Natural number

The natural numbers are the numbers used for counting: $0, 1, 2, 3, \dots$, starting at 0 and going on by adding 1, with no largest. Their set is written $\mathbb{N}$. On this page you take two of them, $a$ and $b$, and one of the four operations: counters carry it out, the number line shows where the result lands, and a table of every $a$ and $b$ from 0 to 20 shows where the result is again a natural number.

There is no universal convention. This page follows the international standard ISO 80000-2, which includes 0: when there are none, the count is 0. Many books start at 1 instead. Where it matters, names that leave no doubt are used instead: the nonnegative integers $\{0, 1, 2, \dots\}$, often written $\mathbb{N}_0$, and the positive integers $\{1, 2, 3, \dots\}$, often written $\mathbb{N}^{+}$ or $\mathbb{Z}^{+}$. Before reading a formula elsewhere, check which one it means.

Putting two collections together and counting gives a count again, and so does $a$ rows of $b$: sums and products of natural numbers are natural numbers, and the addition and multiplication tables are lit everywhere. Taking away works only when there is enough: $a - b$ is a natural number exactly when $a \ge b$, which is the diagonal of the subtraction table and the cells below it.

When $b > 0$, sharing $a$ things equally into $b$ groups comes out exactly only when $b$ divides $a$. Otherwise the natural numbers still say what happens, by division with remainder: $a = qb + r$ with natural numbers $q$, $r$ and $0 \le r < b$. Seven shared into two groups gives 3 each and 1 over, $7 = 3 \times 2 + 1$. Split the 1 left over as well and each group gets $3 + \tfrac{1}{2}$, so the quotient $7 \div 2 = 3.5$ lies between 3 and 4. Division by 0 is another matter. $a \div b$ is the number that gives $a$ when multiplied by $b$, and every number times 0 is 0. So $a \div 0$ has no value when $a \ne 0$, and $0 \div 0$ has no single value. Either way it is undefined: not a number outside the natural numbers, but no result at all.

To make every subtraction possible the numbers are extended to the integers $\mathbb{Z}$, which add $-1, -2, \dots$; to make every division by a nonzero number possible, to the rational numbers $\mathbb{Q}$. Division by 0 stays undefined at every stage.