Definition:Slope/Straight Line

Definition

Let $\LL$ be a straight line embedded in a Cartesian plane.

The slope of $\LL$ is defined as the tangent of the angle that $\LL$ makes with the $x$-axis.

General Form

Let $\LL$ be a straight line embedded in a Cartesian plane.

Let $\LL$ be given by the equation:

$l x + m y + n = 0$

The slope of $\LL$ is defined by means of the ordered pair $\tuple {-l, m}$, where:

for $m \ne 0$, $\psi = \map \arctan {-\dfrac l m}$
for $m = 0$, $\psi = \dfrac \pi 2$

where $\psi$ is the angle that $\LL$ makes with the $x$-axis.

Also defined as

Some sources define the slope of $\LL$ as the actual angle that $\LL$ makes with the $x$-axis, rather than its tangent.

On $\mathsf{Pr} \infty \mathsf{fWiki}$ we specifically mean the tangent of that angle.

Also known as

The slope of a straight line or curve is also sometimes referred to as its gradient.

However, that term has a more generic and abstract meaning than does the concept of slope as given here.

Some sources suggest that the slope of a straight line is the same as its direction, but this is true only in the plane.

Also see

• Results about slope can be found here.