What is the polar representation of a complex number?

The polar representation of a complex number is a way of expressing it in terms of its magnitude and argument.

In the rectangular form of a complex number, it is expressed as a sum of a real part and an imaginary part, such as a + bi. However, in the polar form, it is expressed as r(cosθ + i sinθ), where r is the magnitude of the complex number and θ is its argument.

To find the magnitude of a complex number, we use the formula |z| = √(a^2 + b^2), where a and b are the real and imaginary parts of the complex number respectively. For example, if z = 3 + 4i, then |z| = √(3^2 + 4^2) = 5.

To find the argument of a complex number, we use the formula θ = tan^-1(b/a), where a and b are the real and imaginary parts of the complex number respectively. However, we must be careful to take into account the quadrant in which the complex number lies. For example, if z = -3 + 4i, then θ = tan^-1(4/-3) + π = 2.2143 radians.

Once we have found the magnitude and argument of a complex number, we can express it in polar form as z = r(cosθ + i sinθ). For example, if z = 3 + 4i, then r = 5 and θ = 0.93 radians, so z = 5(cos0.93 + i sin0.93).