The slope of a line corresponds to the idea of the steepness of an inclined plane or the steepness of a stairway. If all the steps of a stairway are uniform, the steepness of the stairway can be defined in terms of the "rise" and "run" of one of the steps. The steepness of a uniform stairway is the number obtained by dividing the rise by the run of one of the steps. If the rise (vertical component) is 6 inches and the run (horizontal component) is 8 inches, then the steepness of the stairway is 6/8 (or 3/4). Likewise, in formal geometry, slope, in rectangular coordinates, is defined as the ratio of the change of the ordinate (vertical component) to the corresponding change of the abscissa (horizontal component) of a point moving along a nonvertical line.
The origin of the term slope as used in mathematics most likely came from the Swedish word for slope "riktningskoefficient" and the Dutch word for slope "richtingscoëfficiënt" that both mean "direction coefficient". According to most mathematical historians, the origin for the symbol "m" that symbolizes slope is unknown. The French word "monter" means "to climb", but there is no evidence to substantiate that this word can be associated with the background of the slope symbol "m". In fact, it is known that the French mathematician René Descartes did not use "m" to reference slope. The earliest known use of "m" for slope is from an 1844 British text by Matthew O'Brien entitled A Treatise on Plane Co-Ordinate Geometry . Later in 1848 George Salmon (1819-1904) referred to O'Brien's 1844 article within his A Treatise on Conic Sections and used the slope-intercept formula "y = mx + b", where "b" is the ordinate (vertical component) of the point where the line intersects the y-axis. It is also known that the four authors Isaac Todhunter in 1855 (Treatise on Plane Co-Ordinate Geometry), George A. Osborne in 1891 (Differential and Integral Calculus), and Arthur M. Harding and George W. Mullins in 1924 (Analytic Geometry) each used "m" to refer to slope in their mathematical writings.
If a straight line, or line segment, is determined by the distinct points A = (x1, y1) and B = (x2, y2) in an x-y plane with the origin at (0, 0) and x1 not equaling x2, then the slope, m, of the line segment AB is given by (y2 - y1) / (x2 - x1). This equation can also be stated m = y / x, where x = x2 - x1 and y = y2 - y1.
As an example for the calculation of slope, if on a line segment within the x-y plane the two endpoints are A = (x1, y1) = (1, 1) and B = (x2, y2) = (3, 2), then the slope, m, of the line AB is (2 - 1) / (3 - 1), or m = 1/2. This value means that the slope of the line increases one-unit step on the y-axis for each two-unit step on the x-axis. As mentioned earlier, a straight line is defined by the slope-intercept equation "y = mx + b". The value of the constant "b" indicates where the line segment intercepts the y-axis. In the previous example one may insert "y = 1", "x = 1", and "m = 1/2" to find that "b = 1/2". This means that at the y-intercept (i.e., where "x = 0") the value of y is 1/2.
The sign of the slope indicates whether the line segment slopes "upward" or "downward". For the x-y coordinate system previously described, a positive slope is considered to slope "upward" from left to right, and a negative slope is considered to slope "downward" as viewed from left to right. If y1 = y2 then the line is horizontal with its slope valued at zero. If x1 = x2 then the line is vertical with an undefined slope.
In functional notation the dependent variable y is a function of the variable x, most often expressed as y = ƒ(x). If the line is not straight (but is a curve) the definition of slope must be modified. At a point (x1, ƒ(x1)) on the curve a straight-line tangent to the curve at that point is constructed. The slope of the curve at point (x1, ƒ(x1)) is then just the slope m of this tangential line. In differential calculus the slope m of the line tangent to a curve at point (x1, ƒ(x1)) is defined formally by: m(x1, ƒ(x1)) = (ƒ(x1 + x) - ƒ(x1)) / x, where the function ƒ is continuous in the neighborhood around the point (x1, ƒ(x1)), and x does not equal zero. The equation given above for the slope m at the point (x1, ƒ(x1)) may also be called the derivative of y with respect to x, and is symbolized in differential calculus by the notation "dy / dx".
As an example of using differentiation to determine slope, consider the curve given by y = ƒ(x) = x2 - 4x + 5. For the ordered pair (x1, ƒ(x1)) the "limit equation" previously defined will yield the slope m at that point; i.e., m(x1, ƒ(x1)). Substituting "x2 - 4x + 5" into the limit equation for ƒ(x) gives: m(x1, ƒ(x1)) = (((x1 + x)2 - 4(x1 + x) + 5) - (x12 - 4x4x1 + 5)) / x. The tools of calculus necessary to determine this differentiation are deferred to the calculus-related articles. Nevertheless, by performing the appropriate calculations, the result is m(x1, ƒ(x1)) = 2x1 - 4. For instance, at the point (x, ƒ(x)) = (5, 10), the slope m(5, 10) = (2 x 5) - 4 = 6. In this case, the value "6" means that the tangential line at the point (x = 5, y = 10) on the curve changes 6 units on the y-axis for a 1 unit change on the x-axis. Since the slope is positive, the tangent line slopes upward from left to right in the x-y plane.
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