# 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**.

## 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.**