The study of the calculus and its many applications depends crucially on the properties of real numbers. The set of all real numbers is often represented

symbolically by writing either R or R. If x is a real number this is often

shown by writing x R or x /R. Here the symbol / is to be read ‘belongs to’. The formal mathematical name for this symbol is the set membership relation

symbol. There are three different types of real numbers:

1 Positive integers or natural numbers 1, 2, 3, . . . 2 Rational numbers (fractions) of the form p /q with p, q integers with no

common factor, such as 1/3, 27/5, /5/16, . . . 3 Irrational numbers or numbers such as 2 which cannot be expressed as a

rational number.