Counting numbers

Counting is older than writing. The Lebombo bone, a baboon fibula from the Border Cave in Eswatini carrying twenty-nine notches, has been dated to roughly 42,000 years ago; the Ishango bone from the Democratic Republic of the Congo, with its three grouped columns of tallies, is perhaps 20,000. What such objects record is disputed — lunar counts, trade, something else entirely — but the method is not. A notch stands for a thing, and the notches are read in order.

Defining what was being counted took far longer. Greek arithmetic began at two, since monas, the unit, was treated as the thing numbers were made of rather than a number itself, and zero was not a quantity at all. Zero arrives as a full number in India: Brahmagupta's Brāhmasphuṭasiddhānta of 628 CE gives rules for arithmetic with it, including the observation that a number minus itself is zero.

The modern footing is nineteenth-century. Hermann Grassmann showed in 1861 that addition and multiplication could be built up recursively from the successor operation alone. Richard Dedekind set out a set-theoretic treatment in Was sind und was sollen die Zahlen? in 1888, and Giuseppe Peano published his axioms the following year — a starting element, a successor function that never repeats itself or returns to the start, and induction. Everything arithmetic says about the counting numbers follows from those.

The disagreement about zero never resolved; it was standardised twice. ISO 80000-2 includes zero in the natural numbers, which is the usual convention in set theory and computer science, while much of number theory still starts at one. Both are current, so careful writing names which it means.

Key properties

• The counting numbers are closed under addition and multiplication: adding or multiplying two of them always gives another. Subtraction and division are not closed, which is what forces the integers and the rationals.
• Every non-empty set of counting numbers has a least member. This well-ordering principle is equivalent to the principle of mathematical induction.
• Peano’s axioms characterise them from a first element and a successor function: no two numbers share a successor, the first element is nobody’s successor, and any property holding at the start and inherited by successors holds everywhere.
• The sum of the first n counting numbers is n(n+1)/2, the nth triangular number: 1 + 2 + 3 + 4 + 5 = 15.
• Every counting number greater than 1 factors into primes in exactly one way up to order — the fundamental theorem of arithmetic.
• The set is infinite but countable, with cardinality ℵ₀. Cantor showed in 1874 that the real numbers are not countable, so not all infinities are the same size.
• Whether zero belongs is a convention, not a fact. ISO 80000-2 includes it; much of number theory does not. The terms "positive integers" and "non-negative integers" are unambiguous where it matters.

Where they turn up

• Array indexing splits on the same question as zero does. Edsger Dijkstra argued for zero-based indexing in his 1982 note Why numbering should start at zero, on the grounds that it makes the subscript range half-open and the length arithmetic subtraction-free; C, Python and Java follow it, while Fortran, MATLAB, R and Lua count from one.
• Gödel’s incompleteness theorems, published in 1931, are statements about exactly this sequence: any consistent formal system strong enough to express arithmetic on the counting numbers contains true statements it cannot prove.
• Street numbering, page numbers, invoice sequences and database auto-increment keys all lean on the same property — that the successor is always available, so a new item can always be added at the end without renumbering what came before.
• The Hilbert hotel, a thought experiment David Hilbert used in a 1925 lecture, turns the countability of this sequence into a paradox: a hotel with a room for every counting number, all occupied, can still take a new guest by moving everyone up one room.