Numbers

Linguistic Definition

A number is a symbol which denotes some quantity.

Example

If we say, "there are 5 books on the table" then the symbol \(5\) is representing the quantity or amount of books.

Types

Following are the types of numbers.

• Natural numbers
• Odd numbers
• Even numbers
• Whole numbers
• Integer numbers
• Rational numbers
• Irrational numbers
• Real numbers
• Complex numbers

Natural Numbers

The set¹ of natural numbers is a union¹ between sets¹ of odd and even numbers.

\[\mathbb N = \mathbb O \, \cup \, \mathbb E\]

\[\mathbb{N} = \{1, 2, 3, \ldots\}\]

Even Numbers

By definition, an even number is such a number which can be divided by \(2\).
Assume \(n\) be any number.
Then \(q\) is an even number if \(q = 2n\) because now \(q\) can be divided by \(2\).

\[\mathbb E = \{2, 4, 6, \ldots\}\]

\[\mathbb E = \{x : x = 2n \quad \land n \in \mathbb Z\}\]

Odd Numbers

An odd number is a number which is not divisible by \(2\).
Intuitively, every even number is succeeded by an odd number.
Therefore

\[\mathbb O = \{x:x = 2n + 1 \quad n \in \mathbb Z\}\]

\[\mathbb O = \{1, 3, 5, \ldots\}\]

Whole Numbers

The set¹ of whole numbers is just the set¹ of natural numbers with the addition of \(0\) element.¹

\[\mathbb W = \{0\} \, \cup \, \mathbb N\]

\[\mathbb W = \{0, 1, 2, \ldots\}\]

Integer Numbers

So far, we have only seen sets¹ of positive numbers.
The set¹ of integers contains all the previous numbers and their negatives.

\[\mathbb Z = \mathbb Z^+ \, \cup \, \{0\} \, \cup \, \mathbb Z^-\]

Where

\[\mathbb Z ^ + = \mathbb N\]

Intuitively,

\[\mathbb Z^- = \{-1, -2, -3, \ldots\}\]

\[\therefore \mathbb Z = \{0, \pm 1, \pm 2, \pm 3, \ldots\}\]

Rational Numbers

The numbers which can be represented as fractions \(\frac p q\) where \(p, q \in \mathbb Z \land q \neq 0\) are called rational numbers.
These are represented by \(\mathbb Q\).

Irrational Numbers

From the definition of rational numbers, these are the numbers which cannot be represented using fractions i.e. \(\frac p q\) where \(p, q \in \mathbb Z \land q \neq 0\).
These are represented by \(\mathbb Q^\prime\).

Examples

• \(\pi\)
• \(e\)

Real Numbers

The set¹ of real numbers is a union¹ between sets¹ of rational and irrational numbers.

\[\mathbb R = \mathbb Q \, \cup \, \mathbb Q^\prime\]

Complex Numbers

Details can be read from here.

References

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1. Read more about sets. ↩↩↩↩↩↩↩↩↩↩↩