# Definition:Slope

## Definition

### Straight Line

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.

### Curve

Let $P$ be a point on a curve $\CC$ embedded in a Cartesian plane.

The **slope** of $\CC$ at $P$ is defined as the slope of the tangent to $\CC$ at $P$.

## Also defined as

### Also defined as: $1$

**Slope** can also be seen defined as the angle the line makes with the horizontal.

### Also defined as: $2$

In the context of geography, the **slope** of a road is often seen defined as the vertical distance travelled with respect to the actual distance travelled along the road.

## Examples

### Arbitrary Example

A **slope** or **gradient** of $1$ in $4$ indicates a vertical distance of $1$ unit for every $4$ units travelled along the road.

This is also indicated as a ratio $1 : 4$ or a fraction $1 / 4$, or can also be expressed as the angle of **slope**, that is, $\arcsin \dfrac 1 4$ or a percentage, that is, a $25 \%$ **gradient**.

## 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

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**gradient**:**1.**(**grade**) - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**gradient**:**1.**(**grade**) - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**slope**