Connecting Algebra and GeometryDate: 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/ |
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