(a) Draw a picture of the complex plane with the complex...

(a) Draw a picture of the complex plane with the complex solutions to $z^6=1$ marked. (b) What is the exact formula (no trigonometric functions allowed) for the primitive sixth root of unity $\zeta_6$ ? (c) Verify explicitly that $1+\zeta_6+\zeta_6^2+\zeta_6^3+\zeta_6^4+\zeta_6^5=0$. (d) Give a geometrical explanation of this identity.

(a) Draw a picture of the complex plane with the complex solutions to $z^6=1$ marked. (b) What is the exact formula (no trigonometric functions allowed) for the primitive sixth root of unity $\zeta_6$ ? (c) Verify explicitly that $1+\zeta_6+\zeta_6^2+\zeta_6^3+\zeta_6^4+\zeta_6^5=0$.
(d) Give a geometrical explanation of this identity.

Key Concepts

Geometric Symmetry

The geometric explanation for many identities involving roots of unity relies on the rotational symmetry in the complex plane. When the roots of unity are plotted, they form vertices of a regular polygon inscribed in the unit circle. This symmetry implies that operations, such as adding all the vertices as vectors, result in cancellation and yield zero, reflecting their balanced distribution around the origin.

Sum of Roots of Unity

An important property of the roots of unity is that their sum equals zero. This results from the symmetrical distribution of these roots on the unit circle, causing their vector representation to cancel out. It also relates to the fact that the minimal polynomial having these roots has its quadratic term evenly balanced about the origin.

Primitive Root of Unity

A primitive nth root of unity is a root of unity that generates all the other roots when raised to the appropriate powers. It is not a power of a smaller root of unity and has full order n. This concept is central in number theory and abstract algebra, reflecting the cyclic structure of the multiplicative group of the roots of unity.

Complex Numbers and the Complex Plane

This concept refers to numbers expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit. The complex plane is a two-dimensional plane in which each complex number corresponds to a unique point with coordinate (a, b). It provides a visual framework to represent and analyze complex numbers, particularly by plotting them along horizontal (real) and vertical (imaginary) axes.

Roots of Unity

Roots of unity are the solutions to the equation z^n = 1 in the complex number system. They are evenly spaced on the unit circle in the complex plane, where each solution represents a rotation by a fixed angle. This concept illustrates the symmetry inherent in these solutions and shows how complex multiplication corresponds to rotations on the circle.

Transcript

00:01 All right, we want to take these complex numbers and rewrite them in a different form.

00:06 So first we're starting off with negative 1 plus i, and we want to write that in polar form.

00:12 So just for the visual, i'll plot it here.

00:15 Here's negative 1 and here's i, then negative 1 plus i is right here.

00:21 Now remember, we need two things to write it in polar form.

00:24 We need r and we need theta.

00:28 So r is just the distance from the origin, right? it's how far it is from the complex number to zero.

00:38 And using the pythagorean theorem, or using the formula for r that we have, which is r squared equals a squared plus b squared, which is really just the pythagorean theorem.

00:48 We're saying that that line is equal to this squared plus this squared.

00:55 So plugging those in, a is negative 1, b is positive 1, so that tells me that r squared is 2, which tells me that r is the square root of 2.

01:13 And then secondly, we need theta, which is the angle between the positive x -axis and that line.

01:22 So i can see here, i can use geometric intuition to realize that that's got to be 135 degrees.

01:30 Or i can use the formula tan theta equals b over a, which would be.

01:37 Tell me that tan theta is negative 1.

01:44 And using the unit circle or using the inverse tangent button on a calculator, i can figure out that it's got to be 135 degrees.

01:54 Okay, so once i have those things, then i just plug it into the polar form, which is r times the cosine of 135 degrees plus i times the sign of a hundred and thirty five degrees plus i times the sign of a 135 degrees.

02:16 And there we have it.

02:18 All right, for the next part, we have a complex number written in polar form, and we want to put it back into rectangular form.

02:27 So again, just for the visual, let's visualize this here...

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