CHAPTER IV 



31. The Graphical Representation of a Complex Quantity. 



Now i = r(cos + i sin 0) where = 90 and r = 1 

 = cos 90 + * sin 90 



Thus i can be represented as the radius, drawn vertically up- 

 wards, of a unit circle. 



i z = (cos 90 + i sin 90) 2 

 = cos 180 + i sin 180 



Then i 2 or 1 can be represented as the radius, drawn hori- 

 zontally to the left, of a unit circle. 



3 = (cos 90 +i sin 90) 3 

 = cos 270 + i sin 270 



Then i 3 or i can be represented as the radius, drawn vertically 

 downwards, of a unit circle. 



i 4 = (cos 90 + i sin 90) 4 

 = cos 360 + i sin 360 



Then i 4 or +1 can be represented as the radius, drawn hori- 

 zontally to the right, of a unit circle. 



We therefore see that each time we multiply by i we are turning 

 the radius of a unit circle, in anti-clockwise direction, through a 

 right angle. Also the odd powers of i always involve the angle 

 90 and odd multiples of 90, while the even powers of i always 

 involve the angle 180 and multiples of 180. Now the odd 

 powers of i give i, and the even powers give 1. Thus, 

 taking the horizontal direction as the direction of measurement 

 for real quantities, we can take the vertical direction as the 

 direction of measurement for imaginary quantities. The complex 

 quantity a + hi can be represented graphically as the sum of two 

 magnitudes, a measured horizontally and b measured vertically, 

 but in this representation we have also to take into consideration 

 the algebraic signs of a and b. 



Case I. Representation of a + bi. 



OC = a measured to the right 



BC = b measured vertically upwards 



OB = Va 2 + b 2 = r 

 /\ 

 BOX = A = in the relation 



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



48 



