If a line passes through distinct points #(x_1, y_1)# and #(x_2, y_2)# then the slope of the line is given by the formula:

slope #m = (Delta y)/(Delta x) = (y_2 - y_1) / (x_2 - x_1)#

If the line is vertical then #x_2 = x_1# so the denominator is #0#.

You can mess with the numbers you are using by adding a 'number' called #oo# which will allow you to express the slope of a vertical line. It can be a useful shorthand, but it does not fix everything and can lead to sloppy reasoning. For example, what is the value of #0 * oo#?

For a more formal approach to using #oo# in an advanced setting you might look at the behaviour of

#f(z) = (az+b)/(cz+d)#

on the Riemann sphere #CC_oo#. Then again, perhaps that's something to look forward to in a few years time.