At a Glance - Associative Properties
Another rule for rearrangement comes from 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 of the property. This rule says that
(x + y) + z = x + (y + z), and
(xy)z = x(yz).
This is another one of these straightforward rules for rearrangement that we use all of the time. It is a fine line between obvious and oblivious, and we wanted to make sure.
Consider when you need to add a lot of numbers together. Say you have 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 will to keep them all.
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9
We usually rearrange this line and group the sums together so that we add numbers that 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
That is 45 total Christmassy things. You must be getting into the holiday spirit. Even if the roof of your car is covered in French hen leavings.