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A Deadly Proof (Video)

3a6060bc7ace07fa75791cd5dac2d46a?s=47 Stuart Kent
February 28, 2017

A Deadly Proof (Video)

A lightning talk on the proof of the irrationality of the square root of 2. Accessible to anyone familiar with basic algebra.

Video: https://youtu.be/Wg8-4Cn1YBI
Venue: Detroit Labs

3a6060bc7ace07fa75791cd5dac2d46a?s=128

Stuart Kent

February 28, 2017
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Transcript

  1. A Deadly Proof

  2. 1 1 √2 Pythagoras: a² + b² = c²

  3. Can √2 be represented concretely?

  4. Pythagoreans whole numbers

  5. 1² = 1 < 2 2² = 4 > 2

    => √2 lies in between 1 and 2
  6. Pythagoreans fractions

  7. What next?

  8. Assume: √2 can be represented as a fraction... √2 =

    p/q
  9. ...where p and q do not share any factors: 18/12

    = 9/6 = 3/2
  10. √2 = p/q

  11. √2 = p/q => 2 = p²/q²

  12. √2 = p/q => 2 = p²/q² => 2q² =

  13. √2 = p/q => 2 = p²/q² => 2q² =

    p² => p² is even
  14. √2 = p/q => 2 = p²/q² => 2q² =

    p² => p² is even => p is even. Write p = 2r.
  15. 2q² = p² and p = 2r

  16. 2q² = p² and p = 2r => 2q² =

    4r²
  17. 2q² = p² and p = 2r => 2q² =

    4r² => q² = 2r²
  18. 2q² = p² and p = 2r => 2q² =

    4r² => q² = 2r² => q² is even
  19. 2q² = p² and p = 2r => 2q² =

    4r² => q² = 2r² => q² is even => q is even. Write q = 2s.
  20. p = 2r and q = 2s

  21. p = 2r and q = 2s => p/q =

    (2r)/(2s)
  22. p = 2r and q = 2s => p/q =

    (2r)/(2s) => p/q reduces to r/s.
  23. "Proof by contradiction"

  24. Conclusion: √2 cannot be represented as a fraction.

  25. Pythagoreans:

  26. RIP Hippasus of Metapontum (maybe)