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

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Presentation on theme: "What is the ratio of the length of the diagonal of a perfect square to an edge?"— Presentation transcript:

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

2

3 The white area in the top square is (a 2 )/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 (a 2 )/2. So the white area in the lower square is 2a 2.

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 (a 2 )/2. So the white area in the lower square is 2a 2. But this area can also be expressed as b 2.

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 (a 2 )/2. So the white area in the lower square is 2a 2. But this area can also be expressed as b 2. Thus, b 2 = 2a 2.

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 (a 2 )/2. So the white area in the lower square is 2a 2. But this area can also be expressed as b 2. Thus, b 2 = 2a 2. 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 2 1/2.

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

10 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 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 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 Proof : Assume, to the contrary, that √ 2 is rational.

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

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28 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 means our assumption that √ 2 is 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 Behold, √ 2 is irrational!


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