If z is a complex number, prove that there exists an r ≥0 and a complex number w with |w|=1 such

step 1 We're asked to prove a statement involving complex numbers. So given any two complex numbers, Z

If z is a complex number, prove that there exists an r ≥0...

Key Concepts

Complex Numbers

A complex number can be written in the form a + bi, where a and b are real numbers and i is the imaginary unit. This concept is fundamental as it extends the one-dimensional real number line into the two-dimensional complex plane, enabling a richer structure that includes both magnitude and direction.

Polar Representation of Complex Numbers

Every complex number can be expressed in polar form as z = r · w, where r is a nonnegative real number representing the modulus (or absolute value) of z, and w is a complex number with modulus 1 representing the direction (or argument) of z. This form provides an intuitive geometric interpretation and simplifies the multiplication and division of complex numbers.

Modulus of a Complex Number

The modulus of a complex number, denoted as r, is the distance from the origin to the point representing the complex number in the complex plane. It is computed as r = ?(a² + b²) for a complex number a + bi, ensuring that r is always a nonnegative real number.

Unit Circle in the Complex Plane

The unit circle consists of all complex numbers that have a modulus of 1. Elements on the unit circle, like the number w in the polar representation, can be seen as purely directional since they only encode an angle or argument and not distance.

Uniqueness in Polar Representation

For any nonzero complex number, the representation as z = r · w is unique up to the choice of the angle, which is determined modulo 2?. In the case of the zero complex number, the representation is not unique because zero multiplied by any unit complex number still gives zero.

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