Computing data with spreadsheets. Enter the following into the corresponding cells: A1: n B1: triangle C1: sqrt

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1 Computing dt with spredsheets Exmple: Computing tringulr numers nd their squre roots. Rell, we showed 1 ` 2 ` `n npn ` 1q{2. Enter the following into the orresponding ells: A1: n B1: tringle C1: sqrt A2: 1 B2: A2*(A2+1)/2 C2: SQRT(B2) A2: =A2+1 B2: A3*(A3+1)/2 C3: SQRT(B3) The result should look like Computing dt with spredsheets Exmple: Computing tringulr numers nd their squre roots. Rell, we showed 1 ` 2 ` `n npn ` 1q{2. Enter the following into the orresponding ells: A1: n B1: tringle C1: sqrt A2: 1 B2: A2*(A2+1)/2 C2: SQRT(B2) A2: =A2+1 B2: A3*(A3+1)/2 C3: SQRT(B3) Next, selet ells A3, B3, nd C3 nd opy.

Finlly, selet ells in s mny rows s you wnt elow tht, in the olumns A, B, nd C, nd pste. Exmple: Computing tringulr numers nd their squre roots.

2 Computing dt with spredsheets Exmple: Computing tringulr numers nd their squre roots. Rell, we showed 1 ` 2 ` `n npn ` 1q{2. Enter the following into the orresponding ells: A1: n B1: tringle C1: sqrt A2: 1 B2: A2*(A2+1)/2 C2: SQRT(B2) A2: =A2+1 B2: A3*(A3+1)/2 C3: SQRT(B3) Next, selet ells A3, B3, nd C3 nd opy. Finlly, selet ells in s mny rows s you wnt elow tht, in the olumns A, B, nd C, nd pste. Exmple: Computing tringulr numers nd their squre roots. Enter the following into the orresponding ells: A1: n B1: tringle C1: sqrt A2: 1 B2: A2*(A2+1)/2 C2: SQRT(B2) A2: =A2+1 B2: A3*(A3+1)/2 C3: SQRT(B3) Next, selet ells A3, B3, nd C3 nd opy. Finlly, selet ells in s mny rows s you wnt elow tht, in the olumns A, B, nd C, nd pste. The spredsheet will extrpolte for you, nd mke new formuls reltive to their shifted positions. If you don t wnt it to djust prt of formul with shifts, dd $ in front of the row numer, ell numer, or oth: A$2, $A2, or $A$2.

3 Pythgoren Triples The Pythgoren theorem sys the lengths of the sides of right tringle stisfy the following: 2 ` 2 2 Geometri proof: Compre the re of the white spes in versus Pythgoren Triples The Pythgoren theorem sys the lengths of the sides of right tringle stisfy the following: 2 ` 2 2 Numer theorists sk: Are there integer solutions? If so, re there infinitely mny integer solutions? Do they ll follow some ommon pttern, or re the solutions rndom? et. Integer solutions: re there,, P Z stisfying 2 ` 2 2? Yes! 3 2 ` , 5 2 ` , 8 2 ` These re lled Pythgoren triples. Trivil solution: 0. (Don t forget to look for the simplest solutions!!)

4 Pythgoren Triples Numer theorists sk: Are there integer solutions? If so, re there infinitely mny integer solutions? Do they ll follow some ommon pttern, or re the solutions rndom? et. Integer solutions: re there,, P Z 0 stisfying 2 ` 2 2? Yes! 3 2 ` , 5 2 ` , 8 2 ` These re lled Pythgoren triples. Trivil solution: 0. (Don t forget to look for the simplest solutions!!) Infinitely mny? Clim: If p,, q is Pythgoren triple, then so is pn, n, nq for ny n P Z 0. So, yes, ut this doesn t generte ll solutions! We ll Pythgoren triple p,, q primitive if,, nd hve no ommon ftors, meningthereisnod P Z 0 suh tht,, nd re ll multiples of d. Ex: (3,4,5) Non-ex: (6,8,10).

5 Primitive Pythgoren triples (PPTs) More exmples: p3, 4, 5q p20, 21, 29q p28, 45, 53q p5, 12, 13q p9, 40, 41q p33, 56, 65q p8, 15, 17q p7, 24, 25q p12, 35, 37q p11, 60, 61q p16, 63, 65q p48, 55, 73q Generi even numers re written 2k with k P Z. Generi odd numers re written 2k ` 1 with k P Z. Squre of n even numer: p2kq 2 2 p2k 2 q lomon int Even! Squre of n odd numer: p2k ` 1q 2 4k 2 ` 4k ` 1 2 p2k looooomooooon 2 ` 2kq `1 integer Odd! Similrly, you n show tht sum of evens nd odds follows the pttern ` even odd even even odd odd odd even Primitive Pythgoren triples (PPTs) You try: For solutions,, P ZZ to 2 ` 2 2, omplete the following tle. even even odd odd even odd even odd Whih ould possily e primitive?

6 Looking for more primitive Pythgoren triples (PPTs) Without loss of generlity (WLOG), ssume is odd nd is even, so is odd. (WLOG here mens tht sine nd re interhngele, we don t need to lso onsider the ses where the reverse is true) If 2 ` 2 2,then p qp ` q. Clims: Assume,, Both nd ` re positive odd integers. 2. There re no divisors ommon to nd `. 3. Both nd ` re perfet squres. Lter we will tlk out the uniqueness of prime ftoriztions. For tody, ssume tht if n for positive integers n,, nd, then there is one-to-one orrespondene etween tprime divisors of n, ounting multipliityu nd tprime divs of, ounting multu\tprime divs of, ounting multu.

7 Looking for more primitive Pythgoren triples (PPTs) Let p,, q e PPT with,, 0. We veshowntht nd ` re positive perfet squres with no ommon divisors. Define s, t P Z 0 y ` s 2 nd t 2.

8 Looking for more primitive Pythgoren triples (PPTs) Let p,, q e PPT with,, 0. We veshowntht nd ` re positive perfet squres with no ommon divisors. Define s, t P Z 0 y ` s 2 nd t 2. Theorem Primitive Pythgoren triples re lssified y positive integers,, nd suh tht st, s2 t 2, nd s2 ` t 2, 2 2 for odd integers s t 1 with no ommon ftors. You try: Exerise 5, prts (), (d), nd/or (e).

1 PYTHAGORAS THEOREM 1. Given a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

1 PYTHAGORAS THEOREM 1. Given a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. 1 PYTHAGORAS THEOREM 1 1 Pythgors Theorem In this setion we will present geometri proof of the fmous theorem of Pythgors. Given right ngled tringle, the squre of the hypotenuse is equl to the sum of the