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slope picture

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.?

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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.

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