## Example 1.2.14 (a): Multiplying geometrically |

Explain geometrically why |

Recall that *Arg(z)* is the principle angle of *z*.
According to our result on
multiplication of complex numbers that implies:

Arg(z*w) = Arg(z) + Arg(w)

If we apply that to our vectors we have:

*Arg( (1+i)*(1-i) ) = Arg(1+i) + Arg(1-i) = /4 - /4 = 0*and a vector with angle 0 lies on the poitive real axis and is thus purely real.*Arg(i*and a vector with angle lies on the negative real axis and is thus purely real.^{2}) = Arg(i*i) = Arg(i) + Arg(i) = 2/2 =*Arg((1+i)*and a vector with angle lies on the negative real axis and is thus purely real.^{4}) = 4 Arg(1+i) = 4/4 =