Complex numbers

Mathematicians customarily write in such numbers as i, and any complex numbers as a+bi.

Ordinary numbers can all be thought of as lying along a single straight line, a continuous stream without gaps in it - what mathematicians call a "continuum". But a typical complex number, a+bi, has no place on the line of ordinary numbers.

When two ordinary numbers are multiplied, the result is a jump along the straight line. When two complex numbers are multiplied, however, the result is a spectacular trapeze like swing within the two-dimensional plane.

The excentric behaviour of the complex numbers is important because it matches perfectly, and therefore serves as a literal translation of the behaviour of many quantities in nature, such as forces, velocities or accelerations, which act in definite directions. When two forces are exerted from different directions on the same point, for instance, their net effect is a third force with a new direction. Diagrammatically the strength and direction of each of the two forces can be represented as the length and direction of a line segment. Each of these two line segments in turn can be represented by a complex number, and the two complex numbers added together will then represent the third force which arises from the combination of the first two.

The line segments that symbolize forces, velocities and the like are called "vectors", and are an essential tool of physics. The fact that they and complex numbers behave alike mathematically makes it possible to analyse complicated situations in which many forces are all acting at once.

Numbers which serve to represent forces, velocities and accelerations acting in more than two dimensions are "hypercomplex numbers" - expressions like a+bi+cj+dk, in which the units i, j, and к when multiplied together, produce minus one.

The most astonishing thing about these hypercomplex numbers is that they flout a basic rule of arithmetic previously thought inviolate. When multiplied together, the same two hypercomplex numbers may produce different results depending on the order in which they are taken; hypercomplex number a times hypercomplex number b does not always equal hypercomplex b times hypercomplex a.