MTH 405 Midterm
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 important part of the equation itself. While you can compute a slope given points, it is more straightforward to count rise and run to get a slope. The geometrical picture of a line gives you slope, tells you whether a line has positive or negative slope, telling you a lot more in less time it takes to calculate them.
The author in the book describes Euclid’s Fifth postulate as being controversial in a sense that it did not follow in line with his other work, it required more than words to fully grasp the intent and direction of the proof, and did not seem as being a certainty. In addition to that, the fifth postulate has avoided several attempts to be proven as a theorem. Euclid himself seemed reluctant to use the parallel postulate, perhaps alluding to his own thoughts on the proof. Perhaps the most famous discovery resulting from the parallel postulate is Euclid’s twenty-ninth proposition dealing with alternate angles. After this little bit of indulgence, Euclid went on to use the parallel postulate in all but one of the remaining propositions contained in the first book. Yet another important theorem that was heavily influenced by the parallel postulate was the theorem that stated that the total size of angles in a triangle equals two right angles, or 180 degrees. The fifth postulate, controversial as it was, went on to provide the groundwork for these two theorems that we now recognize as foundations for geometry.
3. a) x2 +2x + 1
b) 3x2 - 8 – 3
c) x3 – x2 + 2x + 2
d) x4 – 3x2 – 4
In terms of the first question, the Greeks had no concept of integers. While they could grasp the ideas of whole numbers, negative numbers were discouraged and did not feature in any of their works. b. Rational numbers would have come into play early in history. A formula like that for the area of a triangle has a rational number, so early mathematicians would have had a concept of rational numbers. c. The real numbers would have been a step up in...
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