Natural numbers can’t even agree whether zero counts
Natural numbers are the counting sequence—starting at 0 or at 1, depending who you ask—denoted ℕ in bold or blackboard type. They label how many (cardinals) and which place in line (ordinals). Subtraction and division only sometimes stay inside their world.
Everyday talk also says positive integers, non-negative integers, whole numbers or counting numbers, terms that overlap imperfectly as zero's membership shifts. Written with ten digits, numerals can also act as pure labels, like jersey numbers, with no arithmetic meaning; these are called nominal numbers. Genuine naturals, by contrast, line up by size, 1 then 2 then 3, giving what mathematicians call a total order.
Addition and multiplication never leave the set, but their inverses can: take a larger number from a smaller one and you get a negative, and division often leaves a remainder. That is one reason the integers, rationals, reals and complex numbers are each built in turn on top of the naturals. Arithmetic studies how to perform the operations, number theory probes their properties, and much of combinatorics counts patterns and structures defined with natural numbers.
People grasp two linked ideas long before any formal definition: a number can measure how big a collection is, as in seven days in a week, or mark a place in a sequence, as in the third day of the month. The first use is cardinal, the second ordinal.
Cardinality can be pinned down without counting at all. Two finite collections are the same size if their members can be paired off one to one, every apple matched with exactly one orange and none left over; three apples and three oranges share a cardinal number because such a pairing exists. If pairing leaves stragglers in one collection, that collection is the larger, which gives a way to compare sizes before any numerals are written down.
Source: Natural number
