Striking gold

Let $x = \dfrac{a}{b} \gt 1$ . Then the given equation is $x = 1 + \dfrac{1}{x} \Longrightarrow x^{2} - x - 1 = 0 \Longrightarrow x = \dfrac{1 + \sqrt{5}}{2} \approx \boxed{1.618}$ .

Let's solve for a in terms of b:

$\frac{a}{b} = \frac{a+b}{a} \Rightarrow a^2 = ab + b^2 \Rightarrow a = \frac{b \pm \sqrt{b^2 - 4(1)(-b^2)}}{2} = b \cdot \frac{1 \pm \sqrt{5}}{2}.$

Since $a > b > 0$ , then we only admit the positive root $a = b \cdot \frac{1+\sqrt{5}}{2}.$ Hence, $\frac{a}{b} = \boxed{\frac{1+\sqrt{5}}{2}}$ , or the Golden Ratio.

This quantity is known as the "golden ratio" and it appears throughout mathematics.

Like $\pi$ and $e$ , the digits go on and on, to infinity:

$\dfrac{a}{b} = \dfrac{a+b}{a} = 1.6180339887498948420 …$

Rounded to three decimal places gives $\boxed{1.618}$