Definition 6.13. The n th roots of unity are all of the complex numbers z , such that z n = 1, or equivalently, z n - 1 = 0. 27
To make our discussion of roots of unity much simpler, we will introduce some basics about the polar coordinate system. Recall that a polar coordinate has two dimensions: ( r, θ ), where r is the radius (the distance from the origin) and 0 ≤ θ < 2 π is the angle from the horizontal axis going in the counter clockwise direction. One can imagine a polar coordinate system superimposed onto the complex coordinate system. Using this idea, we can convert complex coordinates into polar coordinates. Notice that the radius of a polar coordinate is equal to the distance between the origin and the corresponding complex coordinate pair. Thus, for some z = ( x, y ), | z | = | re iθ | = | r || e iθ | . We will need a better understanding of e iθ to compute | r || e iθ | . To find θ for some complex coordinate pair, we can set up a right triangle between Re( z ), Im( z ) and the modulus of z . Using trigonometry, when | z | = 1 it follows that, the Im( z ) = sin θ , and Re( z ) = cos θ . Therefore z = cos θ + i sin θ . By Euler’s famous formula (see below), we have cos θ + i sin θ = e iθ . This powerful formula allows us to represent complex numbers easily in polar form. Here is the statement of Euler’s formula: Theorem 6.14. Euler 0 s Formula . Let z = x + yi be a complex number and let 0 ≤ θ < 2 π . Then z = re iθ where cos θ = x , sin θ = y and r = | z | . Now we may finish our computation of | e iθ | : | z | = | re iθ | = | r || e iθ | = | r | · 1 since | e iθ | = p cos 2 θ + sin 2 θ = 1 . Thus, | z | = | r | . Note that r is always positive. For n th roots of unity, by Proposition 6.12 (1), we have | z n | = | z | n = 1. Notice that | z | is a real number. Therefore, | z | = 1 for all roots of unity. Example 6.15. Consider the complex number z = - 1 2 + i √ 3 2 . Let us compute 28
the modulus: | z | = v u u t - 1 2 2 + √ 3 2 ! 2 = r 1 4 + 3 4 = 1 As is, z looks somewhat intimidating. However, we convert z into polar form by using cos θ = Re( z ) = - 1 2 . Thus θ = 2 π/ 3, and so the polar form is re iθ = e 2 πi/ 3 . Theorem 6.16. Fundamental Theorem of Algebra . Every non-constant single-variable polynomial with complex coefficients has at least one complex root. Consequently, every non-zero single-variable polynomial with complex co- efficients has exactly as many complex roots as its degree, as long as all multi- plicities of roots are counted. Note that monic polynomials factor over the complex numbers as ( z - r 1 )( z - r 2 ) · · · ( z - r n ) where each r i is a root of the monic polynomial. Theorem 6.16 allows us to easily factor polynomials whose roots are the n th roots of unity. The theorem also allows us to conclude that X n - 1 = 0 has precisely n roots over the complex numbers. Remember that z is an n th root of unity if z n = 1. Converting z into polar form yields z n = ( e iθ ) n = e iθn = 1. We must find values of θ for which e iθn = 1. Using Euler’s formula again, we must find θ values for which e iθn = cos nθ + i sin nθ = 1. Since there is no imaginary component to the number 1, we know sin nθ = 0. Therefore n must sastify cos nθ = 1. These two previous equations are satisfied only when nθ is any multiple of 2 π . However, due to the
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