Connecting Algebra and Geometry

Date: 07/11/2002 at 11:58:16
From: Laura Rightmeier
Subject: connection between algebra and geometry

What are the mathematical connections between algebra and geometry? I 
have come up with graphing equations and the Pythagorean Theorem, but 
that's about it. What else is there?

Date: 07/12/2002 at 08:52:24
From: Doctor Peterson
Subject: Re: connection between algebra and geometry

Hi, Laura.

Just to list some general categories rather than specifics, here are 
some connections I see:

1. Geometric formulas: Area, perimeter, etc. are expressed in 
algebraic form. The Pythagorean theorem can be put in this category.

2. Algebraic proofs of geometric theorems: Many proofs today use 
algebraic manipulation, such as starting with a similarity, 
expressing that as an equation, and deriving further equations to 
show ultimately that two lengths are equal. Many proofs of the 
Pythagorean theorem work this way, and many others use that as one of 
the equations they start with. In particular, those area and volume 
formulas are often found by algebra, including calculus (which is 
really just advanced algebra).

3. Analytic geometry: We can describe a geometric figure in terms of 
coordinates, and use that to do proofs in a way very different from 
the  previous category. For example, rather than starting with a 
similarity, we might start with the equations of two lines and find 
their intersection, to do the same proof.

4. Graphing: Going the other way, we can start with an algebraic 
problem and represent it graphically, solving an equation by finding 
a geometric intersection.

5. Geometric representation of an expression: The Greeks used 
geometry to do all their "algebra"; a quadratic equation, for 
example, would be discussed as a question about squares and 
rectangles rather than powers and products.

If you look through history, as this last item suggests, you will 
find that the line between geometry and algebra is often very thin. I 
find, in looking at questions we get at Dr. Math, that I often can't 
tell whether a student is taking algebra or geometry: is this 
question using algebra to find the answer to a geometry problem, or 
is it using geometry as the basis for an algebra problem?

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/