What is the polar form of a complex number?

The polar form of a complex number is a way of representing a complex number using its magnitude and angle.

In the rectangular form of a complex number, we represent a complex number as a sum of a real part and an imaginary part. For example, if we have a complex number z = 3 + 4i, we can represent it in rectangular form as z = 3 + 4i.

In the polar form of a complex number, we represent a complex number as a magnitude and an angle. The magnitude is the distance from the origin to the complex number in the complex plane, and the angle is the angle that the line connecting the origin to the complex number makes with the positive real axis. We can represent the complex number z = 3 + 4i in polar form as z = 5(cosθ + isinθ), where θ is the angle that the line connecting the origin to the complex number makes with the positive real axis.

To convert a complex number from rectangular form to polar form, we can use the following formulas:

Magnitude: |z| = √(a^2 + b^2), where a is the real part and b is the imaginary part of the complex number.

Angle: θ = tan^-1(b/a), where a is the real part and b is the imaginary part of the complex number.

To convert a complex number from polar form to rectangular form, we can use the following formulas:

Real part: Re(z) = |z|cosθ

Imaginary part: Im(z) = |z|sinθ

Overall, the polar form of a complex number provides a useful way of representing complex numbers, particularly when dealing with multiplication and division of complex numbers.