Number

A number is a unit of quantity in mathematics. Some numbers include:

There are lots of different types of numbers, like the natural numbers for example. The natural set is usually denoted as $\mathbb{N}$ and contains every positive integer, and starts 1, 2, 3, 4, 5... . But this is just an example.

Numbers are abstract mathematical objects, for instance, the number 4 on its own doesn't mean 4 apples, 4 centimeters, it just means 4.

Name of numbers[]

Every three digits a number has after the first digit (assuming it is read left-to-right) the number gets a different name. You can also say that $1000^{n}$ gets a different name depending on the value of n.

Number	Name
$1000^{1}$	Thousand
$1000^{2}$	Million
$1000^{3}$	Billion
$1000^{4}$	Trillion
$1000^{5}$	Quadrillion
$1000^{6}$	Quintillion
$1000^{7}$	Sextillion
$1000^{8}$	Septillion
$1000^{9}$	Octillion
$1000^{10}$	Nonillion
$1000^{11}$	Decillion
$1000^{12}$	Undecillion
$1000^{13}$	Duodecillion
$1000^{14}$	Tredecillion
$1000^{15}$	Quattuordecillion
$1000^{16}$	Quindecillion
$1000^{17}$	Sexdecillion
$1000^{18}$	Septendecillion
$1000^{19}$	Octodecillion
$1000^{20}$	Novemdecillion
$1000^{21}$	Vigintillion
$1000^{22}$	Unvigintillion
$1000^{23}$	Duovigintillion
$1000^{24}$	Tresvigintillion
$1000^{25}$	Quattuorvigintillion
$1000^{26}$	Quinvigintillion
$1000^{27}$	Sexvigintillion
$1000^{28}$	Septenvigintillion
$1000^{29}$	Octovigintillion
$1000^{30}$	Novemvigintillion
$1000^{31}$	Trigintillion
$1000^{32}$	Untrigintillion
$1000^{33}$	Duotrigintillion
$10^{100}$	Googol

Numbers as sets[]

Using Set theory, people have found other ways to construct the natural numbers. Specifically, these numbers can be defined by using recursion with 0 = ∅ or the empty set, and the successor function being defined as $\forall n\in \mathbb {N} ,n+1=n\cup \{n\}$ . In plain English, this formula takes any natural number n, defines it as a set starting at 0 and going up to n-1, so the set has exactly n elements. n+1 represents the length of the list, so if n=2, the set would have n+1 or 3 elements. Zero is already defined with the empty set, so we can write it in two ways, one with the natural numbers, or using just empty sets. The first 5 natural numbers are defined as sets like this:

0: {}, or ∅ 1: {0} or {∅}, 2: {0,1} or {∅,{∅}}, 3: {0,1,2} or {∅,{∅},{∅,{∅}}}, 4: {0,1,2,3} or {∅,{∅},{∅,{∅},{∅,{∅},{∅,{∅}}})

You can clearly see patterns emerge from defining numbers like this!