Complex number

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Hypercomplex Numbers

Delivered by Fengyang Wang on Friday November 18, 2016

There are exactly three distinct two-dimensional unital algebras over the reals, up to isomorphism. Each of these algebras corresponds to a unique geometry, with applications. This talk will develop the concepts needed to understand two-dimensional algebras over the reals, starting from the definitions of key concepts. We will rediscover the familiar complex numbers and generalize its construction to find the other hypercomplex number systems. We will then prove the result that these are the unique hypercomplex number systems, up to isomorphism. Finally, we will discuss possible generalizations to $n$ dimensions. Please ensure that you have a good understanding of fundamental concepts of two-dimensional linear algebra.

We will use Catoni, F., Cannata, R., Catoni, V., & Zampetti, P. (2004) as a reference.

A summary of this talk is available here.