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Complex Numbers |
The product of a real number multiplied by itself is 0 or positive, so the equation x2 = -1 has no solutions in the real number system. If such a solution is desired, new numbers must be invented. Let i = Á be a new number representing a solution of the preceding equation. All numbers of the form a + bi, in which a and b are real numbers, belong to the complex number system. If b is not 0, the complex number is called an imaginary number; if b is not 0 but a is 0, the complex number is called a pure imaginary number; if b is 0, the complex number is a real number.
Imaginary numbers (the term must not be used in a literal sense but in the technical sense just described) are extremely useful in the theory of alternating currents and many other branches of physics and natural science. The relationships of the various types of numbers are illustrated in the number family tree above.
In 1799 the German mathematician Carl Friedrich Gauss proved that every algebraic equation of degree n having the form xn + a1xn-1 + ... + an-1x + an = 0 in which a1, a2..., an are arbitrary complex numbers, is satisfied by at least one complex root (see Equations, Theory of).
Whereas real numbers represent points on a line, complex numbers can be placed in correspondence with the points on a plane. To represent the complex number a + bi geometrically, the x-axis is used as the axis of the real number a, and the y-axis serves as the axis of the pure imaginary bi; the complex number, therefore, corresponds to the point P with the rectangular coordinates a and b. The line, or vector, joining the origin with the point P is the diagonal of a rectangle with the sides a and bi. If the complex number a + bi is multiplied by -1, the vector OP is rotated through 180°, and the point P falls in the third quadrant; a rotation of 90° is achieved by multiplying the complex number by i.
Mystical and magical qualities have been ascribed to numbers both in antiquity and in modern times. The pseudoscience of numerology attempts to interpret the occult by means of the symbolism of numbers, which is based on the doctrine of the Greek philosopher and mathematician Pythagoras that all things are numbers and consist of geometrical figures in various patterns.
