What Is a Theorem and Why Are They Important?

Date: 08/15/97 at 13:09:23
From: Kristin
Subject: Geometry

This is more of a why question. My teacher says that theorems are very 
important when working with Geometry. I don't understand why they 
help us learn. Maybe if I understood them I would be more willing to 
understand and work with them.  

Can you help me?  I am really lost.

Thanks a lot.
Kristin

Date: 08/18/97 at 11:13:26
From: Doctor Ceeks
Subject: Re: Geometry

Hi,

You've asked a very difficult question!

I guess to begin with, it's important to know what a theorem is.

In Geometry (and mathematics, in general), people noticed facts
which were indisputably true.  This is remarkable because in most
other human activities, facts which are indisputably true are rare.

But in math, people noticed definite things which are always true.

Sometimes a person who noticed such a fact would show the fact to
a friend and say, "look, I've just noticed this truth... can you see 
it?" And the friend sometimes would wonder and wonder, and it wouldn't
be so obvious that the fact was true. These facts came to be termed
"theorems."

Here is an example: Take any triangle. The sum of the three angles of 
the triangle is always 180 degrees.

Before the person who first noticed this fact came along, people 
didn't even think that the sum of the angles of a triangle was any 
specific number of degrees. Then some person comes along and says, 
"guess what? If you add up the three angles of a triangle, it ALWAYS 
comes out to be 180 degrees!" This person's friends surely responded 
with skepticism, "really? I don't believe it...are you sure?"

Over the centuries, people have discovered many, many such theorems.

In Geometry, you learn many theorems which are concerned with points, 
lines, triangles, circles, parallelograms, and other figures.

Now, when you learn a theorem, you are learning an absolute truth.
You learn something about the way things are. Later on, you can use 
these facts to notice other things.

If you wonder, "why is this theorem true?" and you think about it
and figure it out (which is what people mean when they say that they
"prove" a theorem), then you have learned a lot. You can read a book
which explains why a theorem is true to figure it out, and by doing
so, you will learn many other things aside from the theorem, because
it takes other facts to prove a theorem.

If, however, you don't read a book to figure out why a theorem is true
and instead figure it all out for yourself... that is, if you convince
yourself beyond a shadow of a doubt that the theorem is true, then you
learn something very precious, which is how to think on your own.

By the way, proving a theorem all on your own isn't easy and even 
people who make a living trying to do that will often need help from 
others to succeed.  Often you get stuck trying to prove a theorem.

Don't forget that the theorems you are learning in less than a year
in Geometry took humanity hundreds of centuries to discover and prove!

To summarize, theorems give you the opportunity to learn new facts and
even have the potential to give you the opportunity to learn how to 
think.

Please write back if you have more questions or if there's a specific
theorem that you don't understand and wish to.

-Doctor Ceeks,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/