Let's begin by showing that two numbers can be added in either order; the total is the same. In other words, A+B = B+A. Don't confuse the following "reasoning" with a proof. The actual proof is part of set theory, and is beyond the scope of this topic.
Imagine you have two bags of marbles and a bowl. You want to know how many marbles you have all together. You empty the two bags into the bowl and count. It doesn't matter which bag you pour in first. Marbles cannot be created or destroyed, so you know you'll have the same amount whether you add A to B or B to A. This is the commutative property of addition, from the word commute, which means to change places.
If you have three bags of marbles, you can pour the first two into the bowl, and then add the third, or you can empty the first bag into the bowl, combine the second and third bags together, then pour that mix into the bowl. In this example, marbles enter the bowl in essentially the same order, A then B then C, but you might group A+B together first, then pile C on top of that, or pile B+C on top of A. We aren't changing the order of the summands, we're just grouping them together in different ways. No matter; the total is the same. This is the associative property of addition.
When a grid is used to illustrate multiplication, it is obviously commutative. Place cookies on a baking tray in a 3 by 4 grid. It doesn't matter if you turn the tray 90°, there are still 12 cookies on the tray. Thus 3×4 = 4×3 = 12.
To explore associativity, we need three dimensions, for the three operands. Place uniform sugar cubes in a box and realize that the number of cubes is the same, even if you turn the box on its side. (A×B)×C = A×(B×C).
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Finally we distribute multiplication over addition.
Let a tray of cookies have 4 rows and 7 columns.
Draw a vertical line to the right of the second column.
We can now add 2 and 5 to get 7 columns, then multiply by 4 rows,
or we can add 4 times 2 + 4 times 5.
The number of cookies is the same.
Written symbolically,
A×(B+C) = A×B + A×C.
These "obvious" properties extend to fractions. Some of the bags of marbles could contain half marbles or quarter marbles; the order of addition still doesn't matter. The rightmost column on a tray of cookies might contain only half cookies; the order of multiplication still doesn't matter. turn the tray; four times six.5 is the same as six.5 times four. |
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Eventually we will explore an abstract structure that is both commutative and associative, just like the integers. This makes it easy to generalize. Whenever we encounter a collection of objects that has the properties of the integers, it behaves like the integers in every way. We get this for free, simply by applying our abstract structure; we don't have to prove it all over again. This is like writing one subroutine in a computer program and calling it from dozens of places, instead of writing the same code over and over again with minor variations. This "abstract" approach is the essence of "modern mathematics", but again, it is beyond the scope of this article. You can read all about it in groups and rings and fields. For now, let's stick with the integers and see where they lead.