PROOFS a a 2 + b 2 = c 2 Familiar? Yes! Obvious? No ! Albert R. - PowerPoint PPT Presentation

Mathematics for Computer Science Getting started: 6.042J/18.062J Pythagorean theorem c b PROOFS a a 2 + b 2 = c 2 Familiar? Yes! Obvious? No ! Albert R. Meyer, 2015 Albert R. Meyer, 2015 proof-intro.1 proof-intro.3 February 4, 2015 February 4, 2015 c c c c × A Cool Proof A Cool Proof c c b b-a c c a Rearrange into: a b (i) a c × c square, and then c (ii) an a × a & a b × b square Albert R. Meyer, 2015 proof-intro.4 Albert R. Meyer, 2015 proof- intro.5 February 4, 2015 February 4, 2015 1

c c c c × × A Cool Proof A Cool Proof a c b b-a b a b-a a b-a � � � � � (b − a) + a Albert R. Meyer, 2015 Albert R. Meyer, 2015 proof-intro.6 proof-intro.7 February 4, 2015 February 4, 2015 A Cool Proof Proof by Picture • elegant and correct a --in this case b a • worrisome in general b-a a � � � � � --hidden assumptions b Albert R. Meyer, 2015 proof-intro.9 Albert R. Meyer, 2015 proof-intro.10 February 4, 2015 February 4, 2015 2

Bogus Proof: Bogus Proof: Getting Rich By Diagram Getting Rich By Diagram 1 1 1 1 10 11 1 1 1 1 10 11 Albert R. Meyer, 2015 Albert R. Meyer, 2015 proof-intro.11 proof-intro.12 February 4, 2015 February 4, 2015 A False Proof: Getting Rich Getting Rich By Diagram The bug: 1 are not right triangles! 1 1 1 1 So the top and bottom line of the 1 “rectangle” is not straight! 10 11 1 10 1 1 10 11 Profit! 1 1 Albert R. Meyer, 2015 proof-intro.13 Albert R. Meyer, 2015 proof-intro.14 February 4, 2015 February 4, 2015 3

Another Bogus Proof Another Bogus Proof Theorem: Every polynomial, Counter-examples: 2 ax +bx +c 0x 2 +0x +1 has 0 roots has two roots over ℂ . 0x 2 +1x +1 has 1 root Proof (by calculation). The roots are: The bug: divide by zero error b 2 - 4ac 2 ::= -b- b 2 - 4ac The fix: require a ≠ 0 -b+ r 1 ::= r and 2a 2a Albert R. Meyer, 2015 Albert R. Meyer, 2015 proof-intro.15 proof-intro.16 February 4, 2015 February 4, 2015 Another Bogus Proof Another Bogus Proof Counter-example: What if D < 0? 1x 2 + 0x + 0 has 1 root. x 2 + 1 has roots i, -i The bug: r 1 = r 2 --ambiguous which is r 1 The fix: require D ≠ 0 where and which is r 2 ? D::= b 2 - 4ac Albert R. Meyer, 2015 proof-intro.17 Albert R. Meyer, 2015 proof-intro.18 February 4, 2015 February 4, 2015 4

1 = -1 ? Consequences of 1 = -1 ½ = - ½ (multiply by ½ ) ambiguity can cause problems: 3 2 2 = 1 (add ) ( ) =-1 1 = 1 = (-1)(-1) = -1 -1 = -1 2 “Since I and the Pope are clearly 2 , Moral: we conclude that I and the Pope are 1 . 1. Be sure rules are properly applied. That is, I am the Pope.” 2. Thoughtless calculation no -- Bertrand Russell substitute for understanding. Albert R. Meyer, 2015 Albert R. Meyer, 2015 proof-intro.19 proof-intro.21 February 4, 2015 February 4, 2015 Consequences of 1 = -1 �������������������������������������������� Bertrand Russell (1872 - 1970) (Picture source: http://www.users.drew.edu/~jlenz/brs.html ) Albert R. Meyer, 2015 proof-intro.22 February 4, 2015 5

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