Types of Numbers

Natural Numbers

The set of natural numbers is represented by the symbol N, and is defined as the set of counting numbers.

That is, the set of natural numbers, N = {1, 2, 3, 4, 5,….}.

Note, zero is not a natural number.

The set of even and odd numbers are two types of natural numbers. Even numbers are those which are exactly divisible by 2.

That is, the set of even numbers = {2, 4, 6, 8, 10, 12,…}

Odd numbers are those which are not exactly divisible by 2.

That is, the set of odd numbers = {1, 3, 5, 7, 9, 11, 13, 15,…}

Whole Numbers

The set of whole numbers is represented by the symbol W, and is defined as the set of natural numbers including zero.

That is, the set of whole numbers, W = {0, 1 , 2, 3, 4, 5….}.

Integers

The set of integers is represented by the symbol Z, and is defined as the set of whole numbers and the negatives.

That is, the set of integers, Z = {… -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5…}

Note zero is neither positive nor negative.

Positive integers are defined as being prime or composite numbers. Prime numbers are numbers which have only two factors (1 and itself).

That is, the set of prime numbers = {2, 3, 5, 7, 11, 13, 17, 19…}.

Note: 1 is not a prime number since it only has one factor (itself).

Composite numbers are numbers which have more than two factors (can be divided by other numbers apart from 1 and itself).

That is, the set of composite numbers = {4, 6, 8, 9, 10, 12, 14, 15, 16}.

Rational Numbers

The set of rational numbers is represented by the symbol Q, and is defined as the set of numbers (whether positive or negative) which can be written as fractions. A fraction is a number written in the form  , where n is the numerator, d is the denominator, and both n and d are integers.

Examples of rational numbers:

That is, the set of rational numbers, Q = { : n and d are integers}

Irrational Numbers

The set of irrational numbers is represented by the symbol I, and is defined as the set of numbers which cannot be expressed as fractions.

Examples of irrational numbers: √3, √99, π=3.14159

Real Numbers

The set of real numbers is represented by the symbol R, and is defined as the set of rational and irrational numbers.

That is, the set of real numbers, R = {rational and irrational numbers}

Relationship between the types of numbers

On reading the definitions of the respective types of numbers, a relationship should become apparent. This relationship is that:

The set of natural numbers is a subset of the set of whole numbers; the set of whole numbers is a subset of the set of integers; the set of integers is a subset of the set of rational numbers; the set of rational numbers is a subset of the set of real numbers.

The above relationship written in terms of symbols is:

N ⊂ W ⊂ Z ⊂ Q ⊂ R

Where, ⊂ means ‘a subset of’.