The imaginary unit i = v-1 has been used by mathematicians for nearly five-hundred years
during which time its physical meaning has been a constant challenge. Unfortunately René
Descartes referred to it as imaginary and the use of the term complex number compounded the
unnecessary mystery associated with this amazing object. Today i = v-1 has found its way into
virtually every branch of mathematics and is widely employed in physics and science from
solving problems in electrical engineering to quantum field theory. John Vince describes the
evolution of the imaginary unit from the roots of quadratic and cubic equations Hamilton's
quaternions Cayley's octonions to Grassmann's geometric algebra. In spite of the aura of
mystery that surrounds the subject John Vince makes the subject accessible and very readable.
The first two chapters cover the imaginary unit and its integration with real numbers. Chapter
3 describes how complex numbers work with matrices and shows how to compute complex
eigenvalues and eigenvectors. Chapters 4 and 5 cover Hamilton's invention of quaternions and
Cayley's development of octonions respectively. Chapter 6 provides a brief introduction to
geometric algebra which possesses many of the imaginary qualities of quaternions but works in
space of any dimension. The second half of the book is devoted to applications of complex
numbers quaternions and geometric algebra. John Vince explains how complex numbers simplify
trigonometric identities wave combinations and phase differences in circuit analysis and how
geometric algebra resolves geometric problems and quaternions rotate 3D vectors. There are two
short chapters on the Riemann hypothesis and the Mandelbrot set both of which use complex
numbers. The last chapter references the role of complex numbers in quantum mechanics and ends
with Schrödinger's famous wave equation. Filled with lots of clear examples and useful
illustrations this compact book provides an excellent introduction to imaginary mathematics
for computer science.
