I'm currently studying linear fractional transformations and cross ratios and came across this in a book (this is translated from Korean, so I apologize if there are any errors or ambiguities):
We can define the cross ratio for complex numbers as:
$$[ z, z_2, z_3, z_4 ] = \dfrac{\dfrac{z - z_3}{z - z_4}}{\dfrac{z_2 - z_3}{z_2 - z_4}}$$
If we think of this as a function of complex number $z$ and express it as $S(z)$, we can easily observe that:
$$\begin{align}S(z_2) & = 1 \\ S(z_3) & = 0 \\ S(z_4) & = \infty\end{align}$$
How was the $\infty$ derived? The other two are easy because you just plug in the values, but plugging in $z_4$ gives:
$$S(z_4) = \left( \dfrac{z_4 - z_3}{z_4 - z_4} \middle/ \dfrac{z_2 - z_3}{z_2 - z_4} \right)$$
This may be due to my lack of background, but how does this result in $\infty$? Thanks.
$z_4 - z_4 = 0$, so you have $0$ at a denominator. Now recall that on the Riemann sphere we have: $$ \frac{1}{0}:= \infty $$ (note that we are not using a sign). This explains why $S(z_4)= \infty$.