1 / 12

Pythagorean Triples

Pythagorean Triples. a 2 + b 2 = c 2. What are they?. Ex. (3, 4, 5)  3 2 + 4 2 = 5 2 (6, 8, 10)  6 2 + 8 2 = 10 2 Is that it? Whole Numbers? 1.5 2 + 2 2 = 2.5 2 a 2 + b 2 = c 2 with a, b, c in Q +. Primitive Pythagorean Triples. (3, 4, 5)  “Primitive”

lysandra-jarvis
Download Presentation

Pythagorean Triples

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Pythagorean Triples a2 + b2 = c2

  2. What are they? Ex. (3, 4, 5)  32 + 42 = 52 (6, 8, 10)  62 + 82 = 102 Is that it? Whole Numbers? 1.52 + 22 = 2.52 a2 + b2 = c2 with a, b, c in Q+

  3. Primitive Pythagorean Triples (3, 4, 5)  “Primitive” a2 + b2 = c2 with a, b, c in Z+ and GCD(a, b, c) = 1 GCD(a, b) = 1 & GCD(a, c) = 1 & GCD(b, c) = 1 Proof: If k|a & k|b then a = kg, and b = kh with k, g, h, in Z+ a2 + b2 = (kg)2 + (kh)2 = k2g2 + k2h2 = k2(g2 +h2) = c2 Thus k|c, so (a, b, c) is NOT primitive

  4. Homework Week 2 (3, 4, 5)  (5, 12, 13)  (7, 24, 25),… (2n + 1, 2n2 + 2n, 2n2 + 2n + 1) with n in Z+ Can all three a, b, and c be even? Can only c be even?

  5. Plimpton 322 185412 + 127092 = 125002 How? (Babylon - 1900- 1600 BC)

  6. c2 – b2 c2 – b2 = a2 (c/a)2 – (b/a)2 = 1 α = c/a, and β = b/a  α2 – β2 = 1 = (α + β)(α – β) (α + β) = m/n and (α – β) = n/m where m > n in Z+ α = ½(m/n + n/m) = (m2 + n2)/2mn = c/a β = ½(m/n – n/m) = (m2 – n2)/2mn = b/a a2 + b2 = c2 a = 2mn, b = m2 – n2, and c = m2 + n2, with m and n relatively prime and m > n in Z+

  7. Homework Week 3 Terminating decimal (x, y) solutions for x2 + y2 = 1? Take line L  y = t(x + 1) x2 + (t(x + 1))2 = 1 x2 + t2(x2 + 2x + 1) = 1 x2 + t2x2 + 2xt2 + t2 = 1 (t2 + 1)x2 + 2t2x + (t2 – 1) = 0 x = -1 , and x = (1 – t2)/(1 + t2) y = 2t/(1 + t2) ([(1 – t2)/(1 + t2)], [2t/(1 + t2)])

  8. More fun! Take ([(1 – t2)/(1 + t2)], [2t/(1 + t2)]) Let t = n/m with n, m in Z+ x = (m2 – n2)/(m2 + n2) and y = 2nm/(m2 + n2) [(m2 – n2)/(m2 + n2)]2 + [2nm/(m2 + n2)]2 = 1 (m2 – n2)2 + (2nm)2 = (m2 + n2)2 a = m2 – n2, b = 2nm, and c = m2 + n2

  9. Cool Theorems If (a, b, c) is a primitive Pythagorean triple • then a and b have opposite parity • then c must be odd • then one of a or b is divisible by 3 • then one of a, b, or c is divisible by 5 • where b is even, then b is divisible by 4

  10. Integer Triangles If b is even, then the area is divisible by 6 (a, b, c) is primitive if and only if ∆ABC has the smallest area among the integer right triangles similar to ∆ABC. The length of the radius of the circle inscribed in an integer triangle is always an integer

  11. Interesting “Nature of Mathematics” by Karl Smith where a/b is reduced Ex. m = 3 82 + 152 = 172

  12. PSAT Use Ans 1 (2, 3) Ans 2 (18, 3)

More Related