Why does the slope of a vertical line not exist? Can you provide a geometric interpretation?

Vertical lines have an undefined slope because the horizontal change is 0 — you cannot divide a number by 0. The slope of a line is a measure of its steepness. Vertical lines have NO SLOPE.

Vertical lines cross the x-axis where y = 0. Can this also be a reason why vertical lines have no slope? I understand the steepness explanation.

Considering lines making angle $\theta$ with the positive x-axis. Its slope is $tan(\theta)$. The limit, as $\theta$ goes to $\frac{\pi}{2}%, $tan(\theta)$ goes to infinity.

(Above image scraped from Purplemath. To see it in context, try [here].) No. The line \(y = x\) also crosses the \(x\)-axis at the origin (that is, where \(y = 0\)), but this line has a slope of \(m = 1\), not zero. P.S. Hi, Sologuitar!

A vertical slope has a "steepness" that is VERTICAL. Ask a mountain climber if a vertical slope exists. They will say yes. They problem is division by zero. The problem is mathematics. Consider a horizontal slope. It exists and is definable in mathematics. Yet what is its "steepness"? Some will say a number cannot be "vertical". How then can a number "horizontal". As again horizontal slopes are definable.

NUMBERS are neither horizontal nor vertical. Those are geometric concepts. While lines are geometric and can have slope, can be horizontal or vertical, numbers are not. Yes, lines have a "steepness" and can be vertical that is not what we are discussing here. The "slope" of a line is a number measuing the steepness. It is the tangent of the angle the line makes with the horizontal (or cotangent of the angle with vertical. Neither tangent of 90 degrees nor cotangent of 0 degrees is defined- there is no number associated with them. A horizontal line has slope tan(0)= cot(90)= 0 but a vertical line has NO slope because tan(90) and cot(0) are not defined.

So if they were defined... 0 = horizontal a/a = "a slope" if a=/=0 a = "vertical" for any a=/= 0 again the only thing holding mathematics back from defining a vertical slope is division by zero.