Structure of the Real Number System - SS1 Mathematics Lesson Note

All numbers in existence are one of two kinds, complex numbers (which we shall not treat) and real numbers. Real Numbers are in turn made up of rational and irrational numbers. Rational numbers are real numbers that can be expressed as a ratio of two integers (whole numbers) such as \(\frac{2}{3}\), \(- \frac{4}{9}\), \(\frac{27}{6}\), \(2\) which is \(\frac{2}{1}\), \(\sqrt{}4\) which is \(2 = \ \frac{2}{1}\) whilst irrational numbers cannot be expressed as ratio of two integers such as \(\sqrt{}2\), \(\sqrt{}5\) and \(\pi\). Rational Numbers are further divided into integers and non-integers.

Integers are made up of positive whole numbers, zero and negative whole numbers. Non-integers are fractions. Fractions can be expressed in either vulgar \((\frac{3}{5})\), decimal \((0.6)\) or percentage form \((60\%)\), where \((\frac{3}{5},\ 0.6\ and\ 60\%\) are all the same fractional quantity\()\).

Positive integers start at \(1,\ 2,\ 3,\ 4,\ 5,\ 6\ \ldots\) and continue to infinity \((\infty)\), meaning they go on forever. These positive integers are known as Counting or Natural Numbers.

Within the family of natural numbers, there are even (numbers that can be divided by 2 without remainder), odd (numbers that cannot be divided by 2 without remainder), prime (numbers that are only divisible by themselves and 1), and composite numbers (numbers that are not prime).