Properties of the modulus - complex variables question

[jaejoon89]

Given
f(z) = (z+1) / (z-1) for z not equal to 1

My teacher wrote
|f(z)| = |x+1+iy| / |x-1+iy| = sqrt((x+1)^2 +1) / sqrt((x-1)^2 + 1)

How do the values within the modulus work out to the right hand side? I can't figure it out.

 

[snipez90]

That's the definition of the modulus. You square the real part and then you square the imaginary part and add the two together and finally you take the square root. But I think there is an error since the imaginary part of z = x + iy is y, not 1.

 

[futurebird]

$$f(z) = \frac{z+1}{z-1}$$

$$|f(x+iy)| = \frac{|x+iy+1|}{|x+iy-1|}$$

$$= \frac{|x+1+iy|}{|x-1+iy|}$$
$$= \frac{\sqrt{(x+1)^2+(iy)^2}}{\sqrt{(x-1)^2+(iy)^2}}$$
$$= \frac{\sqrt{(x^2+2x+1)+(-y^2)}}{\sqrt{(x^2-2x+1)+(-y^2)}}$$
$$= \frac{\sqrt{(x^2+2x+1)+(-y^2)}}{\sqrt{(x^2-2x+1)+(-y^2)}}$$

?