Why does a vertical line have a SLOPE (steepness) of undefined?

Question

The curve shows :

• x goes 1 to right
• y goes up by 3

the SLOPE is defined as the ratio of rise to the step

SLOPE = rise/step = 3/1 = 3

Now that :

• Horizontal line has SLOPE of 0.
• Vertical line has SLOPE of undefined.

It makes sense for me to imagine horizontal line has SLOPE of 0 since there is no rise at all. But why the vertical has SLOPE of undefined.?

Answer 1

Because $dx$ (step) for a vertical line is 0 and the expression for slope $$m=\frac{dy}{dx}$$ results in the bogeyman of mathematics, division by zero. That's why.

However, in some cases we define the slope of a vertical line as $\infty$. This implies $\frac 1{\infty}=0$ and vice versa, and is useful in projective geometry among other areas of mathematics.