CHAPTER XXIV.
Imaginary quantities.
266. An imaginary quantity is an indicated even root of a negative quantity. In distinction from imaginary quantities all other quantities are spoken of as real quantities. Although from the rule of signs it is evident that a negative quantity cannot have a real square root, yet quantities represented by symbols of the form -a, -1, are of frequent occurrence in mathematical investigations, and their use leads to valuable results. We therefore proceed to explain in what sense such roots are to be regarded.
When the quantity under the radical sign is negative, we can no longer consider the symbol \sqrt{} as indicating a possible arithmetical operation; but just as \sqrt{a} may be defined as a symbol which obeys the relation \sqrt{a} \times \sqrt{a} =a, so we shall define \sqrt{-a} to be such that \sqrt{-a} \times \sqrt{-a}=-a, and we shall accept the meaning to which this assumption leads us.
It will be found that this definition will enable us to bring imaginary quantities under the dominion of ordinary algebraic rules, and that through their use results may be obtained which can be relied on with as much certainty as others which depend solely on the use of real quantities.
267. Any imaginary expression not involving the operation of raising to a power indicated by an exponent that is an irrational or imaginary expression, can be reduced to the form a + b \sqrt{-1} , which may be taken as the general type of all imaginary expressions. Here a and b are real quantities, but not necessarily rational. An imaginary expression in
