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

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

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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

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1

February 4, 2015 Albert R. Meyer, 2015

Mathematics for Computer Science 6.042J/18.062J

PROOFS

proof-intro.1

February 4, 2015 Albert R. Meyer, 2015

Getting started: Pythagorean theorem

c b

Familiar? Obvious?

proof-intro.3

a

Yes! No!

a2 + b2 = c2

February 4, 2015 Albert R. Meyer, 2015

× c c c c

A Cool Proof

Rearrange into: (i) a c × c square, and then (ii) an a×a & a b×b square

proof-intro.4

c b a

February 4, 2015

proof-

c c c a b c b-a

Albert R. Meyer, 2015

A Cool Proof

intro.5

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2

February 4, 2015 Albert R. Meyer, 2015

A Cool Proof

b c a

proof-intro.6

c c × c c ×

b-a b-a

February 4, 2015 Albert R. Meyer, 2015

A Cool Proof

proof-intro.7

b a a b-a

(b−a)+a

  • February 4, 2015
Albert R. Meyer, 2015

A Cool Proof

proof-intro.9

b a a a b-a

b

  • February 4, 2015
Albert R. Meyer, 2015

Proof by Picture

  • elegant and correct
  • -in this case
  • worrisome in general
  • -hidden assumptions

proof-intro.10

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February 4, 2015 Albert R. Meyer, 2015

Bogus Proof: Getting Rich By Diagram

proof-intro.11

February 4, 2015 Albert R. Meyer, 2015

Bogus Proof: Getting Rich By Diagram

proof-intro.12

11 10 11

1 1 1 1 1 1 1 1

10

February 4, 2015 Albert R. Meyer, 2015

A False Proof: Getting Rich By Diagram

proof-intro.13

11 10 11

1 1 1 1

10

1 1

Profit!

February 4, 2015 Albert R. Meyer, 2015

Getting Rich

The bug: are not right triangles!

So the top and bottom line of the “rectangle” is not straight!

proof-intro.14

10 1

1 1 1 1

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February 4, 2015 Albert R. Meyer, 2015

Another Bogus Proof

Theorem: Every polynomial,

has two roots over ℂ.

  • b+

b2 - 4ac r

1 ::=

proof-intro.15

2a r

2 ::= -b- b2 - 4ac

2a

Proof (by calculation). The roots are:

and

2

ax +bx +c

February 4, 2015 Albert R. Meyer, 2015

Another Bogus Proof Counter-examples: 0x2 +0x +1 has 0 roots 0x2 +1x +1 has 1 root The bug: divide by zero error The fix: require a ≠ 0

proof-intro.16

February 4, 2015 Albert R. Meyer, 2015

Another Bogus Proof

Counter-example:

1x2 + 0x + 0 has 1 root.

The bug: r1 = r2 The fix: require D ≠ 0 where

D::= b2 - 4ac

proof-intro.17

February 4, 2015 Albert R. Meyer, 2015

Another Bogus Proof

What if D < 0? x2 + 1 has roots i, -i

  • -ambiguous which is r1

and which is r2?

proof-intro.18

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February 4, 2015 Albert R. Meyer, 2015

ambiguity can cause problems: 1 = 1

proof-intro.19

= (-1)(-1) =

  • 1 -1 =
  • 1

1 = -1 ?

2

( ) =-1

Moral:

  • 1. Be sure rules are properly applied.
  • 2. Thoughtless calculation no

substitute for understanding.

February 4, 2015 Albert R. Meyer, 2015

Consequences of 1 = -1

½ = -½ (multiply by ½)

2 = 1 (add ) “Since I and the Pope are clearly 2, we conclude that I and the Pope are 1. That is, I am the Pope.”

  • - Bertrand Russell

proof-intro.21

3 2

February 4, 2015 Albert R. Meyer, 2015

Consequences of 1 = -1

Bertrand Russell (1872 - 1970)

(Picture source: http://www.users.drew.edu/~jlenz/brs.html)

proof-intro.22

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6.042J / 18.062J Mathematics for Computer Science

Spring 2015 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.