Let $z$ be a complex number. The number $1$ is written on a board. You perform a series of moves, where in each move you may either
-
replace the number $w$ written on the board with $zw$,
or -
replace the number $w$ with a different complex number $w'$ so that
$$\max(\lvert\operatorname{Re} w\rvert, \lvert\operatorname{Im} w\rvert) = \max(\lvert\operatorname{Re} w'\rvert, \lvert\operatorname{Im} w'\rvert).$$
After some time, a positive real number less than $0.001$ is written on the board. The set of all $z$ for which this is possible forms a region $A$ in the complex plane. What is the area of $A$?
So I'm baffled on where to begin this problem. I suppose defining $z$ = a + bi. I do not know where to go from here.
I understand the max notation and the $\rm{Re}$ (Real) and $\rm{Im}$ (Imaginary) notation.
This problem was sent to me by my math coach, so this may be from a book.
Best Answer
The region $A$ looks like a 4-leaf clover, and has area $$2 + \pi = 8\int_0^{\pi/4} \cos^2(\pi/4 - \theta) d\theta.$$ A picture of the region is at the bottom of the answer. A nice idea pointed out in a comment below is that one can also compute the area without an integral by dissecting the region into a square of side length $\sqrt{2}$ and two circles of radius $r = 1/\sqrt{2}$.
Here's an outline of how to determine the region $A$. Polar coordinates are well-suited for getting our hands on the complex multiplication option in the game, and we work mostly in them. Writing every detail would be rather lengthy — I'm happy to explain any of the steps that are unclear.
The region is bounded by the below curve, plotted in Desmos.