## Roots of Unity

on Jim Blowers' Mathematics Page

What's the cube root of 1? 1. That's correct. But there are two other cube roots of 1, as mentioned in my article below on "Fractals from Newton's Method". They are complex numbers, and on the complex plane, they form a triangle. It's the same with fourth roots of 1 (1, -1, i, and -i), fifth roots of 1, and so forth. As you can see from the Fractals paper, the expression for the fifth roots of 1 is suprisingly complex. So I tried finding the seventh roots of 1. This involved breaking down an equation into quadratics and solving a cubic equation. The result is below in "The Seventh Roots of Unity". Take a look at it and admire its complexity, and maybe even check my work. There they are. These roots each fill a quarter of a page, but every single one of them, when multiplied by itself 7 times, gives 1.

I am now wondering about the eleventh roots of unity. The result is a quintic equation which is solvable in radicals. However, there is no formula for it, so I am not sure how to find the radical expression, but I imagine it is complex looking.

I have been interested in mathematics since I was a small child. I jumped a grade in high school in mathematics and took calculus in my senior year. I then went to the University of Rochester, and then to Northwestern University, where I earned a Ph.D. in the subject with a thesis in the area of cohomology of groups and topology.

On this page I will highlight some areas of mathematics that interest me a lot. I think these subjects will also interest the average person. These are a list of what is available from this page: .

1. Converting between binary, decimal, and hexadecimal notations.
2. Knowing binary and hexadecimal notations can be vital in this era's emphasis on computers. Some short-cut methods for converting back and forth.
3. Car and Goats Problem. The famous "Monty Hall Problem" can't be solved due to insufficient information. See why.
4. Eliminate the Number Glut! Why am I identified by hundreds of digits when only ten need suffice?
5. The End Plus One Why things will never end.
6. Fractals from Newton's Method. Or how applying a solution technique incorrectly can lead to beautiful patterns.
7. Hamlet is a Big Number. Shakespeare's Hamlet is a big number! Find out why.
8. Hamlet Part II: Complicated Numbers. Including numbers of all complexities from countably complicated to Ultimately Indescribable.
9. Hats and Hangar Queens
10. . Guess the color of your hat, in two different ways, one old and one new.
11. Logarithms Keep Doc Brown in Perspective. The biggest love odyssey of them all was Doc Brown's trip to 1885 in the movie series Back to the Future where he found Clara, who he said was "one in a googolplex". But was she really? Logarithms tell the answer.
12. Mattresses, Contra Dancing, and Quilts. How group theory describes mattress flipping, contra dancing, and some quilts.
13. Pop Quizzes and Trust. What do you do when your math professor calls for a pop quiz?
14. NEW Seventh Roots of Unity. There are seven of these, and these are not the easiest to find.
15.  Supper Groups and Plane Geometry. Miniaturizing geometry creates schedules for people to invite each other to their homes for dinner.
16. Weaving Paper Polyhedra. Mathematical models pretty enough for the Christmas tree. I give a workshop on this.
Jim Blowers