The Imaginary in Geometry 3 



tiplies both black and red vectors by the same real number without 

 changing any direction. It is a simple magnification (fig. 3). 



Fig. 3. 



The multiplication by a complex number is more complicated. 

 Consider, therefore, first the multiplication of (1, i) by 



e iO ==cos0-f- 2 sin 6. 



The product is (cos 6-\-i sin 6, i cos 9 — sin 6). An inspection 

 of fig. 4 shows that this is merely the double vector of (1, i) 



