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# Hamilton's Quaternions

by

Paul Friedman
Union College

October 19, 2009
4:15 pm
Bailey Hall 201

Refreshments will be served at 4 pm in Bailey 204

## Abstract:

One can view the set of complex numbers,$\mathbb{C}$ as a set of ordered pairs of real numbers under the identification $a+bi \leftrightarrow (a,b)$. In this way, multiplication of two complex numbers can be interpreted as a way to multiply ordered pairs: $(a,b)\cdot (c,d)=(ac-bd, ad+bc)$. This viewpoint was developed in 1835 by Sir William Rowan Hamilton. Related to this, Hamilton then asked how one ordered triples $(a, b, c)$ could be multiplied in analogy to couples. In this talk, we will discuss Hamilton's attempts to multiply triples and why he inevitably failed! However, initial failures often lead to greater successes. Though unable to multiply triples, Hamilton was able to determine how to multiply ordered quadruplets, leading to his discovery (invention?) of the quaternions , named $\mathbb{H}$ in his honor. We will then compare and contrast the field $\mathbb{C}$ with (the not quite a field) $\mathbb{H}$, with some interesting results.

For additional information, send e-mail to math@union.edu or call (518) 388-6246.
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