Abstract Algebra
MTH 405 Midterm
16/3/2011
1. A specific area for the area of various polygons is the one for the area of a regular polygon. The setup and initial steps to creating the proof require a geometric approach that would otherwise make proving a big challenge. For example, a polygon with n sides is broken up into a collection of n congruent triangles, this geometric setup is key in reaching an easy solution for the area. The algebraic aspect comes into play when it comes to deriving the equation for the area. It is a simple yet important step in the whole proof, the icing on the cake, algebra and geometry playing equal parts. In a similar manner, algebraic formulas can also be derived from geometric diagrams. A good example would be the conics. One can’t imagine the conics without their respective geometric diagrams. Not only is geometry tied into algebra in that sense, but the fact that the curves had been under scrutiny by the Greeks, the greatest exponents of geometry, shows their inclination toward some algebra. Numerical approximations for pi also heavily featured geometry in an otherwise arithmetic branch of mathematics. Pi itself is a constant, unlikely, one would think, to have anything remotely to do with geometry. But attempts to find and approximate the value of pi have used geometry. Ptolemy of Alexandria used a 360 sided polygon inscribed within a circle to arrive at an estimate for pi that was very accurate for his time. Algebraic equations largely make use of geometric arguments because there is such a strong union between them both. Most algebraic equations require some form of geometrical interpretation to make sense of things and it is a tool that is still features heavily today. Most minds find it easier to process images that to make sense of figures, which is why algebra relies on geometry to explain, and geometry relies on algebra to process. The equation of a line is a simple enough example of this. A line has a slope, which is the
16/3/2011
1. A specific area for the area of various polygons is the one for the area of a regular polygon. The setup and initial steps to creating the proof require a geometric approach that would otherwise make proving a big challenge. For example, a polygon with n sides is broken up into a collection of n congruent triangles, this geometric setup is key in reaching an easy solution for the area. The algebraic aspect comes into play when it comes to deriving the equation for the area. It is a simple yet important step in the whole proof, the icing on the cake, algebra and geometry playing equal parts. In a similar manner, algebraic formulas can also be derived from geometric diagrams. A good example would be the conics. One can’t imagine the conics without their respective geometric diagrams. Not only is geometry tied into algebra in that sense, but the fact that the curves had been under scrutiny by the Greeks, the greatest exponents of geometry, shows their inclination toward some algebra. Numerical approximations for pi also heavily featured geometry in an otherwise arithmetic branch of mathematics. Pi itself is a constant, unlikely, one would think, to have anything remotely to do with geometry. But attempts to find and approximate the value of pi have used geometry. Ptolemy of Alexandria used a 360 sided polygon inscribed within a circle to arrive at an estimate for pi that was very accurate for his time. Algebraic equations largely make use of geometric arguments because there is such a strong union between them both. Most algebraic equations require some form of geometrical interpretation to make sense of things and it is a tool that is still features heavily today. Most minds find it easier to process images that to make sense of figures, which is why algebra relies on geometry to explain, and geometry relies on algebra to process. The equation of a line is a simple enough example of this. A line has a slope, which is the