Most of this chapter should be familiar, but it is important that you really understand all of the material, which is largely a series of definitions.

Numbers

Real numbers are numbers that can be fitted into a place on the number scale (Fig. 1.1). The other kinds of numbers are complex (or imaginary) numbers, which cannot be fitted onto this scale, but lie above or below the line. They are of the general form a + ib, where a and b are real numbers but i is the square root of −1.

Real numbers can be divided into:

1. Integers: these are whole numbers, positive or negative, such as 7, 341, −56.

2. Rational numbers: these can be expressed precisely as the ratio of two integers. All integers are rational (they can be written as n / 1) and many non-integers are also rational, such as 3 / 4, 2.5 (= 5 / 2), −7.36 (= −736 / 100).

3. Irrational numbers: these cannot be precisely expressed as the ratio of two integers; examples are π (which is not exactly 22 / 7 nor any other ratio of integers) and the square roots of all prime numbers (except 1). Note that a number that has to be written as a recurring decimal is not irrational: 0.333 333 … is exactly 1 / 3; and 0.142 857 142 857 142 857 … is 1 / 7. Also, all approximations are rational: if we give π the approximate value of 3.142 this is 3142 / 1000, a rational number.