Difference between revisions of "Introduction" - Rhea

Revision as of 15:37, 2 December 2018

Introduction

The golden ratio is a ratio such that, given two quantities a and b, (a+b)/a=a/b We can solve this equation to find an explicit quantity for the ratio. LHS=a/a+b/a=1+b/a 1+b/a=a/b We set the ratio equal to a certain quantity given by r. r≡a/b Then we can solve for the ratio numerically. 1+1/r=r r+1=r^2 We can see from the above result that the golden ratio can also be described as a ratio such that in order to get the square of the ratio, you add one to the ratio. r^2-r-1=0 We can then apply the quadratic formula to solve for the roots of the equation. r=(1±√(1^2-4(1)(-1) ))/2=(1±√5)/2 The positive root is then the golden ratio. (1+√5)/2=1.618…≡ϕ The golden ratio, ϕ, is sometimes also called the golden mean or the golden section. The golden ratio can be frequently observed in man-made objects, though they are generally “imperfectly golden” – that is, the ratio is approximately the golden ratio, but not exactly. Some everyday examples include: credit cards, w/h=1.604, and laptop screens, w/h=1.602 (Tannenbaum 392). Visualizations of the golden ratio can be seen below (Weisstein):

Back to Daniel's Honor Project

Back to MA265 Fall 2011 Prof. Walther

Back to MA265

@@ Line 7: / Line 7: @@
 ----
-I, and I assume other students, often find learning experience enhanced when the context of the study, i.e. origin and the application, clearly visible. (e.g. Students like psychology class - it's about themselves) And it is with this intent of identifying the context of [http://en.wikipedia.org/wiki/Linear_algebra linear algebra] I sought to compile articles and notes for the project.
+The golden ratio is a ratio such that, given two quantities a and b,
+(a+b)/a=a/b
+We can solve this equation to find an explicit quantity for the ratio.
+LHS=a/a+b/a=1+b/a
++b/a=a/b
+We set the ratio equal to a certain quantity given by r.
+r≡a/b
+Then we can solve for the ratio numerically.
++1/r=r
+r+1=r^2
+We can see from the above result that the golden ratio can also be described as a ratio such that in order to get the square of the ratio, you add one to the ratio.
+r^2-r-1=0
+We can then apply the quadratic formula to solve for the roots of the equation.
+r=(1±√(1^2-4(1)(-1) ))/2=(1±√5)/2
+The positive root is then the golden ratio.
+(1+√5)/2=1.618…≡ϕ
+The golden ratio, ϕ, is sometimes also called the golden mean or the golden section. The golden ratio can be frequently observed in man-made objects, though they are generally “imperfectly golden” – that is, the ratio is approximately the golden ratio, but not exactly. Some everyday examples include: credit cards, w/h=1.604, and laptop screens, w/h=1.602 (Tannenbaum 392).
+Visualizations of the golden ratio can be seen below (Weisstein):
-I sought forth to identify an individual responsible for the invention of linear algebra, a progenitor of the principles of linear algebra as Newton was to Calculus [http://en.wikipedia.org/wiki/Leibniz_and_Newton_calculus_controversy (or was he?)]. I no longer think of doing so. Theorems and rules are attributed to single/group of mathematicians; linear algebra, I think, cannot. Rather, linear algebra appears to be a fundamental faculty in mathematics, and as I lend Professor Uli's words, "Linear algebra to mathematician is what addition and multiplication is to non-mathematicians." Perhaps as one cannot find a sole author of a particular language, one cannot find one for linear algebra.
-In the search for application, few applications of matrix theory are readily noted, including its use in the Google's PageRank algorithm and alternate representation of geometry/graphs. But I'm rather unsatisfied with such selections. What I mean by the previous statement is that I may have used and will use the term linear algebra and mathematics indiscriminately in my deciding the applications of linear algebra because I do not understand both well. I doubt that my chosen applications of linear algebra will be an application of linear algebra and nothing else, and am afraid that if I take such singular position, I will not be satisfied.  I don't plan on weighing too much of my time on the applications.
-Then, what will I actually do if inventor of algebra is undefinable and the application of linear algebra is unidentifiable? I will compile brief biography of intriguing mathematicians who may have had some say in the advancement of linear algebra. I may also try to write few articles on linear algebra and its historical development. But most importantly, I seek to write about mathematics- from a very normal college student's point of view. That would be my very own context in learning linear algebra.
-It is quite hard to make any statement regarding mathematics since I have only slightest idea of its nature, but I would like to attempt, for the sake of introducing what I think of mathematics:
-Perhaps the entire construct of mathematics may have begun with the sole purpose of application- to count the number of fruit, , share harvests, collect taxes. Mathematics was a reflection of the physical world into a numerical form that our logical minds can construe, and it was useful once the findings in this alternative universe was translated back into the language of the physical world. I think that mathematics has matured to the point where it has become more than a mirror image of the physical world, and reveals things that cannot be translated in the language of the physical world. It is complete.
-I find this following quote intriguing:
-"In adolescence, I hated life and was continually on the verge of suicide, from which, however, I was restrained by the desire to know more mathematics." - Bertrand Russell
-I don't really appreciate the study of mathematics as he does; I think I only like the idea of appreciating mathematics as I read the works and stories of great mathematicians. However, I do generally like mathematics. And I will find it pleasant to like mathematics much more, and this project will be one of many attempts to appreciate mathematics.
--LTY 5 Oct 2011
 ----
 [[Honors_Project|Back to Daniel's Honor Project]]

Difference between revisions of "Introduction" - Rhea

Revision as of 15:37, 2 December 2018

Introduction

Alumni Liaison