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

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 = p²
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)