Global InformationSlope information
In mathematics, the slope or gradient of a line is a number that describes the direction and steepness of the line.[1] Often denoted by the letter m, slope is calculated as the ratio of the vertical change to the horizontal change ("rise over run") between two distinct points on the line, giving the same number for any choice of points. A line descending left-to-right has negative rise and negative slope. The line may be physical – as set by a road surveyor, pictorial as in a diagram of a road or roof, or abstract.
The steepness, incline, or grade of a line is the absolute value of its slope: greater absolute value indicates a steeper line. Direction is defined as follows:
- An increasing line goes up from left to right, and has positive slope: .
- A decreasing line goes down from left to right, and has negative slope: .
- A horizontal line (the graph of a constant function) has zero slope: .
- A vertical line has undefined or infinite slope (see below).
If two points of a road have altitudes y1 and y2, the rise is the difference (y2 − y1) = Δy. Neglecting the Earth's curvature, if the two points have horizontal distance x1 and x2 from a fixed point, the run is (x2 − x1) = Δx. The slope between the two points is the difference ratio:
This is equivalent to the grade or gradient in geography and civil engineering. Through trigonometry, the slope m of a line is related to its angle of inclination θ by the tangent function
Thus, a 45° rising line has slope m = +1, and a 45° falling line has slope m = −1.
Generalizing this, differential calculus defines the slope of a curve at a point as the slope of its tangent line at that point. When the curve is approximated by a series of points, the slope of the curve may be approximated by the slope of the secant line between two nearby points. When the curve is given as the graph of an algebraic expression, calculus gives formulas for the slope at each point. Slope is thus one of the central ideas of calculus and its applications to design.
There seems to be no clear answer as to why the letter m is used for slope, but it first appears in English in O'Brien (1844)[2] who introduced the equation of a line as "y = mx + b", and it can also be found in Todhunter (1888)[3] who wrote "y = mx + c".[4]
- ^ Clapham, C.; Nicholson, J. (2009). "Oxford Concise Dictionary of Mathematics, Gradient" (PDF). Addison-Wesley. p. 348. Archived from the original (PDF) on 29 October 2013. Retrieved 1 September 2013.
- ^ O'Brien, M. (1844), A Treatise on Plane Co-Ordinate Geometry or the Application of the Method of Co-Ordinates in the Solution of Problems in Plane Geometry, Cambridge, England: Deightons
- ^ Todhunter, I. (1888), Treatise on Plane Co-Ordinate Geometry as Applied to the Straight Line and Conic Sections, London: Macmillan
- ^ Weisstein, Eric W. "Slope". MathWorld--A Wolfram Web Resource. Archived from the original on 6 December 2016. Retrieved 30 October 2016.