What is the ratio of the length of the diagonal of a perfect square to an edge?

1. What is the ratio of the length of the diagonal of a perfect square to an edge?

2. What is the ratio of the length of the diagonal of a perfect square to an edge?

3. What is the ratio of the length of the diagonal of a perfect square to an edge? The white area in the top square is (a2)/2.

4. What is the ratio of the length of the diagonal of a perfect square to an edge? The white area in the top square is (a2)/2. So the white area in the lower square is 2a2.

5. What is the ratio of the length of the diagonal of a perfect square to an edge? The white area in the top square is (a2)/2. So the white area in the lower square is 2a2. But this area can also be expressed as b2.

6. What is the ratio of the length of the diagonal of a perfect square to an edge? The white area in the top square is (a2)/2. So the white area in the lower square is 2a2. But this area can also be expressed as b2. Thus, b2 = 2a2.

7. What is the ratio of the length of the diagonal of a perfect square to an edge? The white area in the top square is (a2)/2. So the white area in the lower square is 2a2. But this area can also be expressed as b2. Thus, b2 = 2a2. Or, (b/a)2 = 2.

8. We conclude that the ratio of the diagonal to the edge of a square is the square root of 2, which can be written as √2 or 21/2.

9. So √2 is with us whenever a perfect square is.

10. So √2 is with us whenever a perfect square is. • For a period of time, the ancient Greek mathematicians believed any two distances are commensurate (can be co-measured).

11. So √2 is with us whenever a perfect square is. • For a period of time, the ancient Greek mathematicians believed any two distances are commensurate (can be co-measured). • For a perfect square this means a unit of measurement can be found so that the side and diagonal of the square are both integer multiples of the unit.

12. This means √2 would be the ratio of two integers.

13. This means √2 would be the ratio of two integers. • A ratio of two integers is called a rational number.

14. This means √2 would be the ratio of two integers. • A ratio of two integers is called a rational number. • To their great surprise, the Greeks discovered √2 is not rational.

15. This means √2 would be the ratio of two integers. • A ratio of two integers is called a rational number. • To their great surprise, the Greeks discovered √2 is not rational. • Real numbers that are not rational are now called irrational.

16. This means √2 would be the ratio of two integers. • A ratio of two integers is called a rational number. • To their great surprise, the Greeks discovered √2 is not rational. • Real numbers that are not rational are now called irrational. • We believe √2 was the very first number known to be irrational. This discovery forced a rethinking of what “number” means.

17. We will present a proof that √2 is not rational.

18. We will present a proof that √2 is not rational. • Proving a negative statement usually must be done by assuming the logical opposite and arriving at a contradictory conclusion.

19. We will present a proof that √2 is not rational. • Proving a negative statement usually must be done by assuming the logical opposite and arriving at a contradictory conclusion. • Such an argument is called a proof by contradiction.

20. Theorem: There is no rational number whose square is 2.

21. Theorem: There is no rational number whose square is 2. Proof : Assume, to the contrary, that √2is rational.

22. Theorem: There is no rational number whose square is 2. Proof : Assume, to the contrary, that √2is rational. So we can write√2= n/m with n and m positive integers.

23. Theorem: There is no rational number whose square is 2. Proof : Assume, to the contrary, that √2is rational. So we can write√2= n/m with n and m positive integers. Among all the fractions representing √2, we select the one with smallest denominator.

24. So if √2 is rational (√2= n/m) then an isosceles right triangle with legs of length m will have hypotenuse of length n= √2m. n = √2m

25. So if √2 is rational (√2= n/m) then an isosceles right triangle with legs of length m will have hypotenuse of length n= √2m. Moreover, for a fixed unit, we can take ΔABC to be the smallest isosceles right triangle with integer length sides. n = √2m

26. Now, for the basic trick. Bisect the angle at A and fold the edge AB along the edge AC.

27. Now, for the basic trick. Bisect the angle at A and fold the edge AB along the edge AC.

28. Now, for the basic trick. Bisect the angle at A and fold the edge AB along the edge AC. This creates a new triangle ΔDEC with the angle at E being a right angle and the angle at C still being 45⁰.

29. Now, for the basic trick. Bisect the angle at A and fold the edge AB along the edge AC. This creates a new triangle ΔDEC with the angle at E being a right angle and the angle at C still being 45⁰. AE=AB=m

30. Now, for the basic trick. Bisect the angle at A and fold the edge AB along the edge AC. This creates a new triangle ΔDEC with the angle at E being a right angle and the angle at C still being 45⁰. AE=AB=n EC=AC-AE

31. Now, for the basic trick. Bisect the angle at A and fold the edge AB along the edge AC. This creates a new triangle ΔDEC with the angle at E being a right angle and the angle at C still being 45⁰. AE=AB=n EC=AC-AE=n-m

32. Now, for the basic trick. Bisect the angle at A and fold the edge AB along the edge AC. This creates a new triangle ΔDEC with the angle at E being a right angle and the angle at C still being 45⁰. AE=AB=n EC=AC-AE=n-m BD=DE

33. Now, for the basic trick. Bisect the angle at A and fold the edge AB along the edge AC. This creates a new triangle ΔDEC with the angle at E being a right angle and the angle at C still being 45⁰. AE=AB=n EC=AC-AE=n-m BD=DE=EC=n-m

34. But, if BD=DE=EC=n-m and BC=m,

35. But, if BD=DE=EC=n-m and BC=m, then DC=BC-BD

36. But, if BD=DE=EC=n-m and BC=m, then DC=BC-BD=m-(n-m)

37. But, if BD=DE=EC=n-m and BC=m, then DC=BC-BD=m-(n-m)=2m-n.

38. But, if BD=DE=EC=n-m and BC=m, then DC=BC-BD=m-(n-m)=2m-n.

39. But, if BD=DE=EC=n-m and BC=m, then DC=BC-BD=m-(n-m)=2m-n. Since n and m are integers, n-m and 2m-n are integers and ΔDEC is an isosceles right triangle with integer side lengths smaller than ΔABC .

40. This contradicts our choice of ΔABC as the smallest isosceles right triangle with integer side lengths for a given fixed unit of length.

41. This contradicts our choice of ΔABC as the smallest isosceles right triangle with integer side lengths for a given fixed unit of length. This means our assumption that √2is rational is false. Thus there is no rational number whose square is 2. QED

42. This beautiful proof was adapted from Tom Apostol: “Irrationality of the Square Root of Two: A Geometric Proof”, American Mathematical Monthly,107, 841-842 (2000).

43. This beautiful proof was adapted from Tom Apostol: “Irrationality of the Square Root of Two: A Geometric Proof”, American Mathematical Monthly,107, 841-842 (2000). Behold, √2is irrational!