The four basic arithmetical operations are addition, subtraction,
multiplication, and division. These operations are sufficient to
perform most of our everyday calculations. One might wonder what
kind of number system is sufficient to allow us to carry out such
arithmetical operations. Note that many systems are not sufficient.
For example, the set of the natural numbers
supports addition and multiplication fully, but when you subtract
or divide a natural number from or by another, the result is often
not a natural number (for example, ^{5}/_{2} = 2.5 is
not a natural number, nor is 3 − 5 = −2).

We can express this "deficiency" in the natural numbers by using
the concepts of additive and multiplicative inverses. The *additive
inverse* of a number is the number that, when added to the number,
produces
zero.
The *multiplicative inverse* of a number is the
number that, when multiplied by the number, produces one.

A system of numbers wherein every number has an additive inverse
and every number except for zero has a multiplicative inverse, and
also obeys some basic mathematical laws, is called a *field*.
The rational numbers are an example of
a field.

A field is a type of ring; every field is a ring, but not every ring is a field. A field is a commutative ring with unity with at least two elements, where every element except the identity element has an inverse with respect to the second operation (in terms of the rational numbers, the second operation is multiplication, and the inverse of the second operation is division).