natural number
Natural numbers are the numbers you first learn as a child by pointing and counting: one apple, two apples, three apples. They are the everyday answer to the question “how many?” and they march upward without ever stopping: 1, 2, 3, 4, 5, and so on forever.
More precisely, the natural numbers are the positive whole quantities used for enumerating objects. Each one has a next number (its successor), so the list never ends — there is no largest natural number. They are closed under addition and multiplication, meaning the sum or product of two natural numbers is again a natural number, but they are not closed under subtraction or division (3 − 5 and 2 ÷ 4 leave the set).
Be careful: there is a genuine convention clash about zero. Many school textbooks and number theorists start the naturals at 1, while many logicians and computer scientists start them at 0. Neither is “wrong” — just check which convention a book is using. In this glossary we treat the natural numbers as starting at 1, and we call the set with zero added the whole numbers.
Counting the letters in “algebra” gives the natural numbers 1, 2, 3, 4, 5, 6, 7 — one for each letter, ending at 7.
Natural numbers answer “how many?”
The symbol for the set of natural numbers is N. Whether 0 belongs to N is a matter of convention, not of mathematical fact.