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Purpose of Algebra


Date: 10/15/96 at 8:23:59
From: Landon Stuckey
Subject: Purpose of Algebra

What is the purpose of algebra and who came up with the idea of 
algebra?

Sincerely,
Landon Stuckey          


Date: 10/16/96 at 14:0:42
From: Doctor Ceeks
Subject: Re: Purpose of Algebra

Hi,

Your second question is much easier to answer, so I'll start with that
one.  The idea of algebra has occurred to many in the past, and it is
unclear who was the very first to come up with the idea.

Okay, now to your first question.

Algebra is useful for many things.

It's like this.  You're studying some objects and how they behave and 
interact.  You notice some patterns that happen over and over.  You 
try to eliminate all the unnecessary details and grasp the essence of 
what is going on. You find out that you can describe what is going on 
by explaining some rules. You discover that these rules describe 
many, many things and imply some things you never thought of before.
You end up explaining lots of new things.  You have then created an 
algebra and used it with purpose.

(Technically, the word "algebra" is now used by mathematicians to mean 
a very special kind of thing, but that won't matter unless you go to 
graduate school in mathematics.)

Here's an example of the above.

You're playing with a square. You notice that you can move the square 
some ways without making it seem like you moved the square. For 
instance, you can flip it around one of its diagonals. Or, you can 
rotate it 90 degrees clockwise (or counter-clockwise). You notice 
that sometimes, you can follow such a motion by another such motion 
and get something new which doesn't affect the square. 

You want to understand these motions. You decide to simplify things 
by simply denoting the rotation by 90 degrees clockwise by R and the 
90 degree counter-clockwise rotation by L. You decide to write RR for 
the motion you get when you do R twice. You find that RRRR is really 
the same as doing absolutely nothing. You notice that RRR is the same 
as L, and LLL=R. You find that RR=LL. You find that you can't get the 
flip about the diagonal using just R and L, so you call it F.

You keep exploring, and "algebra" just naturally pours forth! 
Eventually you may discover everything you can do to a square. You 
may see that with just R and F, you can get everything! You may try 
to do this for a regular pentagon, then a regular hexagon...even, a 
regular n-gon! Are they really so different?

If you have fun doing this, and you do it a lot, welcome to the world
of algebra... you're an algebraist.

Usually, people first encounter algebra when they study the algebra
of numbers. So you may be more used to seeing things like 3x+5=y, 
where x and y stand for numbers. The algebraic rules numbers obey are

  commutativity:     x+y = y+x, 
  associativity: (x+y)+z = x+(y+z), and 
 distributivity:  x(y+z) = xy+xz.  

There's also a special number, called zero, and denoted 0.  It has the 
property that x+0=x.)

-Doctor Ceeks,  The Math Forum
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