Let z = a + bı . Consider the complex number z plotted in the complex plane. The distance between z and the origin is called the modulus of z (symbolized | z| ) and is equal to . The angle θ between the positive real axis and the ray containing z whose endpoint is the origin is called the argument of z , and is equal to arctan() . The polar form of a complex number z = a + bı is this: z = r(cos(θ) + ısin(θ)) , where r = | z| and θ is the argument of z . Polar form is sometimes called trigonometric form as well.
The polar form of a complex number is especially useful when we're working with powers and roots of a complex number. First, we'll look at the multiplication and division rules for complex numbers in polar form. Let z _{1} = r _{1}(cos(θ _{1}) + ısin(θ _{1}))andz _{2} = r _{2}(cos(θ _{2}) + ısin(θ _{2})) be complex numbers in polar form.
multiplicationanddivision
z _{1}×z _{2} = r _{1} r _{2}[cos(θ _{1} + θ _{2}) + ısin(θ _{1} + θ _{2})] |
= [cos(θ _{1} - θ _{2}) + ısin(θ _{1} - θ _{2})] |
The power of a complex number is given by an equation known as De Moivre's Theorem: Let z = r(cos(θ) + ısin(θ) . Then z ^{n} = [r(cos(θ) + ısin(θ)]^{n} = r ^{n}(cos(nθ) + ısin(nθ) , where n is any positive integer.
The roots of a complex number are also given by a formula. A complex number a + bı is an n th root of a complex number z if z = (a + bı)^{n} , where n is a positive integer. A complex number z = r(cos(θ) + ısin(θ) has exactly n n th roots given by the equation [cos() + ısin()] , where n is any positive integer, and k = 0, 1, 2,..., n - 2, n - 1 .
When the nth roots of a complex number are graphed in the complex plane, they all lie on the same circle with radius r ^{ }1n . They are also all evenly spaced around the circle, like spokes on a bike. This is because the arguments of consecutive roots differ by a measure of radians.