First Math Teacher, Pythagorean Theorem

Date: 6 Jan 1995 12:04:13 -0500
From: Jessica Corbin
Subject: The very first math teacher

Dear Dr. Math,

Hi - my name is Jessica and I'm a student in Monta Vista High School's 
Internet class. For our final exam, part of our assignment is to write to 
either a scientist, geologist, or a mathemetician. I have a couple of 
questions for you.

First off. . .since I was in first grade, in my very first math class I 
wondered who the first math teacher was. How did he/she learn everything 
they knew? Who taught them? It's kind of like the question about the 
chicken and the egg. It's okay if you're unsure, but I didn't think that 
a common math book would have the answer and it may be a fun question 
to answer.

My other question is about the Pythagorean Theorem. I still don't 
understand how Pythagoras figured out that a to the second power plus b to the 
second power equals c to the second power in a right triangle.  Do you 
think he just substituted numbers and said to himself "Wow I just figured 
out a new theorem!"?

Sincerely,
Jessica Corbin

Date: 10 Jan 1995 04:53:18 -0500
From: Dr. Sydney
Subject: Re: The very first math teacher

Dear Jessica, 

Hello!  Thanks for writing Dr. Math!  You asked some great
questions.  

There isn't really a clear-cut answer to your first question.  You
see, there were "teachers" of math long before there were even schools.
Math is a subject that has been around just about as long as human 
beings!  There really was no ONE person who first came up 
with "math." It was kind of an evolutionary kind of thing.  Some say math 
started with the study of patterns, while others say the origins of math 
can be found in counting.  Regardless, ideas in math developed through 
discussion and communication among thinkers, where there was no one 
designated  "teacher."  So, as you can see there really is no "first math 
teacher."  From the beginning of math history, there have been countless 
teachers, all seeking to better understand the deep, wonderful, complicated 
field of mathematics.  

I'm glad you asked about proving the Pythagorean theorem.  There are 
actually many proofs of the Pythagorean Theorem, so I'll just show 
you one of the most famous proofs which was done by a mathematician 
named Euclid who lived around 300 B.C.E.

First, draw a right triangle ACB, with the right angle at C, and
then draw squares along each side of the triangle.  Side AC has
a square sticking out of it with sides of the same length as side AC.  The
other vertices of the square are K and H.

Likewise there is a square with sides the same length as side CB 
- the vertices of this square are G and F, and there is a square
with sides the same length as AB whose vertices are D and E.  Now drop a
perpendicular line from C to the line CE.  The point where this line
intersects AB is X' and the point where this line intersects CE is X. 

Now, you should have a picture.  The basic idea of the proof is to 
show that the sum of the areas of the two smaller squares (the
ones with side lengths equal to side lengths of the legs of the triangle) is
equal to the area of the biggest square.  If we show this is true, we've
proven the Pythagorean Theorem. Do you see why?

I'll give you the basic steps, and you can try to figure out the reasons 
for the steps. Okay, here we go!

First show the area of the square with length AC is twice the area of
the triangle ABK.  Second, show the triangles ABK and ACD are 
congruent. Third, show the area of rectangle ADXX' is twice the 
area of triangle ACD, thus the area of the square with sides the 
length of AC is equal to the area of rectangle ADXX'.  

Follow similar steps to show that area of the square with sides the
length of BC is equal to the area of rectangle BX'XE.  Then that means that
the sum of the areas of the two squares with sides of length AC and BC is
equal to the area of the square with side length AB.

I'm not sure how much or how recently you've had geometry, but the
reasoning should follow from a high school geometry course.  I would 
suggest drawing a picture, labeling everything, and then trying the steps of
Euclid's proof.  Write back if you get stuck somewhere!

As it turns out, Pythagoreas was NOT the first person to figure out
that a^2 + b^2 = c^2 where and be are legs and c is the hypotenuse of a
right triangle.  Around 2000 B.C.E. , the ancient Egyptians discovered the
"magic 3-4-5" triangle (that is, a triangle whose legs have lengths 3 and 4 
and whose hypotenuse has length 5).  Later, around 500 B.C.E., the 
Pythagoreans, a group of Greek philosophers, began to think about the 
triangle's sides as sides of 3 squares, as Euclid does in the proof.

I hope this helps.  Please do write back if you have any more
questions!  Happy New Year to you!

--Sydney, Dr. "math rocks" Foster