2 4] NUMBER. 5 



Since even powers of all real numbers are positive, there is no 

 real number that has a square equal to 1. If we further extend 

 the idea of numbers, so as to call that a number whose square is 

 - 1, we have a means of satisfying the equation. If we denote the 

 new number by i, defined by the equation i 2 = 1, we may multiply 

 it by any real number, positive or negative, integral, fractional, or 

 irrational, and thus get a class of new numbers, known as pure 

 imaginary numbers. Evidently no imaginary number is equal to 

 a real number, for the quotient of two real numbers is always real. 



If we consider the sum of a real and an imaginary number, we 

 arrive at the conception of a complex number (in the narrow 

 sense). Two complex numbers are equal when their real parts are 

 equal and their imaginary parts also. Any equation containing 

 complex numbers is accordingly equivalent to two equations con- 

 taining only real numbers. In particular the equation 



a4-bi = 0, 



where a and b are real, is equivalent to the two, 



a = and 6 = 0. 



A complex number vanishes only when its real and imaginary 

 parts both vanish. 



4. Complex Numbers in the Extended Sense. As we 



have formed numbers by multiplying the real and imaginary units 

 1 and i by all real numbers and forming sums therefrom, so we 

 may still further extend the notion of numbers to include sums of 

 terms each formed by multiplying any number n of different units 

 by real numbers. Such numbers are complex numbers in the 

 extended sense, a number involving n units 'e s being an w-fold 

 number. The units may have any properties by which we wish to 

 define them. If they are all independent of one another, it is 

 obvious that two complex numbers are equal only when composed 

 of the same number of each unit e s , so that any equation contain- 

 ing all the units is equivalent to n equations containing only real 

 numbers. In particular, a complex number 



a = a^ + 2 e 2 + a n en, 



vanishes only when the coefficient a. 8 of each unit e s is zero. 



Two complex quantities satisfy the associative and commuta- 

 tive laws with respect to addition, and accordingly the sum of 

 a = OL& + OA + a n e n and 6 = 



