The Pythagorean Theorem - Educational Outreach

More documents

Recommendations

Info

However, Euler’s Conjecture did not stand the test oftime as Fermat’s Last Theorem did. In 1966, Lander andParker found a counterexample for n 5 :5 5 5 527 84 110133144Two counterexamples for n 4 followed in 1988. The firstwas discovered by Noam Elkies of Harvard. The second, thesmallest possible for a quartet of numbers raised to thefourth power, was discovered by Roger Frye of ThinkingMachines Corporation.4442,682,44015,465,639187,96044495,800 217,519 414,560 5422,48120,615,67344Today, power sums—both equal and non-equal—provide asource of mathematical recreation of serious and not-soseriousamateurs alike. A typical ‘challenge problem’ mightbe as follows:Find six positive integerssumsumuu77 7 7 v w x yv w x y zu , v,w,x,y,z satisfying the power7 z7where the associated linear is minimal.Table 3.7 on the next page displays just a few of theremarkable examples of the various types of power sums.With this table, we close Chapter 3 and our examination ofselect key spin-offs of the Pythagorean Theorem.→134
# EXPRESSION6 3 4 53 3371 3 7 13 3407 4 0 71 2135 1 3 51 2175 1 7 54 4 41634 1 6 3 43 4 33435 3 4 3 51 2598 5 9 85 5 5 554,748 5 4 7 4 83 3 3 312333435364758392169 13 and 961 3122025 45 and 20 25 4510211312 4913 17 and 4 9 13 171676 1 6 7 6 1 6 7 621233 12 332990100 990 100294122353 9412 23531 2 3 4 5 62646798 2 6 4 6 7 9 81 2 32427 2 4 2 73 3221859 22 18 593343 (3 4)1 2 3 4 5 4 3 213142152162177184193205Table 3.7: Power Sums135
Page 1:
The PythagoreanTheoremCrown Jewel o
Page 4:
The Pythagorean TheoremCrown Jewel
Page 7 and 8:
Table of ContentsList of Tables and
Page 9 and 10:
List of Tables and FiguresTablesNum
Page 11 and 12:
Figures…continuedNumber and Title
Page 13 and 14:
List of Proofs and DevelopmentsSect
Page 15 and 16:
PrefaceThe Pythagorean Theorem has
Page 17 and 18:
1 : 2x2x2****22 13x 7 13x 7 0 : a
Page 19 and 20:
1) Consider the Squares“If it was
Page 21 and 22:
s090045 0ss45450450090Figure 1.3: O
Page 23 and 24:
Notice that the right-triangle prop
Page 25 and 26:
The middle square (minus the donut
Page 27 and 28:
Our proof in Chapter 1 has been by
Page 29 and 30:
2) Four Thousand Years of Discovery
Page 31 and 32:
The proof Pythagoras is thought to
Page 33 and 34:
cabcFigure 2.4: Annotated Square wi
Page 35 and 36:
Figure 2.6 is the diagram for a sec
Page 37 and 38:
1x:b2:Area(CBA)3:Area( ABC)Area(ABC
Page 39 and 40:
Euclid’s proof of the Pythagorean
Page 41 and 42:
First we establish that the two tri
Page 43 and 44:
1 2y b b: y b c c2:A3:A4:Aunshaded
Page 45 and 46:
XB'A090 CBFigure 2.12: Euclid’s C
Page 47 and 48:
2.3) Liu Hui Packs the SquaresLiu H
Page 49 and 50:
One of my favorites was Boxel, a ga
Page 51 and 52:
2.4) Kurrah Transforms the Bride’
Page 53 and 54:
Note: as is the occasional custom i
Page 55 and 56:
2.5) Bhaskara Unleashes the Power o
Page 57 and 58:
2.6) Leonardo da Vinci’s Magnific
Page 59 and 60:
In Figure 2.24, we enlarge Step 5 a
Page 61 and 62:
But, this is precisely the nature o
Page 63 and 64:
2.8) Henry Perigal’s TombstoneHen
Page 65 and 66:
In the century following Perigal, b
Page 67 and 68:
Returning to Henry Perigal, Figure
Page 69 and 70:
Figure 2.32 is President Garfield
Page 71 and 72:
Even though the Cartesian coordinat
Page 73 and 74:
B 2← γ →α y βx C-xC 2Figure
Page 75 and 76:
Being polynomial in form, the funct
Page 77 and 78:
Thus, the critical point ( C 2 ,0)i
Page 79 and 80:
Sincexcpwas chosen on an arbitrary
Page 81 and 82:
∂F/∂x = ∂F/∂y = 0x(C - x) 2
Page 83 and 84:
Define a new function21 A2 A3 2xcp
Page 85 and 86: 2.11) Shear, Shape, and AreaOur las
Page 87 and 88: Step 1: Cut the big square into two
Page 89 and 90: In Table 2.3, we briefly summarize
Page 91 and 92: We can formally state this similari
Page 93 and 94: What we will do in Section 3.2 is p
Page 95 and 96: There are three Pythagorean-triple
Page 97 and 98: Thus, if we multiply any given prim
Page 99 and 100: The last equality shows that the in
Page 101 and 102: A Pythagorean Quartet is a set of f
Page 103 and 104: 3.5) Pythagoras and the Three Means
Page 105 and 106: yxy MHxab2ab2 y x FC y ab2ab 2ab y
Page 107 and 108: Hero was the founder of the Higher
Page 109 and 110: 5 : Solve for area using the formul
Page 111 and 112: 4 : Construct the parallelogram wit
Page 113 and 114: We already have introduced Thabit i
Page 115 and 116: Some historians believe Stewart’s
Page 117 and 118: In pre-computer days, both all four
Page 119 and 120: One example of a periodic process i
Page 121 and 122: The six trigonometric functions are
Page 123 and 124: The following four relationships ar
Page 125 and 126: As with any set of identities, the
Page 127 and 128: Figure 3.17 can be used as a jumpin
Page 129 and 130: The remaining two pillars are the L
Page 131 and 132: a2Law of Cosines c2 b2 2bccos()The
Page 133 and 134: An example of a Diophantine equatio
Page 138 and 139: 4) Pearls of Fun and WonderPearls o
Page 140 and 141: We have achieved five-digit accurac
Page 142 and 143: 140212213116)74370116)(74370(370)74
Page 144 and 145: X X X X O X X O X XO X X O O OO X O
Page 146 and 147: 4.3) Earth, Moon, Sun, and StarsIn
Page 148 and 149: The four-step solution follows. All
Page 150 and 151: Figure 4.9 depicts Eratosthenes’
Page 152 and 153: ---High above the earth, we see the
Page 154 and 155: The commonly accepted value is abou
Page 156 and 157: 1BC : tan( BAC)BABC BAtan(BAC)0BC
Page 158 and 159: BD ACtan( 21 )Even for the closest
Page 160 and 161: 3w 2h s w 5 s w 0.38196s&5 1s
Page 162 and 163: Figure 4.16: Triangular Phi---The h
Page 164 and 165: What remains to be done is to devel
Page 166 and 167: 3 : h32 4 h222 2 24C 2 h33162 2 2
Page 168 and 169: Epilogue: The Crown and the Jewels
Page 170 and 171: Not only has it endured, but the Py
Page 172 and 173: A] Greek AlphabetGREEK LETTERUpper
Page 174 and 175: C] Geometric FoundationsThe Paralle
Page 176 and 177: 8. Congruent Triangles: Two triangl
Page 178 and 179: 3. Rectangle: A bh : P 2b 2h, b &
Page 180 and 181: Recreational Mathematics11. Pickove
Page 182 and 183: GGolden RatioDefined and algebraica
Page 184 and 185: Puzzles (cont)Pythagorean Magic Squ
Page 186 and 187:
Pythagorean Theorem ProofsAlgebraic
Page 188:
Trigonometry (cont)Eratosthenes mea
show all

The Pythagorean Theorem - Educational Outreach

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?