Final answer:
Imaginary numbers are used to represent the square root of negative numbers, while complex numbers are used to provide solutions to polynomial equations and represent quantities that involve oscillation or rotation.
Step-by-step explanation:
The purposes of imaginary and complex numbers lie in their ability to extend the number system beyond the set of real numbers. Imaginary numbers are used to represent the square root of negative numbers, which cannot be expressed as real numbers. Complex numbers, on the other hand, consist of both a real part and an imaginary part, and they are used in various areas of mathematics, including algebra, calculus, and physics.
One of the main purposes of complex numbers is to provide solutions to polynomial equations. For example, the equation x^2 + 1 = 0 has no real solutions, but it does have complex solutions, which are ±i, where i is the imaginary unit.
Complex numbers are also used in representing quantities that involve oscillation or rotation, such as in electrical engineering and quantum mechanics. They are often used in the form of complex exponentials to describe wave phenomena, such as electromagnetic waves or quantum wave functions.