Polar coordinates
The representation of a complex number as a sum of a real and imaginary number, z = x + iy, is called its Cartesian representation.Recall from trigonometry that if x, y, r are real numbers and r^{ 2} = x^{ 2} + y^{ 2}, then there is a unique number θ with 0 ≤ θ < 2π such that
sin(θ) = y / r.
That number is
θ | = | cos^{ -1}(x / r), |
= | sin^{ -1}(y / r) | |
= | tan^{ -1}(y / x) |
We can therefore express any complex number z = x + iy as
z | = | | z | (x / | z | + iy / | z |) |
= | | z | (cos θ + i sin θ) | |
= | | z | e^{ iθ} , |
where θ = tan^{ -1}(y / x). The angle or argument θ is measured in radians, and it is written as arg(z). So we have the polar representation of any complex number z as
The two representations are related by
and
The values x and y are called the Cartesian coordinates of z, while r and θ are its polar coordinates. Note that r is real and r ^{3} 0.
Note that for any integer K,
This is because
and
Thus, the polar coordinates (r, θ) and (r, θ + 2Kπ) for any integer K represent the same complex number. Thus, the polar representation is not unique; by convention, a unique polar representation can be obtained by requiring that the angle given by a value of θ satisfying 0 ≤ θ < 2π or -π < θ ≤ π.
Example 1
The polar representation of the number 1 is 1 = 1 e^{ i}0. Notice that it is also true that 1 = 1 e^{ i}2π, because the sine and cosine are periodic with period 2π. The polar representation of the number -1 is -1 = 1 e^{ i}π. Again, it is true that -1 = 1 e^{ i}3π, or, in fact, -1 = 1 e^{ i}π + K2π for any integer K. |
Products
Products of complex numbers represented in polar coordinates are easy to compute. If z_{i } = r_{i }e^{ i}θ_{i }, thenThus, the magnitude of a product is a product of magnitudes, and the angle of a product is the sum of the angles,
arg(z_{1 }z_{2 }) = arg(z_{1 }) + arg(z_{2 })
Example 2
We can use the polar representation to find the n distinct roots of the
equation z^{ n} = 1. Write z = re^{
i}θ, and 1 = e^{
i}2kπ, so
which gives r = 1 and θ = 2kπ / n, k = 0, 1, ... , n - 1. These are called the n roots of unity. |
Figure: The 5 roots of unity.