Elementary operations with complex numbers. When real numbers are combined by addition, subtraction, multiplication, or division with non-vanishing divisor, the results are real numbers; such numbers therefore form a closed system for these operations. But this is not always true when we pass to root extraction. No real number can be the square root of a negative real number.

The situation is analogous to one which exists for the number system composed of the positive integers. Here we have a system closed for addition, but in which subtraction of a number from one not greater than itself is impossible. When it seems desirable to allow such an operation, the difficulty is met by the enlargement of the number system so as to include zero and the negative integers. In order that division with non-vanishing divisor may always produce a number within the system, we pass from the system of positive and negative integers to the system of rational numbers which includes fractions as well as integers. The totality of all real numbers constitutes a still more inclusive system which is closed for the additional operation of passing to a finite limit.

We shall find that complex numbers include the real numbers and form a system which permits root extraction as well as the other operations we have noted.