Natural number

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the natural numbers are the numbers <a href="/facts/0/4NBRtKHc">0</a>, <a href="/facts/1/m3wpSfMf">1</a>, <a href="/facts/2/u5WC44t9">2</a>, <a href="/facts/3/6QzW9RkE">3</a>, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers 0, 1, 2, 3, ..., while others start with 1, defining them as the positive integers 1, 2, 3, ... . Some authors acknowledge both definitions whenever convenient. Sometimes, the whole numbers are the natural numbers as well as zero. In other cases, the whole numbers refer to all of the <a href="/facts/Integer/PUwwrawx">integers</a>, including negative integers. The counting numbers are another term for the natural numbers, particularly in primary education, and are ambiguous as well although typically start at 1.
The natural numbers are used for counting things, like "there are six coins on the table", in which case they are called <a href="/facts/Cardinal_number/ccoKwU4R">cardinal numbers</a>. They are also used to put things in order, like "this is the third largest city in the country", which are called <a href="/facts/Ordinal_number/GelFLjJt">ordinal numbers</a>. Natural numbers are also used as labels, like <a href="/facts/Number_(sports)/PUAmHg5b">jersey numbers</a> on a sports team, where they serve as <a href="/facts/Nominal_number/vcEbjAY4">nominal numbers</a> and do not have mathematical properties.
The natural numbers form a <a href="/facts/Set_(mathematics)/BfucfHMq">set</a>, commonly symbolized as a bold N or <a href="/facts/Blackboard_bold/CGTmvA8H">blackboard bold</a> ⁠
 
 
 
 
 N
 
 
 
 {\displaystyle \mathbb {N} }
 
⁠. Many other <a href="/facts/Number_set/BfucfHMq">number sets</a> are built from the natural numbers. For example, the <a href="/facts/Integer/PUwwrawx">integers</a> are made by adding 0 and negative numbers. The <a href="/facts/Rational_number/VJQmyY05">rational numbers</a> add fractions, and the <a href="/facts/Real_number/R02gw5Pb">real numbers</a> add all infinite decimals. <a href="/facts/Complex_number/2w7mqI7e">Complex numbers</a> add the <a href="/facts/Imaginary_unit/1SCllQlU">square root of −1</a>. This chain of extensions canonically <a href="/facts/Embedding/HKeVKc42">embeds</a> the natural numbers in the other number systems.
Natural numbers are studied in different areas of math. <a href="/facts/Number_theory/zhoa7zrc">Number theory</a> looks at things like how numbers divide evenly (<a href="/facts/Divisibility/DKNa2GvC">divisibility</a>), or how <a href="/facts/Prime_number/L20FLkgn">prime numbers</a> are spread out. <a href="/facts/Combinatorics/k14xUoKT">Combinatorics</a> studies counting and arranging numbered objects, such as <a href="/facts/Partition_(number_theory)/LtyXWqwH">partitions</a> and <a href="/facts/Enumerative_combinatorics/AtqWU2bj">enumerations</a>.

Natural number open-in-new

Natural number