Number systems

Numbers can be grouped into different sets.
Many standard sets of numbers can be indicated using specialized set notations such as N, Z, Q, and R.

Natural numbers are Positive integers, including 0. These are numbers used to count things.

Integer numbers are whole numbers, including both positive and negative values.

Rational numbers are any number that can be represented as a fraction, including an integer.

Irrational numbers are any number that cannot be represented as a fraction, including π and square roots of non-square numbers.

Real numbers are the collection of all rational and irrational numbers.

Ordinal numbers are numbers used to count the order that something appears; for example, 1st, 2nd, 3rd, and so on.

Natural numbers

The set of natural numbers, N, is the set of positive integers. A handy way to remember this is that, when humans first started counting, these were the numbers that were obvious. 0 people, one person, two people, three people, and so on.
N = { 0 , 1 , 2 , 3 , 4 , … }

The set of natural numbers is infinitely large, so we use the notation ‘…’ to show that the set carries on forever.

Integer numbers

The set of integer numbers, Z, is the set of all integers, both positive and negative.
Z = { … , −3 , −2 , −1 , 0 , 1 , 2 , 3 , … }

Rational numbers

The set of rational numbers, Q, is the set of all numbers that can be represented as fractions. This includes integers since all integers can be written as a fraction of 1 (for example, 3 = 3/1).
Q includes { … , −2 , −4/3 , 0 , 1/100 , … }

Irrational numbers

The set of irrational numbers is the set of all real numbers which are not rational numbers (in other words, numbers that cannot be represented as fractions). It does not use a specialized set notation, though it can be considered as the set of all real numbers, minus all rational numbers.

The set of irrational numbers includes { … , π ,√2, e , … }

Real numbers

The set of real numbers, R, contains all rational and irrational numbers. It does not include imaginary numbers, such as i or √−2.

It is important to recognize the relationships between the sets. For example, the set of natural numbers is a subset of the set of integer numbers; the set of integer numbers is a subset of the set of rational numbers; the set of rational numbers and the set of irrational numbers are both subsets of the set of real numbers.

Ordinal numbers

Ordinal numbers are used to describe the order in which numbers appear. For example, in the ordered set {apple, banana, clementine}, ordinal numbers can be used to indicate that ‘clementine’ is the third item in the set.

Counting and measurement

Natural numbers are used for counting. This is because the numbers are discrete and relate to physically countable objects.
Real numbers are used for measurement. This is because the numbers are continuous and can be expressed in more flexible terms (for example, calculating a hypotenuse using a square root, or a measurement of a circle using π)