(a) Explain in detail why the n^th roots of 1 lie on the vertices of a regular n-gon. What is t

(a) Explain in detail why the n^th roots of 1 lie on the...

Key Concepts

Complex Numbers in Polar Form

Every complex number can be expressed in polar form as r·e^(i?), where r denotes the magnitude and ? the argument. This representation is crucial because it allows finding the roots of a complex number by taking the nth root of the magnitude and dividing the angle by n, leading to points that are evenly distributed in angle.

De Moivre's Theorem

De Moivre's theorem states that for any real number ? and integer n, (e^(i?))^n = e^(in?). This theorem is used to derive the formula for the nth roots of a complex number by equating magnitudes and dividing the argument by n, showing how complex roots are computed systematically.

nth Roots and Roots of Unity

Finding the nth roots of a complex number involves solving the equation z^n = c, which yields n distinct solutions. In particular, when c = 1 (the roots of unity), the formula simplifies to e^(2?ik/n) for k = 0, 1, ..., n-1, showing that these roots are spaced apart by an angle of 2?/n on the unit circle.

Regular n-gon in the Complex Plane

When plotting the nth roots of unity in the complex plane, the points form the vertices of a regular n-gon. This occurs because the roots are equally spaced by an angular increment of 2?/n, ensuring that the shape is symmetrical and each side of the polygon has equal length.

Generalization to Non-zero Complex Numbers

The property of equiangular spacing applies not only to the roots of unity but also to the nth roots of any non-zero complex number. By representing any non-zero complex number in polar form, its nth roots can be obtained by taking the nth root of its magnitude and dividing its argument by n. This results in a rotated and scaled version of the regular n-gon formed by the roots of unity.

Transcript

00:01 In this video, we're going to assume that a complex number z is a non -real complex number.

00:07 Therefore, in its polar form, r equal to r -e to the theta i, theta, the angle measurement, is neither zero degrees nor 180 degrees.

00:18 So this point does not lie on the horizontal axis in a complex number plane.

00:24 And let's explain why none of the nth roots of z are also going to be along the x -axis.

00:30 To explain this, let's actually assume, for the sake of contradiction, that the nth roots of this complex number z do indeed lie on the x -f axis or the horizontal axis.

00:56 So assuming that our nth roots of z do lie on, or at least one nth rate of z lies on the real axis or horizontal axis, this would mean that the angle measurement associated with that complex root is equal to either zero or 180 degrees.

01:13 According to demois theorem, if the angle measurement associated with z is theta, then the angle measurements for the nth root of z would be theta plus 360k divided by n.

01:32 So those would be the angle measurements for the nth root of z, evaluated at different integers or different whole numbers for k.

01:41 So if we're assuming that our nth root is on the x -axis, then for one of our, at least one of our values for k, we have either zero or 180 degrees for this angle measurement.

02:00 So then solving for theta, let's see what happens.

02:05 Let's put this into two equations, actually.

02:08 Theta plus 360k divided by n equals zero.

02:13 And theta plus 360k divided by n equals 180 degrees.

02:22 With our first equation, multiplying both sides by n, our root, gives us theta plus 360k equals zero.

02:34 So now let's think about where that angle measurement would be.

02:38 Theta is a non -real complex number, so it might be, you know, somewhere over here.

02:45 Maybe it is in quadrant one.

02:47 If we add 360 degrees to that, which is what it says in this equation, that means we're going around the circle all the way, 360 degrees.

03:06 And different values of k, we might be going around the circle once, twice, three times, etc., some whole number of circles.

03:13 Which means that we're not at zero degrees.

03:16 We're at whatever angle measurement would be for z...

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