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Complex Numbers

Polar Form of Complex Numbers

Problems

Problems

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

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