Question

If product of two # = 0, show that one factor is 0?

Answer 1

where z=x+iy, I let z_1=x_1+iy_1 and similarly z_2=x_2+iy_2. Then I multiplied z_1*z_2 together and got

Answer 2

\[z_1*z_1 =x_1x_2 + ix_1y_2 +ix_2y_1 - y_1_y_2 \]

Answer 3

so I can say that in order for the entire equation to be 0 at least either z_1 or z_2 has to be 0?

Answer 4

if z1*z2 = 0,

and z1*z2 is a complex number, then

z1*z2 = 0+0i

Answer 5

yeah but how do we proof that? I don't get the 'show one factor must be 0' part

Answer 6

prove*

Answer 7

z1*z2 = 0+0i

x1*x2 + i*x1*y2 + i*x2*y1 - y1*y2 = 0+0i

(x1*x2 - y1*y2) + (x1*y2 + x2*y1)*i = 0+0i

this implies that

x1*x2 - y1*y2 = 0

and

x1*y2 + x2*y1 = 0

Answer 8

Oh I got it now thanks so much!

Answer 9

from there you have to show that either x1,y1 are 0 or x2,y2 are zero

Answer 10

yw

Answer 11

There's a straightforward relationship for the magnitude of the product of two complex numbers. Let \(z_1=r_1e^{i\theta_1}\) and \(z_2=r_2e^{i\theta_2}\). We see \(z_1z_2=r_1r_2e^{i(\theta_1+\theta_2)}\) and thus \(|z_1z_1|=r_1r_2\). If \(z_1z_2=0\) we know \(|z_1z_2|=0\) and thus \(r_1r_2=0\) (these are real numbers). We know that either \(r_1,r_2=0\) and thus either \(z_1,z_2=0\) since \(z=0e^{i\theta}=0\)