Proof Without Words: The Golden Ratio Roger B. Nelsen ([email protected]), Lewis & Clark College, Portland, OR Theorem. If x > 0 and x = 1 + 1/x, then x = ϕ = (1 +

√ 5)/2.

Proof. 1/x

x 1/x

1 x

1 1 1

1/x

x

1 x

2x – 1

1/x

√ 1+ 5 . (2x − 1) = 5 =⇒ x = ϕ = 2 2

Exercise. Show that ϕ 2 + (1/ϕ)2 = 3. (Hint: In the figure, create a square with area 3 by drawing a diagonal in each of the four rectangles.) Summary. We employ a square with area 5 to determine the golden ratio (without using the quadratic formula). http://dx.doi.org/10.4169/college.math.j.47.2.108 MSC: 97G40

108

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Proof Without Words: The Golden Ratio 108

Proof Without Words: The Golden Ratio. Roger B. Nelsen ([email protected]), Lewis & Clark College, Portland, OR. Theorem. If x > 0 and x = 1 + 1/x, then x = ϕ ...
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