# What is the slope of a vertical line?

Jun 17, 2015

The slope of a vertical line is undefined like $\frac{1}{0}$ is undefined.

#### Explanation:

If a line passes through distinct points $\left({x}_{1} , {y}_{1}\right)$ and $\left({x}_{2} , {y}_{2}\right)$ then the slope of the line is given by the formula:

slope $m = \frac{\Delta y}{\Delta x} = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}}$

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 $\infty$ 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 $0 \cdot \infty$?

For a more formal approach to using $\infty$ in an advanced setting you might look at the behaviour of

$f \left(z\right) = \frac{a z + b}{c z + d}$

on the Riemann sphere ${\mathbb{C}}_{\infty}$. Then again, perhaps that's something to look forward to in a few years time.