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
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: .
- Converting between binary, decimal, and hexadecimal
Knowing binary and hexadecimal notations can be vital in this era's emphasis
on computers. Some short-cut methods for converting back and forth.
- Car and Goats Problem. The famous "Monty
Hall Problem" can't be solved due to insufficient information. See why.
- Eliminate the Number Glut! Why am I identified
by hundreds of digits when only ten need suffice?
- The End Plus One Why things will never
- Fractals from Newton's Method. Or how applying
a solution technique incorrectly can lead to beautiful patterns.
- Hamlet is a Big Number. Shakespeare's
Hamlet is a big number! Find out why.
- Hamlet Part II:
Complicated Numbers. Including numbers of all complexities from countably
complicated to Ultimately Indescribable.
- Hats and Hangar Queens
. Guess the color of your hat, in two different ways, one old and one new.
- 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.
- Mattresses, Contra Dancing, and Quilts. How
group theory describes mattress flipping, contra dancing, and some quilts.
- Pop Quizzes and Trust. What do you do when your
math professor calls for a pop quiz?
- NEW Seventh
Roots of Unity. There are seven of these, and these are not the easiest
- Supper Groups and Plane Geometry. Miniaturizing
geometry creates schedules for people to invite each other to their homes
- Weaving Paper Polyhedra. Mathematical models
pretty enough for the Christmas tree. I give a workshop on this.
Page updated 2002 August 30