The Golden Ratio

The Golden Ratio is a fascinating number which can be found almost everywhere –from nature to architecture to art. To 18 decimal places, it has a value of 1. 618033988749894848 but is usually shortened to 1. 618 much like ? is usually rounded off to 3. 1416 (Powis, n. d. ). Signified by the letter Phi (? ), the Golden Ratio can be simply defined as “to square it, you just add 1” (Knott, 2007). Written in mathematical equation, this definition becomes ? 2 = ? + 1. When the resulting quadratic equation ? 2- ? – 1=0 is solved, there are two solutions: 1. 6180339887… and -0. 6180339887…. Notice that the two solutions have identical decimal parts. The positive…

Mathematics & Natural Sciences with absolute certainty (TOK)

Write an essay outlining your personal response to this topic. “Mathematicians have the concept of rigorous proof, which leads to knowing something with complete certainty. Consider the extent to which complete certainty might be achievable in mathematics and the natural sciences.” To what extent can man use mathematics and the natural sciences to embrace the concept of achieving absolute certainty? It is not possible for humans to achieve absolute certainty in knowledge using mathematics and the natural sciences. This is because mathematics is a creation of man to organize and communicate highly complex concepts and theories to others through a kind of language which goes beyond the spoken or written word. Many people believe the written word to be more…