TL;DR: Natural numbers are the counting numbers: 1, 2, 3, 4, … Whether zero belongs is a genuine disagreement — mathematicians usually include it, school curricula often start at 1. They are the numbers used for counting and ordering, they never run out, and they are closed under addition and multiplication but not subtraction, which is exactly why negative numbers had to be invented.
The two definitions, and why both exist
There is no single standard, and the confusion is worth clearing up before it causes trouble.
Starting at 1 — 1, 2, 3, … — is the everyday definition. It matches what counting means: you count objects starting from one. School textbooks, particularly in the UK and much of Europe, use this.
Starting at 0 — 0, 1, 2, 3, … — is the set-theoretic definition. Zero is the number of elements in an empty set, and building the naturals from sets naturally produces 0 first. Most formal mathematics uses this.
Neither is wrong. What matters is which convention a particular document uses, because the difference changes answers in set notation, in programming, and in any context where “the first natural number” matters.
What makes them special
Four properties, each of which explains something about how arithmetic behaves.
They are closed under addition and multiplication. Add two natural numbers and you get a natural number. Multiply them and the same is true. This is not true of every operation, which is the point.
They are not closed under subtraction. 3 − 5 has no answer within the naturals. That gap is why integers exist — the negatives were invented precisely to make subtraction always possible.
They are not closed under division. 7 ÷ 2 does not land in the naturals either, which is why fractions exist.
They are well-ordered. Every non-empty set of natural numbers has a smallest element. That sounds trivial and is the foundation of mathematical induction, which is the main proof technique for statements about counting.
The pattern here is worth noticing: each extension of the number system — integers, rationals, reals, complex — exists because some operation failed to stay inside the previous set.
Natural, whole, integer, rational
The terms nest, and mixing them up produces errors in word problems.
Natural numbers — counting numbers, 1, 2, 3, … (possibly including 0).
Whole numbers — the naturals plus zero. If your definition of natural already includes zero, these two are identical, which is why some curricula avoid the term entirely.
Integers — whole numbers and their negatives: …, −2, −1, 0, 1, 2, …
Rational numbers — anything expressible as a fraction of two integers, including terminating and repeating decimals. 0.5 and 0.333… are both rational.
Irrational numbers — real numbers that cannot be written as such a fraction: π, √2, e.
The nesting is: naturals ⊂ integers ⊂ rationals ⊂ reals. Every natural number is an integer; not every integer is natural.
The infinite part
The natural numbers are infinite, but the kind of infinity matters.
They are countably infinite, meaning you can list them in order with a first element and no gaps. That sounds like it should be true of every infinite set, and it is not: the real numbers cannot be listed this way, which is the substance of Cantor’s diagonal argument and the reason there is more than one size of infinity.
Countability has a practical consequence that surprises people. The set of all even numbers is the same size as the set of all natural numbers, because you can pair them up — 1 with 2, 2 with 4, 3 with 6. “Half as many” is meaningless for infinite sets. Matching by pairing is the definition of equal size.
That result is unintuitive, and it is why the naturals are used as the reference for counting infinite sets at all.
Why anyone outside a maths class cares
Three practical places the distinction matters.
Indexing. If you convert letters to numbers, A as 1 is using the natural numbers in their counting sense. A as 0 uses zero-based indexing, which is what programming languages do. Mixing them shifts every value by one, and the error is invisible because the output still looks like letters — this is the same 1-based versus 0-based problem that causes off-by-one bugs everywhere.
Data types. A column typed as an unsigned integer accepts only naturals. That is a real constraint with a real failure mode: subtracting below zero either errors or wraps to a huge number, depending on the system.
Counting agreements. “The number of items” is a natural. A count of −1 or 0.5 signals a bug rather than a measurement. Treating a count as a float invites rounding errors that accumulate.
If you are working with letter values and need to check which indexing convention a puzzle used, converting letters to numbers side by side makes the shift visible immediately — an A of 1 against an A of 0 identifies the convention in one look.
Common mistakes
Calling zero a natural number without saying which definition you use. In an exam or a spec, state it.
Assuming natural numbers include negatives. They do not; that is the integers.
Treating “whole number” as a distinct set when zero is included in the naturals. Under the 0-based definition, the terms coincide.
Assuming subtraction is always possible. It is not, within the naturals, and that is the historical reason the negatives exist rather than a technicality.
The short version
Natural numbers are the counting numbers, with zero included or excluded depending on convention. They are closed under addition and multiplication but not subtraction or division, which is why integers and fractions exist. They are countably infinite, indexing conventions shift by one, and the nested order is natural, integer, rational, real.
