The Pythagorean Theorem - Educational Outreach

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The rope was shown in a context suggesting its use as aworkman’s tool for creating right angles, done via thefashioning of a 3-4-5 right triangle. Thus, the Egyptianshad a mechanical device for demonstrating the converse ofthe Pythagorean Theorem for the 3-4-5 special case:3 2 2 20 4 5 90 .Not only did the Egyptians know of specificinstances of the Pythagorean Theorem, but also theBabylonians and Chinese some 1000 years beforePythagoras definitively institutionalized the general resultcirca 500 BCE. And to be fair to the Egyptians, Pythagorashimself, who was born on the island of Samos in 572 BCE,traveled to Egypt at the age of 23 and spent 21 years thereas a student before returning to Greece. While in Egypt,Pythagoras studied a number of things under the guidanceof Egyptian priests, including geometry. Table 2.1 brieflysummarizes what is known about the Pythagorean Theorembefore Pythagoras.Date Culture Person Evidence2000BCE1500BCE1100BCE520BCEEgyptianBabylonian&ChaldeanChineseGreekUnknownUnknownTschou-GunPythagorasTable 2.1: Prior to PythagorasWorkman’s rope forfashioning a3-4-5 triangleRules for righttriangles written onclay tablets alongwith geometricdiagramsWritten geometriccharacterizations ofright anglesGeneralized resultand deductivelyproved28
The proof Pythagoras is thought to have actuallyused is shown in Figure 2.2. It is a visual proof in that noalgebraic language is used to support numerically thedeductive argument. In the top diagram, the ancientobserver would note that removing the eight congruent righttriangles, four from each identical master square, brings themagnificent sum-of-squares equality into immediate view.Figure 2.2: The First Proof by Pythagoras29
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: 2) Four Thousand Years of Discovery
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
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Define a new function21 A2 A3 2xcp
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2.11) Shear, Shape, and AreaOur las
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Step 1: Cut the big square into two
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In Table 2.3, we briefly summarize
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We can formally state this similari
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What we will do in Section 3.2 is p
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There are three Pythagorean-triple
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Thus, if we multiply any given prim
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The last equality shows that the in
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A Pythagorean Quartet is a set of f
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3.5) Pythagoras and the Three Means
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yxy MHxab2ab2 y x FC y ab2ab 2ab y
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Hero was the founder of the Higher
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5 : Solve for area using the formul
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4 : Construct the parallelogram wit
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We already have introduced Thabit i
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Some historians believe Stewart’s
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In pre-computer days, both all four
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One example of a periodic process i
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The six trigonometric functions are
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The following four relationships ar
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As with any set of identities, the
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Figure 3.17 can be used as a jumpin
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The remaining two pillars are the L
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a2Law of Cosines c2 b2 2bccos()The
Page 133 and 134:
An example of a Diophantine equatio
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However, Euler’s Conjecture did n
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4) Pearls of Fun and WonderPearls o
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We have achieved five-digit accurac
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140212213116)74370116)(74370(370)74
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X X X X O X X O X XO X X O O OO X O
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4.3) Earth, Moon, Sun, and StarsIn
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The four-step solution follows. All
Page 150 and 151:
Figure 4.9 depicts Eratosthenes’
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---High above the earth, we see the
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The commonly accepted value is abou
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1BC : tan( BAC)BABC BAtan(BAC)0BC
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BD ACtan( 21 )Even for the closest
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3w 2h s w 5 s w 0.38196s&5 1s
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Figure 4.16: Triangular Phi---The h
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What remains to be done is to devel
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3 : h32 4 h222 2 24C 2 h33162 2 2
Page 168 and 169:
Epilogue: The Crown and the Jewels
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Not only has it endured, but the Py
Page 172 and 173:
A] Greek AlphabetGREEK LETTERUpper
Page 174 and 175:
C] Geometric FoundationsThe Paralle
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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
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The Pythagorean Theorem - Educational Outreach

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