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 a straight line $\LL$ 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.
The word grade can sometimes be seen, but this is discouraged as it has a number of meanings.
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.
Sources
- 1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text {II}$. The Straight Line: $2$.
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): slope: 1.
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): slope: 1.