Question

Does a vertical line have infinite slope?

Answer 1

A few thoughts...

Explanation:

Conventionally, when dealing with numbers, functions and limits we would add two objects to the real line, namely `+oo` "infinity" and `-oo` "minus infinity".

If a line passes through two points `(x_1, y_1)` and `(x_2, y_2)` then its slope `m`, is given by the formula:

`m = "rise"/"run" = (Delta y)/(Delta x) = (y_2-y_1)/(x_2-x_1)`

If `x_2 = x_1` then the denominator of `(y_2-y_1)/(x_2-x_1)` is `0`.

Conventionally, we say that division by `0` is undefined. If the numerator is non-zero then we might be tempted to say that the slope is infinite, but is it `+oo` or `-oo` ?

If you imagine a non-vertical line gradually twisting until it becomes vertical then its slope would either gradually increase without limit or gradually decrease without limit. If you rotated it past vertical then its slope would suddenly jump from very large and positive to very large and negative or vice versa.

There is one circumstance in which we can meaningfully and unambiguously say that the slope of a vertical line is infinite. That is when we do not distinguish between `+oo` and `-oo`.

The real projective line `RR_oo` is effectively the result of bending the real line into a circle and joining it with a single point called `oo`. Then we can define:

`1/0 = oo`

`1/oo = 0`

Some expressions are indeterminate, for example:

`oo - oo`

`oo/oo`

`0 * oo`

So `oo` does not fully behave like a number, but it is nevertheless a useful concept.

Apart from using the real projective line, it is normally best to simply describe the slope of a vertical line as undefined.

Answer 2

A line perfectly perpendicular to the `x` axis has `"no slope."`

Explanation:

When I was in high school, the math teacher tried to help us distinguish between
A slope of `0`  –   compared to   –  `"no slope"`

He said
▸ Slope is something that happens to floors.
▸ Walls don't have slope.

`0` slope ... a perfectly flat floor

`"No slope"` ... a wall, not even a floor at all

He said "`"slope"`" is not a property of vertical lines.

I'm not so sure that this is a mathematically rigorous answer, but I certainly remembered it for all this time.