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M4005 Geometry

April 3, 2006

Introduction to complex numbers.

The imaginary number i, which has the property that ii = -1.

Complex numbers can be written in the form a + ib, where a and b are real numbers.

The complex numbers can be put into correspondence with points in the plane. Looking at them this way, adding a fixed complex constant b to an arbitraty complex number z is a translation:

T(z) = z + b (a translation).

Multiplying by a fixed complex constant m is a dilation together with a rotation about 0.

W(z) = mz (a rotation together with a dilation by a factor of |m| ).

In class, we did some experiments where we plotted collections of complex points, multiplied tham all by a fixed complex and looked at the resulting configuration.

The next class will include a more careful and rigorous treatment.