The Pythagorean Theorem - Educational Outreach

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So what does poetry have to do with mathematics?Any mathematics text can be likened to a poetry text. In it,the author is interspersing two languages: a language ofqualification (English in the case of this book) and alanguage of quantification (the universal language ofalgebra). The way these two languages are interspersed isvery similar to that of the poetry text. When we aredescribing, we use English prose interspersed with anillustrative phrase or two of algebra. When it is time to doan extensive derivation or problem-solving activity—usingthe concise algebraic language—then the whole page (or twoor three pages!) may consist of nothing but algebra. Algebrathen becomes the alternate language of choice used tounfold the idea or solution. The Pythagorean Theorem followsthis general pattern, which is illustrated below by adiscussion of the well-known quadratic formula.2Let ax bx c 0 be a quadratic equation writtenin the standard form as shown with a 0 . Thenax2 bx c 0has two solutions (including complex andmultiple) given by the formula below, called the quadraticformula.x b b2 4ac2a .To solve a quadratic equation, using the quadratic formula,one needs to apply the following four steps considered to bea solution process.1. Rewrite the quadratic equation in standard form.2. Identify the two coefficients and constantterm a , b,&c .3. Apply the formula and solve for the two x values.4. Check your two answers in the original equation.To illustrate this four-step process, we will solve thequadratic equation 2x 2 13x 7 .14
1 : 2x2x2****22 13x 7 13x 7 0 : a 2, b 13,c 7****3 ( 13) : x 13 x ( 13)169 5642(2)13 225 1315x 4 41x { ,7}****422 4(2)( 7) : This step is left to the reader.Taking a look at the text between the two happy-facesymbols , we first see the usual mixture of algebra andprose common to math texts. The quadratic formula itself,being a major algebraic result, is presented first as a standaloneresult. If an associated process, such as solving aquadratic equation, is best described by a sequence ofenumerated steps, the steps will be presented in indented,enumerated fashion as shown. Appendix B provides adetailed list of all mathematical symbols used in this bookalong with explanations.Regarding other formats, italicized 9-font text isused throughout the book to convey special cautionarynotes to the reader, items of historical or personal interest,etc. Rather than footnote these items, I have chosen toplace them within the text exactly at the place where theyaugment the overall discussion.15
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: PrefaceThe Pythagorean Theorem has
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
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Figure 2.32 is President Garfield
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Even though the Cartesian coordinat
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B 2← γ →α y βx C-xC 2Figure
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Being polynomial in form, the funct
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Thus, the critical point ( C 2 ,0)i
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Sincexcpwas chosen on an arbitrary
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∂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
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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
Page 162 and 163:
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
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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
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The Pythagorean Theorem - Educational Outreach

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