The Golden Ratio

What just happened?

You set the division point of the segment so that the longer part $a$ makes up about $0.618$ of the whole. Then — and only then — the two different ratios describing this division become equal. That is precisely the golden ratio.

Definition of the golden ratio:

We divide the segment into a longer part $a$ and a shorter part $b$ so that the ratio of the whole to the longer part equals the ratio of the longer part to the shorter:

The first known written definition was given by Euclid in his work "Elements" (c. 300 BC) as the "division of a segment in extreme and mean ratio".

The golden number φ:

The value of this ratio is the golden number $\varphi$ — an irrational number. From the equality $\tfrac{a+b}{a}=\tfrac{a}{b}$ after substituting $\varphi=a/b$ we get a quadratic equation whose positive solution is precisely $\varphi=(1+\sqrt5)/2$.

The invariant — the equation of the golden number:

This short equation fully characterizes $\varphi$. The same number appears as the limit of the ratios of consecutive terms of the Fibonacci sequence (1, 1, 2, 3, 5, 8, 13…) — two seemingly distant worlds, geometry and sequences, lead to the same constant.

Where does it appear?

Associated with the golden ratio are the golden rectangle (with side ratio $\varphi$) and the golden spiral. Since antiquity these proportions have been considered especially harmonious — they are found in art, architecture (proportions of buildings) and nature (leaf arrangement, shells, flowers), although their "ubiquity" is sometimes exaggerated.

Why you need this for the exam:

The golden ratio is a classic context for quadratic equations and proportions. Remember the definition $\tfrac{a+b}{a}=\tfrac{a}{b}$ and that it leads to $\varphi^2=\varphi+1$, that is $\varphi=(1+\sqrt5)/2\approx1.618$.

Remember: the golden ratio is a division of a segment in which $\tfrac{a+b}{a}=\tfrac{a}{b}=\varphi$; the golden number $\varphi=(1+\sqrt5)/2\approx1.618$ satisfies $\varphi^2=\varphi+1$.