How are polar coordinates useful for working with complex numbers (hint: think about powers and roots)?

Answer 1

Put simply, the polar representation of complex numbers allows us to raise them to powers and to take roots very easily!

Cartesian Form

Complex numbers are often first taught in Cartesian Form.

z = a + bi

'z' is the complex number, 'i' is the imaginary unit, 'a' is the real part, and 'b' is the imaginary part. This form lends itself very nicely to the 2D cartesian plane. However, it can be very tedious to raise these complex numbers to powers or to take roots of them.

Polar Form

Complex numbers can also be represented in Polar Form.

z = r(cosθ + i·sinθ) or z = re^(iθ)

'z' is the complex number, 'i' is the imaginary unit, 'e' is the Euler constant, 'r' is the magnitude of the complex number, and 'θ' is the angle from the positive horizontal axis. These forms lend themselves very nicely to the 2D polar plane. Remember, the magnitude and angle can be calculated from these equations:

r = √(a²+b²) and θ = arctan(b/a)

Powers and Roots

As stated previously, the polar form of complex numbers allows us to raise them to powers and take roots very easily, using De Moivre's Theorem.

To get the n-th power of a complex number 'z', use one of these equations:

zⁿ = rⁿ(cos(nθ) + i⋅sin(nθ)) or z = rⁿe^(inθ)

To get the n-th roots of a complex number 'z', use one of these equations:

ⁿ√(z)= ⁿ√(r)(cos((θ+2kπ)/n) + i⋅sin((θ+2kπ)/n)) or ⁿ√(z)= ⁿ√(r)e^(i(θ+2kπ)/n)

It is important to note that for n-th roots, there will be 'n' distinct solutions. The way to obtain all solutions is to use different values for the variable 'k'. The variable 'k' should be set to 0, 1, ... , all the way up to n - 1. For example, if 'n' is equal to 5, set 'k' equal to 0, 1, 2, 3, and 4.

That's it! In summary, polar form lets us easily compute powers and roots of complex numbers, making these operations far simpler than in Cartesian form.