Final answer:
To determine how many points have a modulus of 4, we need to consider the complex numbers in the form a + bi, where a and b are real numbers. The modulus of a complex number is the distance from the origin to the point represented by the complex number in the complex plane.
Step-by-step explanation:
To determine how many points have a modulus of 4, we need to consider the complex numbers in the form a + bi, where a and b are real numbers. The modulus of a complex number is the distance from the origin to the point represented by the complex number in the complex plane. It can be calculated using the formula |z| = sqrt(a^2 + b^2), where z is the complex number. For a modulus of 4, we have the equation |z| = 4, and by substituting the values in the equation, we get:
sqrt(a^2 + b^2) = 4
Squaring both sides of the equation, we get:
a^2 + b^2 = 16
This equation represents a circle with a radius of 4 centered at the origin. Therefore, any point on the circle with a modulus of 4 will satisfy the equation a^2 + b^2 = 16. The points on the circle can be both real and imaginary.