50 



PRACTICAL MATHEMATICS 



Case IF. Representation of a bi. 



OC = a measured to the right 



BC = b measured vertically downwards 



OB = VoM~P - r 

 /\ 

 BOX = = 360 - A in the relation 



a bi = r(cos 6 + i sin 0) 



It is evident from the above that a complex quantity can also 

 be represented by the radial line OB, which makes an angle 6 

 with the initial line OX. The real part is the projection of OB 

 on the horizontal axis, and the imaginary part is the projection 

 of OB on the vertical axis. 



32. The Use of i as an Operator. It has already been shown 

 that i = cos 90 + i sin 90 



i z = cos 180 + i sin 180 

 i s = cos 270 + i sin 270 

 i 4 = cos 360 + i sin 360 



Hence if we work with a circle of unit radius and commence 

 with the perpendicular radius as the initial line, the effect of 

 raising i to a power is equivalent to turning this initial radius 

 through a certain number of right angles. This number is fixed 

 by the power. 



Also, if we commence with a horizontal radius in this circle, 

 the process of multiplying by i is represented by turning that 



-_JL!^_ 



' fc~Cbs/it BSinHt" 



FIG. 15. 



horizontal radius anti -clockwise to the vertical position ; while 

 multiplying by i would be represented by turning it clockwise 

 to the vertical position. If we operate in the same way on a 

 quantity a, then the result is expressed by turning a horizontal 

 radius of length a, anti -clockwise or clockwise, to the vertical 

 position. 



The quantity b sin pt measured horizontally can be represented 

 as the horizontal projection of a radial line of length b, inclined 

 at an angle pt to the initial vertical line. Then operating on this 

 by i would have the effect of turning this radial line through a 

 right angle. The horizontal projection of the radial line in its 



