Positions of a woman and her dog at any time on the complex plane are z and 2z^2013 + z + 1, respectively. How many turns does the dog
make around the origin if the woman goes once along the circle |z| = 1?
To determine the number of turns the dog makes around the origin, we need to find the number of times the dog's position repeats after the woman completes one loop along the circle |z| = 1.
Let's break down the problem step by step:
1. The woman's position on the complex plane is given by z.
2. The woman goes once along the circle |z| = 1. This means that her position covers a distance of 1 unit along the circle.
3. The dog's position on the complex plane is given by 2z^2013 + z + 1.
4. To find the number of turns the dog makes around the origin, we need to observe how the dog's position changes with respect to the woman's position.
Let's start by considering the effect of each term in the expression for the dog's position:
1. The term "z" represents a linear transformation of the woman's position. It has no effect on the number of turns the dog makes around the origin.
2. The term "2z^2013" represents a power transformation of the woman's position. Since the exponent is an odd number (2013), it introduces a "rotation" effect. In other words, when the woman completes one loop along the circle |z| = 1, the dog's position undergoes 2013 rotations around the origin.
3. The term "1" is a constant term. It does not affect the number of turns the dog makes around the origin.
Based on the above analysis, we can conclude that the dog makes 2013 turns around the origin when the woman completes one loop along the circle |z| = 1.
Thus, the number of turns the dog makes around the origin is equal to 2013.