February 14, 2007

How the four operations become two

Awhile back I posted about fraction division; that post made me think of reciprocals, algebra, and even properties of addition and multiplication.

You see, when you study abstract algebra, you get to concept of a group, or a ring, or a field - and all of those consist of a set of elements and one (group) or TWO (ring or field) operations.

Yet, in school math, we always study the FOUR basic operations.

Why are two operations enough in higher-level algebra? Where did they throw subtraction and division?

Addition and multiplication


Did you ever notice these similarities between addition and multiplication?

  • Both addition and multiplication are commutative:
    a + b = b + a and ab = ba for all real numbers a, b.
    Division and subtraction are not.

  • Boht addition and multiplication are associative:
    (a + b) + c = a + (b + c).
    Division and subtraction are not.


  • There exists an element so that adding it to a number doesn't change it (ADDITIVE IDENTITY). We call it zero:
    0 + x and x + 0 both end up being just x.

  • There also exists an element so that multiplying by it doesn't change the number (MULTIPLICATIVE IDENTITY). This is number one:
    1x and x1 both are just x.


You might a lready know these well... but how about the INVERSE:

  • For any real number x, there is a real number so that when you add the two, you will get the additive identity element or zero.(THE ADDITIVE INVERSE, or OPPOSITE)

  • For any real number x (excepting zero), there is another real number so that when you multiply those, you will get the multiplicative identity element or one. (THE MULTIPLICATIVE INVERSE, or RECIPROCAL)


For example, for 8 we have −8: if you add them, you get zero.
For -5.4 you have 5.4; if you add them, you get zero.

And with multiplication:
For 8 we have 1/8. If you multiply them, you get 1.
For -5.4 or -54/10, we have -10/54: if you multiply them, you get 1.

Of course we normally call those the opposite (with addition) and the reciprocal (with multiplication).

But notice how both are based on the similar principle: for each number you can find a "pair" so that when you do the operation (either add or multiply), you get that operation's identity element (zero or one).

Nothing like that is true at all about subtraction and division.


Getting "rid" of subtraction and division


Now, you don't really have to get rid of these two operations, but just observe how it CAN be done:

Simply define "division" as multiplying by the reciprocal... (doesn't that sound a bell?)

...and...

define "subtraction" as adding the opposite (that should sound a bell as well).

You are actually using these in elementary math as well. Just remember the fraction division; you're told to "invert and MULTIPLY" - to change the division to multiplication by the reciprocal.

Or, remember the rule about subtracting negative numbers; you're told that 8 − (−9) is actually adding 9... You're changing subtraction into addition of the opposite (opposite of −9 is 9).

But did you ever consider; that same thing is TRUE of EVERY DIVISION problem and EVERY SUBTRACTION problem!

For example:

24 ÷ 4 is the same as 24 × 1/4.
0.08 ÷ 0.02 is the same as 0.08 × 100/2.

10 − 3 is the same as 10 + (−3).
−1/5 − (−1/2) is the same as −1/5 + 1/2.

In both cases, the change is made in a similar way: the operation is changed to mult/add, and the number is changed to its multiplicative or additive inverse.

So instead of four operations, we can get by with two. And the two that are left have some nice properties (like we saw above).

I think all this is really neat!

(Don't worry though; just keep using subtraction and division in your everyday life and teach them to students, too. I just feel the TEACHER should know about all this, and then bring little bits and pieces of this underlying algebra structure in his/her teaching when appropriate.)
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