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Polar Form of Complex Numbers

Polar Form of Complex Numbers

Polar Form of Complex Numbers

Polar Form of Complex Numbers

Polar Form of Complex Numbers

Polar Form of Complex Numbers

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)]    

These equations arise from the sum and difference formulas for the trigonometric functions sine and cosine.

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() + ısin() , 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.

Figure %: The roots of a complex number graphed in the coordinate plane