Tuesday, August 12, 2008

Forget Aunt Sally, I've got a GEMS for you!

No, I don't want you to neglect your relatives. You probably do that enough on your own without my encouragement. As a matter of fact, you ought to contact some of those relatives that you have been neglecting that would love to hear from you.

Actually, what I'm talking about is that mnemonic for remembering the order of operations in algebra: PEMDAS, or Please Excuse My Dear Aunt Sally. Math teachers offer that up as an easy way to remember Parentheses, Exponents, Multiplication, Division, Addition and Subtraction.

Instead of PEMDAS, I propose a shorter mnemonic, GEMS. Before I explain what GEMS stands for, I have to show you a different way of looking at the basic arithmetic operations.

In basic arithmetic, we learn to associate numbers with amounts of things: 4 apples, 3 oranges, 5 peanut butter sandwiches, for example. In algebra and geometry, numbers are associated with displacement. In algebra, the displacement is along a line, such as an x-, y-, or z-axis. In geometry, degrees and radians are displacement along a circle.

For now, let's just focus on the linear displacement.

You can draw a number line, with the left end representing zero (0). Now you can represent addition graphically with this number line. Let's say you want to add 3 and 5. First you count 3 units to the right of zero, then from that location, you can count 5 more units to the right to arrive at the answer of 8.

Now, I know that is a very simple illustration, but it shows how addition works linearly. Using this linear concept, we can see that the other basic operations are forms of addition.

Now draw another number line, with zero at the center. Numbers to the left of zero are negative. This allows us to see subtraction as negative addition, or the addition of a negative. Let's say we want to subtract 5 from 3. We count 3 units to the right of 0 to 3, then count 5 units to the left of 3 to arrive at the answer of -2.

If you've taken basic algebra, I've not really introduced anything new yet. What I am going to introduce that is new is the way to look at multiplication and division as multiplex operations.

Let's say you are going to use the number line for multiplying 4 and 3. Starting from zero, you could count to the right 4 units, then 4 more units, and finally another 4 units. In other words, you add three 4's. You could also count 3 units to the right 4 times. Multiplication is repeated, or multiple addition. One factor gives the number of times the other number is added.

If multiplication is multiple addition, then division is multiple subtraction. How many times can you subtract 4 from 12? Find 12 on the number line and count 4 units to the left, and repeat until you reach zero.

So, now I can say that GEMS stands for Groupings, Exponents, Multiplex, and Simplex.

Grouping symbols are parentheses, ( ), brackets, [ ], and braces, { }.

Exponents include powers, roots, and logarithms.

Multiplex operations are multiplication and division.

Simplex operations are addition and subtraction.

Now isn't GEMS easier to remember than PEMDAS?

If your name is Sally, and you are somebody's aunt, please don't take this personally.

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