the reciprocal of n, and write it as (1/n). So far all we know is that n(1/n)=1.

However, we can quickly discern that reciprocals should keep multiplication commutative and associative. If we think of m*(1/n) as separating m things into n even groups, we can think of (1/n)*m = 1*(1/n)*m as cutting one thing into n pieces, then taking m of those pieces. Suppose you have two apples to share between three people, each person should get 2(1/3) of an apple. But on easy way to make sure that this happens is that you could share each apple between all three people individually, then each person gets 1*(1/3)*2 of an apple. While the pieces may appear different, the amount of apple they represent will be the same.

Suppose you have two cartons of a dozen eggs each and you use all the eggs in one of the cartons to make a cake. Now suppose you have the same two cartons, but you decide to take half the eggs from each carton for your cake, either way you use 12 eggs. The first way you have (2*1/2)*12 eggs and the second you have 2*(1/2*12) eggs.

Fractions:

Because writing m(1/n) all the time seems like a waste of space, we condense this to m/n. This is what we call a fraction. It has a numerator, which is the number before the slash, and a denominator, which is the number after the slash. If you write a fraction vertically the numerator is on top and the denominator is on bottom.

Suppose you want to multiply to fractions, what is (m/n)*(a/b)? We know we may rewrite this as m(1/n)*a(1/b) because this is how fractions are defined. Since multiplication remains commutative, this is certainly ma(1/n)*(1/b) so we need to consider what it means to separate something into n pieces then further separate into b pieces. Suppose we try to put something back together after it has been separated into n pieces then each piece into b further pieces. Each of the n pieces is in b smaller pieces, so the total number of pieces is n*b. Thus separating something into n pieces then each piece into b pieces is the same as separating the whole thing into n*b pieces. This leads us to the rather convenient conclusion that (1/n)*(1/b) ought to be 1/(nb). So (m/n)*(a/b) = (ma)/(nb). Unfortunately, addition will not work out so conveniently as we will soon see.