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