IMAGINARY QUANTITY. An imagi nary or complex quantity is an expression con sisting of a pair of real numbers, or numbers which can be represented by positive or nega tive terminating or non-terminating decimals. The pair consisting of a and b is written a + ib, which is in general different from b ia. A complex quantity of the form a + 1.0, is said to be real, but must not be confused with a number which is real in the primary sense, notwithstanding the many common prop erties which the two possess. A complex quan tity of the form 0+ ib is called a pure imagi nary. If x be the complex quantity a + ib and y is the complex quantity c + id, x-Fy is defined as (a -I- c)-1- t(b d), and .ry as (ac—bd)-1-i(bc + ad). Most of the theorems of arithmetic and algebra suffer no loss of validity by their transference to complex quan tities, while, as was proved by Gauss, every algebraic equation in complex algebra has at least one root. For this reason, complex alge bra is the algorithm of mathematicians par ex cellence.
0+ ii, or i, as it is written, has the prop erty that its square is —1, so that it is often written V—I. It cannot be defined as
however, for in ordinary algebra is meaningless, and operating with meaningless symbols yield sheer nonsense. On the other hand, if we start with complex algebra, the symbol i is usually one of the initial concepts.
The so-called Argand representation of the complex number (x + iy) is the point (x, v) on a Cartesian co-ordinate-plane. If the angle between the x-axis and the line from the origin to (x, y) be 0, and the distance from the origin to (x, y) be P. we get the result that x sy = p (cos 0 i sin e). It may be shown that [p (oos i sin 8)] [p' (cos 0 + i sin 01] =pp [cos (6 + 0') i sin (6 + OTI, so that the multiplication of two points on the Argand diagram is attained by adding their O's or arguments and multiplying their p's or ampli tudes. The addition of two points results in a point which forms a parallelogram together with them and the origin.