Making Arrangements

Okay, I've used some of this earlier, but to increase my knowledge, I  have had to "relearn" he technical terms.  While the labels we put on don't really matter, the concepts do.

The most important thing I like to stress about math is knowing how things work.  However, if someone ever asks, there are three "properties" in math that can make things a whole lot easier for those wishing to take control of the numbers they face every day.

Commutative Property

A+B = B+A

A*B = B*A

Yes, this is the very first thing of which I said I am quite certain.  It is one of the undeniable facts of nature.  When it comes to numbers, it makes no difference what order they are in when adding or multiplying.  If you are ever asked what this property is called, just remember the "commute" to work (or school, or wherever).  The numbers are just "moved around."

For this reason, those really scary tables with scores of numbers in them can be reduced in half!  It is a very convenient and time saving fact.  Besides that, it helps you around most multiplication roadblocks.  When memorizing the tables, it is no coincidence that you get the idea that you've seen that "answer" before.  I like to just use the bottom triangular half of the table.

One number says it all on the multiplication table: 49

I almost made that number a separate blog.  It is the square of the number 7, which means it is 7x7.  Anyone who has ever memorized the dreaded "Times Tables" has faced the difficulty of "the sevens."  It is hard to visualize, and difficult to "count by," multiples of 7.  They just about HAVE to be memorized.  That is why the commutative property is so cool.  You don't have to use the "sevens" if you use the OTHER number instead.  Except for one time.  Memorize the fact that 7x7 = 49.

Associative Property

(A+B)+C = A+(B+C)

The associative property is an extension of the commutative property.  It takes advantage of the fact that we can work only with two numbers at a time.  By getting to "round" numbers one can be more confident with the answer.

2+47+64+31+72+99+44+63 = (47+63)+(31+99)+(64+2)+44 
= 110 + 130 + 66 + 44 = 240 + 110 = 350

7 x 9 x 3 x 2 x 5 x 6 = (7x3)x(9x6)x(2x5) = 21 x 54 x 10 = 540 x 21 = ????

So, with multiplication, associative properties only get you so far.  As numbers get larger, another concept is needed to reach the answer. Old fashioned arithmetic has us stack the numbers and then distribute the task using on digit at a time.  That brings us to the next property.

Distributive Property

A(B+C) = AxB + AxC

This property, in which multiplication is spread out over several steps, is the method used practically on scrap paper across the world.  Take the unsolved product of  "540 x 21" for example.

540

x21

The process of distribution is not evident, but it happens as you multiply by 1 and then by 20.  Written in distributive form it looks like this (54)(20 +1). 

(540)(20+1) = (540*20)+(540*1) = 10800 + 540 = 11,340 

Where the distributive property comes in real handy is when calculating products near 10 (8,9,11 and 12).  Though 11 and 12 are derivative of basic facts, they can be bypassed when using a little math. Some math "short cuts" are as follows:

3 = 2 + 1

7 = 5 + 2

8 = 10-2

9 = 10-1

11= 10+1

12 = 10+2

So the take away is that using this property of multiplication, one can "divide and conquer."  One example and then I will let the reader's brain rest.  I will now "randomly" chose two large numbers to multiply.

Let me see, lets do a three digit number by a two digit number, but a little harder than the one above.  I'm randomly picking "794" and "69."

So, a slip of a finger (I meant "8") and blind luck give me 794 x 69.

794 x 69 = (800 - 6)(70 - 1) = [(800x70) -  (6x70)] - 800 -(-6) {I confess, I took a short cut here}

= 56000 - 420 - 800 + 6 = 56,006 - 1220 = 54,786

I really didn't mean to be that complicated, but this is about "what I know."  So using both distributive and associative properties, I took the long way around.  Using rounding, I was able to redistribute the numbers and work inside my head.  

Using the old fashioned arithmetic would be faster, but I showed that I know these three handy properties of numbers!

So, what do I know?

AxB = BxA

A+B = A+B

(A+B)+C = A+(B+C)

A(B+C) = AxB + AxC

Next: Inverse Behavior