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in the Engineering Auditorium room 189

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The real numbers, which make up the number line, are familiar to everyone and are used everywhere in science and mathematics. But they are so ... well, one-dimensional. They lack glamour and also some useful properties. For example, there is no room on the number line for square roots of negative numbers.

It turns out that points in hte plane can be added and multiplied just as well as points on the line, and they all have square roots. These two dimensional numbers — called "complex" numbers — are useful not only in arithmetic, but also in geometry, engineering, and physics.

So two dimensions are better than one. Why stop there? Why not three, four, or more dimensions? Surprisingly, there are no reasonable three-dimensional numbers. The only dimensions which come close to having "numbers" are the dimensions four and eight, so good dimensions are 1, 2, 4, 8.

This talk will explain how it is possible for numbers to be two-dimensional but not three dimensional, and will describe the four- and eight-dimensional "numbers" (quaternions and octonions) and where they come from.

John Stillwell was born in Melbourne, Australia, and educated at Melbourne University and MIT. From 1970 to 2001 he taught at Monash University in Australia, but since 2002 he has been Professor of Mathematics at the University of San Francisco.

He has written numerous books on mathematics, the best known of which isMathematics and Its History(2^{ nd }edition, Springer-Verlag, 2002).

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