Mathematics for Physicists

1 Real numbers, variables and functions

In this chapter we introduce some simple ideas about real numbers, i.e. the ordinary numbers used in arithmetic and measurements, real variables and algebraic functions of a single variable. This discussion will be extended in Chapter 2 by considering some important examples in more detail: polynomials, trigonometric functions, exponentials, logarithms and hyperbolic functions. Much of the material in these first two chapters will probably already be familiar to many readers and so is covered briefly, but even if this is the case, it is useful revision and sets the scene for later chapters.

1.1 Real numbers

This section starts from the basic rules of arithmetic and introduces a number of essential techniques for manipulating real numerical quantities. We also briefly consider number systems other than the decimal system.

1.1.1 Rules of arithmetic: rational and irrational numbers

The first contact with mathematics is usually via counting, using the positive integers 1, 2, 3, 4, … (also called natural numbers). Later, fractional numbers such as , etc. and negative numbers −1, −3, , , etc. are introduced, together with the rules for combining ...