Moreover, if we multiply a real number a by a complex number x+iy,wegeta(x+iy)=ax+i(ay). In terms of points in the complex plane, we have that(x, y)is moved to(ax, ay), or to write it another way, a(x, y)=(ax, ay).
15. Multiplication by i rotates a complex number by a right angle
The kinds ofoperations that enjoy these two properties are known as linear and are ofparamount importance throughout all mathematics. Here, I wish only to draw to your attention to the fact that the effect ofsuch an operation L is determined by its action on the two points(1,0)and(0,1), for let us suppose that L(1,0)=(a, b)and L(0,1)=(c, d). Then for any point(x, y)we have(x, y)=x(1,0)+y(0,1), and so using the properties ofa linear operation we obtain: