I did a little research on Omar Khayyam this weekend. I had never heard of this man before our class and I was very interested in his work with cubic equations that we talked about. Khayyam did some awesome work with cubic equations by using geometry in his book

*Treatise on Demonstration of Problems of Algebra*(1070). He also wrote about a lot of other mathematical discoveries like the triangular array of binomial coefficients, known to us as Pascal's triangle. (Pascal's book on the triangle wasn't published until 1665. So, why in the world is it named after Pascal? Something to think about…)

Here is a quote that Professor Golden commented :

“When I want to understand what is happening today or try to decide what will happen tomorrow, I look back.”

― Omar Khayyám

Anyway, Khayyam was able to create a parabola and circle and find solutions of cubic equations using the intersection of the two. And here is whats really crazy: He did all this before we had graphing calculators and computers and all that!

Below is a neat animation from Geogebra that illustrates Khayyam's use of circles and parabolas.

The animation is from https://tube.geogebra.org/m/rTV3y4Bb and gives the description:

"Khayyam showed that the solution could be found from the intersection point of a parabola with a semi-circle. By varying the scale of the parabola and the diameter of the circle you can solve the cubic equation for any positive value of b and c."

Using the red diamonds A and D the equation can be changed to find the solutions to different cubic equations.

Khayyam was clearly a very intelligent man. It totally blows my mind that he did all this graphing before any type of graphing technology was invented.

Heres a picture of a manuscript from Khayyam's book that shows some of his drawings on the subject. This is all pretty awesome if you ask me.
As you can see, Khayyam was a pretty kick-butt guy, but we have only looked at one of his great mathematical ideas. Now lets talk about the binomial theorem. According to my research, Omar Khayyam had a pretty decent understanding of the binomial theorem. People know this because he shows his understanding of extracting roots from binomials in some of his mathematical works. So you might ask: what exactly is the binomial theorem? |

If we want to see how the binomial theorem works we can try something like:

**a**counts down from 6 for each term and the exponent for

**b**counts up from 0 to 6 for each term. Each terms exponents will sum to 6.

You can use this expansion rule for any positive integer n, or you can use the equation for the binomial theorem which is: