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Associative Property

Another rule for rearrangement shows up when we add or multiply more than two terms. If you ever have occasion to introduce these terms to anyone in a business setting, you can make yourself sound über-professional by referring to them as your "associates." Hence the name: the associative property. Here's what it looks like for addition:

(x + y) + z = x + (y + z)

And it also works for multiplication:

(xy)z = x(yz)

This is another one of these straightforward rules for rearrangement that we use all the time. It's a fine line between obvious and oblivious, and we wanted to make sure.

Sample Problem

Say you've got nine ladies dancing, eight maids a-milking, seven swans a-swimming, six geese a-laying, five golden rings, four calling birds, three French hens, two turtle doves, and a partridge in a pear tree, and you want to know how many total things that is. Not to mention where in the world you're gonna keep them all.

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9

Now we can rearrange all those terms and group together the numbers are straightforward to add. We rewrite using commutativity and associativity to get:

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 =
(1 + 9) + (2 + 8) + (3 + 7) + (4 + 6) + 5 =
10 + 10 + 10 + 10 + 5 = 45

That's 45 total Christmas-y things. You must be getting into the holiday spirit. Even if the roof of your car is covered in French hen droppings.

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