Pythagorean Theorem

PythagoreanTheorem

  1. The : History and Motivation.

Pythagoraswas an ancient philosopher who spent most of his life exploring theuniverse. He managed to crisscross the ancient Babylon, India andeven Egypt (Pythagoras of Samos 1). This influenced his way ofthinking. Pythagoras had a group of other followers who believed inhis way of thinking. Together they formed the group calledPythagoreans. According to their philosophy, everything that existedon the planet had an underlying mathematical concept (Morris 1).

Evolutionof algebra, as a result of modernization in 1600CE, caused thegeometric statement to assume its algebraic form. Pythagoras devisedthe popular theorem as a geometric statement in about 2,500 years ago(Morris 1). He later managed to prove the basis of this equation andthus immortalizing its contemporary existence and use.

Hestated that: x + y = z where z is the hypotenuse and x and yrepresents the vertical sides of a right angle triangle (Morris 1).

Thisis because, in a right angle triangle, the length of one side of thetriangle that is represented by numerical value z, is a sum of theother two other sides that are represented by numerical values x andy. The longer side of the right angle triangle is called thehypotenuse. Hypotenuse stems from Greek words hypo that meansbeneath, and teinen, meaning ‘to stretch’ (Morris 1). Thetheorem, therefore, makes sense when the triangle is placed with thehypotenuse at the bottom as outlined in Euclid`s Element (Maor 36).

  1. The context of Pythagoras theorem

Pythagoraswas, however, not the first inventor of this mathematical theory.Initially, about a thousand years before Pythagoras, the theorem waswell in use amongst the Chinese and Babylonians. They had numerouscollections of geometric facts for solving problems. They had theability to solve several numerical problems even though theirapproach was based on determining areas using broken lines andrectangular shapes. As archived from Egyptian and Babylonian tablets,problems were solved even though there was no clear-cut methodologyto their approach (Edwards 7).

Pythagorastheorem was recognized for its beautiful nature in 1895. Thisrecognition was done by the talented mathematician Lewis Carrol(Pythagoras of Samos 1). He was amazed at the splendid beauty thatthe Pythagoras theorem displays because of its versatile nature tosolve many geometric problems. This notion was later assumed in 2004when Physics World Journal shed more recognition on the theorem. Thisis because Pythagoras theorem was voted as the fourth most beautifulformulae by its avid readers. According to modern mathematicians,the alluded beauty of Pythagoras theorem lied on the criteria ofsymmetry that it employs (Maor 36). This is mainly because thetriangle’s three altitudes always join at one point even duringangle bisectors and medians. This aspect of the equation has anallured elegance because of the symmetry nature. This is true inmodern eyes the triangle has been recognized as one horizontal legwith the other two sides oriented vertically. There is, therefore, away by which the hypotenuse can leap out as a square at odd angles.

Froma different approach, the elegance of the theorem is also embedded onthe outstretched proofs that have cropped up in support of itsprincipals over the centuries. One eccentric mathematical teachercalled Elisha Scott Loomis who lived between 1852 and 1940, spent herlifetime collecting these proofs and condensed them up in a bookcalled The Pythagorean Propositions (Maor 36).

Loomisthe teacher in Master in Mathematics tasked the students in his classto come up with equations that prove this theorem. Overwhelmingequations were formulated to this effect because the Pythagorastheorem cuts across various topics in mathematics that includedissociation, algebra, triangles, and vectors. The appeal also arisesfrom its popularity and frequency for use in the field of mathematicsto solve problems. In trigonometry, where several formulae areavailable, application of has been used.

sin2x +cos2x= 1, 1 + tan2x= csc 2x,and 1 + cos2x= csc 2x. (Morris 1)

Theseequations are derivatives of the . The same caseapplies to algebra in property and calculus. Therefore the theoremdominates supremacy in solving equations. (Cartier, Dhombes,Heinzmann and Villani 56)

Inphysics, the theorem was demonstrated when colored liquid was pouredto freely flow between squares that were built on a hypotenuse.

Inthe modern days, has been polished into theequation

an+bn=cn

When‘n’ is 1, the equation becomes basic and simple to solve.However, in most cases n =2 and applies to a wide range of interestsfor solving problems. The Pythagoras theorem normally solvesequations with three integral values. In problems where two integersare known, it is always easy to determine the third integer becauseof the assumption made that the three integers represent the lengthsof a right angle triangle. Numerous examples of triple integerinclude (x, y, z) plots on graphs, (3, 4, 5) and (5, 12, 13).However, this equation does not hold for cases where the value of ‘n’is greater than two because it is impractical to divide anything tothe power that is similar to its second denomination. This argumentwas pointed out and proved by Fermat. It has thus been immortalizedeven though it has a skeptical backdrop (Maor 3).

  1. Consequences of Pythagoras theorem

Calculus,which is analytical geometry, has numerous mathematical problems thatcan be solved without the use of Pythagoras theorem. These includecoordinate systems, equations that are linear either originating froma straight line or linear equations of parallel slopes. This isbecause they are irrational in nature and embrace the aspect ofcomplete autonomy and dominance on constituent sides. There will,however, be no error arising in the solutions of these problemsbecause no one side can precede another. For instance, when a chordCD is drawn inside a circle passing through point P. equations thatarise such as PC x PD are constant for any other cord that will passthrough point P, each having individual dominance and henceself-defined (Martinetti and D`Andrea 470). This explains thatPythagoras theorem is undemocratic in nature and only applies tospecific special cases involving right angle triangle. It also bringsout the fact that the hypotenuse has a different unique role whencompared to the remaining two sides (Martinetti and D’Andrea 470).

Physicsis currently experiencing a revolution where engineering productionis digital in nature. Audio instruments, computers, videos and otherelectronics have transformed to the use of whole number sequencesduring their operation. This evolution fully endorses the philosophythat ‘all are numbers,` the basis of Pythagoras theorem. Quantummechanics has also quantized states of matter and adopted the theory(Allein 13).

WorksCited.

&quotPythagorasof Samos.&quot PythagorasBiography.N.p., n.d.http://www-groups.dcs.st-andrews.ac.uk/~history/Biographies/Pythagoras.html. Accessed 06 November 2016

Allein,Jont,.’ Concept in Ingineering Mathematics’: Part1: Discovery of Number Systems.12 December. 2015, Pg 2-72.

CartierPierre, Dhombes Jean, Heinzmann Gerhard and Villani Cedric. Theorigin of mathematics.Springers. 2016.

Edwards.C. H. Thehistorical development of the calculus.Springer science and business media., 2012.

Maor,Eli. ThePythagorean theorem: a 4000-year history.Princeton University Press, 2010.

MartinettiPierre and D’Andrea Francesco. ‘Letters in mathematical physics’:OnPythagoras theorem for products of spectral triples.Springer.2013.

Morris,Stephanie J. &quotThe .&quot The.Stephanie J. Morris, n.d.http://jwilson.coe.uga.edu/emt669/Student.Folders/Morris.Stephanie/EMT.669/Essay.1/Pythagorean.html. Accessed on 06 November 2016