The Pythagorean Triple Essay Example for Free

The Pythagorean Triple EssayPythagoras a Greek philosopher and mathematician is very noted for its Pythagorean Theorem. This theorem states that if a, b and c are sides of a right triangle then a2 + b2 = c2 (Morris, 1997).The study of the Pythagorean triples started long before Pythagoras knew how to solve it. There were evidences that Babylonians ingest lists of the triples written in a tablet. This would only mean that Babylonians may have known a method on how to produce such triples (Silverman, 16 26). Pythagorean triple is a set of number consisting of three natural numbers that can suit the Pythagorean equation a2 + b2 = c2. Some of the known triples are 3, 4, 5 and 5, 12, 13 (Bogomolny, 1996). How can we derive such triples?If we multiple the Pythagorean radiation diagram by 2 then we generate another formula 2a2 + 2b2 = 2c2. This only marrow that if we multiply 2 to the Pythagorean triple 3, 4, 5 and 5, 12, 13 then we can get another set of Pythagorean triple. The a nswer to that is triple 6, 8, 10 and 10, 24, 26. To check whether the said triple are Pythagorean triple, we can substitute it to the original formula a2 + b2 = c2.Check is 6, 8, 10 Pythagorean triple?62 + 82 = 10236 + 64 = 100100 = 100Thus 6, 8 and 10 repay the Pythagorean equation. 6, 8, 10 is a Pythagorean triple.Check is 10, 24, 26 satisfy the Pythagorean equation?102 + 242 = 262100 + 576 = 676676 = 676Thus 10, 24, 26 satisfy the Pythagorean equation. 10, 24, 26 is a Pythagorean triple.If we multiply the Pythagorean equation by 3 and using the first 2 Pythagorean triple mentioned above, we can yield another set of Pythagorean triple. Thus we can formulate a general formula that can produce different sets of Pythagorean triple. We can generate an infinite number of Pythagorean triple by using the Pythagorean triple 3, 4, 5. If we multiple d, where k is an integer, to that triple we will yield different sets of Pythagorean triple all the time.d*(3, 4, 5) where d is an integer.Che ck if k is constitute to 4 we get a triple 12, 16, and 20. Is this a Pythagorean triple?By substitution,122 + 162 = 202144 + 256 = 400400 = 400Thus 12, 16, 20 satisfy the Pythagorean equation. 12, 16, 20 is a Pythagorean triple.Check if k is equal to 5 we get a triple 15, 20, 25. Is this a Pythagorean triple?By substitution,152 + 202 = 252225 + 400 = 625625 = 625Thus 15, 20, 25 satisfy the Pythagorean equation. 15, 20, 25 is a Pythagorean triple.But the formula given above is just a formula for getting the multiples of the Pythagorean triple. But is there a general formula in getting these triples? There are formulas that can solve each and all Pythagorean triple that one can ever imagine. One formula that can give us the triples is a = st, b = (s2 + t2)/2 and c = (s2 t2)/2 (. A simple derivation of these formula will come from the main formula a2 + b2 = c2 (Silverman, 16 26). This is a shorten way to derive the formula from theorem 2.1(Pythagorean triples).a2 + b2 = c2 with a i s odd, b is even and a, b and c have no common factors.a2 = c2 b2 by additive propertya2 = (c b)(c + b) by factoring (difference of two squares)by checking32 = (5 4)(5 + 4) = 1*952 = (13 12)(13 + 12) = 1*2572 = (25 24)(25 + 24) = 1*49