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In mathematics, the slope or gradient of a
line Line, lines, The Line, or LINE may refer to: Arts, entertainment, and media Films * ''Lines'' (film), a 2016 Greek film * ''The Line'' (2017 film) * ''The Line'' (2009 film) * ''The Line'', a 2009 independent film by Nancy Schwartzman Lite ...
is a number that describes both the ''direction'' and the ''steepness'' of the line. Slope is often denoted by the letter ''m''; there is no clear answer to the question why the letter ''m'' is used for slope, but its earliest use in English appears in O'Brien (1844) who wrote the equation of a straight line as and it can also be found in Todhunter (1888) who wrote it as "''y'' = ''mx'' + ''c''". Slope is calculated by finding the ratio of the "vertical change" to the "horizontal change" between (any) two distinct points on a line. Sometimes the ratio is expressed as a quotient ("rise over run"), giving the same number for every two distinct points on the same line. A line that is decreasing has a negative "rise". The line may be practical - as set by a road surveyor, or in a diagram that models a road or a roof either as a description or as a plan. The ''steepness'', incline, or grade of a line is measured by the
absolute value In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

of the slope. A slope with a greater absolute value indicates a steeper line. The ''direction'' of a
line Line, lines, The Line, or LINE may refer to: Arts, entertainment, and media Films * ''Lines'' (film), a 2016 Greek film * ''The Line'' (2017 film) * ''The Line'' (2009 film) * ''The Line'', a 2009 independent film by Nancy Schwartzman Lite ...
is either increasing, decreasing, horizontal or vertical. *A line is increasing if it goes up from left to right. The slope is positive, i.e. $m>0$. *A line is decreasing if it goes down from left to right. The slope is negative, i.e. $m<0$. *If a line is horizontal the slope is zero. This is a
constant function 270px, Constant function ''y''=4 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (m ...

. *If a line is vertical the slope is ''undefined'' (see below). The rise of a road between two points is the difference between the altitude of the road at those two points, say ''y''1 and ''y''2, or in other words, the rise is (''y''2 − ''y''1) = Δ''y''. For relatively short distances, where the earth's curvature may be neglected, the run is the difference in distance from a fixed point measured along a level, horizontal line, or in other words, the run is (''x''2 − ''x''1) = Δ''x''. Here the slope of the road between the two points is simply described as the ratio of the altitude change to the horizontal distance between any two points on the line. In mathematical language, the slope ''m'' of the line is :$m=\frac.$ The concept of slope applies directly to
grade Grade or grading may refer to: Arts and entertainment * Grade (band) Grade is a melodic hardcore band from Canada, often credited as pioneers in blending metallic hardcore with the hon and melody of emo, and - most notably - the alternating scr ...
s or
gradient In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Prod ...

s in
geography Geography (from Ancient Greek, Greek: , ''geographia'', literally "earth description") is a field of science devoted to the study of the lands, features, inhabitants, and phenomena of the Earth and Solar System, planets. The first person t ...

and
civil engineering Civil engineering is a professional engineering Regulation and licensure in engineering is established by various jurisdictions of the world to encourage public welfare, safety, well-being and other interests of the general public and to defin ...
. Through
trigonometry Trigonometry (from ', "triangle" and ', "measure") is a branch of that studies relationships between side lengths and s of s. The field emerged in the during the 3rd century BC from applications of to . The Greeks focused on the , while ...

, the slope ''m'' of a line is related to its angle of incline ''θ'' by the
tangent function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

:$m = \tan \left(\theta\right)$ Thus, a 45° rising line has a slope of +1 and a 45° falling line has a slope of −1. As a generalization of this practical description, the mathematics of
differential calculus In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
defines the slope of a
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (geo ...

at a point as the slope of the
tangent line In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space ...

at that point. When the curve is given by a series of points in a diagram or in a list of the coordinates of points, the slope may be calculated not at a point but between any two given points. When the curve is given as a continuous function, perhaps as an algebraic formula, then the differential calculus provides rules giving a formula for the slope of the curve at any point in the middle of the curve. This generalization of the concept of slope allows very complex constructions to be planned and built that go well beyond static structures that are either horizontals or verticals, but can change in time, move in curves, and change depending on the rate of change of other factors. Thereby, the simple idea of slope becomes one of the main basis of the modern world in terms of both technology and the built environment.

# Definition

The slope of a line in the plane containing the ''x'' and ''y'' axes is generally represented by the letter ''m'', and is defined as the change in the ''y'' coordinate divided by the corresponding change in the ''x'' coordinate, between two distinct points on the line. This is described by the following equation: :$m = \frac = \frac= \frac.$ (The Greek letter ''
delta Delta commonly refers to: * Delta (letter) (Δ or δ), a letter of the Greek alphabet * River delta, a landform at the mouth of a river * D (NATO phonetic alphabet: "Delta"), the fourth letter of the modern English alphabet * Delta Air Lines, an Ame ...
'', Δ, is commonly used in mathematics to mean "difference" or "change".) Given two points $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$, the change in $x$ from one to the other is $x_2-x_1$ (''run''), while the change in $y$ is $y_2-y_1$ (''rise''). Substituting both quantities into the above equation generates the formula: :$m = \frac.$ The formula fails for a vertical line, parallel to the $y$ axis (see
Division by zero In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
), where the slope can be taken as
infinite Infinite may refer to: Mathematics *Infinite set, a set that is not a finite set *Infinity, an abstract concept describing something without any limit Music *Infinite (band), a South Korean boy band *''Infinite'' (EP), debut EP of American musi ...

, so the slope of a vertical line is considered undefined.

## Examples

Suppose a line runs through two points: ''P'' = (1, 2) and ''Q'' = (13, 8). By dividing the difference in $y$-coordinates by the difference in $x$-coordinates, one can obtain the slope of the line: :$m = \frac = \frac = \frac = \frac = \frac$. :Since the slope is positive, the direction of the line is increasing. Since , ''m'', < 1, the incline is not very steep (incline < 45°). As another example, consider a line which runs through the points (4, 15) and (3, 21). Then, the slope of the line is :$m = \frac = \frac = -6.$ :Since the slope is negative, the direction of the line is decreasing. Since , ''m'', > 1, this decline is fairly steep (decline > 45°).

# Algebra and geometry

*If $y$' is a
linear function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...

of $x$', then the coefficient of $x$ is the slope of the line created by plotting the function. Therefore, if the equation of the line is given in the form ::$y = mx + b$ :then $m$ is the slope. This form of a line's equation is called the ''slope-intercept form'', because $b$ can be interpreted as the
y-intercept Image:Y-intercept.svg, 300px, Graph ''y''=''ƒ''(''x'') with the ''x''-axis as the horizontal axis and the ''y''-axis as the vertical axis. The ''y''-intercept of ''ƒ''(''x'') is indicated by the red dot at (''x''=0, ''y''=1). In analytic ...

of the line, that is, the $y$-coordinate where the line intersects the $y$-axis. *If the slope $m$ of a line and a point $\left(x_1,y_1\right)$ on the line are both known, then the equation of the line can be found using the point-slope formula: ::$y - y_1 = m\left(x - x_1\right).$ *The slope of the line defined by the
linear equation In mathematics, a linear equation is an equation that may be put in the form :a_1x_1+\cdots +a_nx_n+b=0, where x_1, \ldots, x_n are the variable (mathematics), variables (or unknown (mathematics), unknowns), and b, a_1, \ldots, a_n are the coeffi ...

::$ax + by +c = 0$ :is ::$-\frac$. *Two lines are
parallel Parallel may refer to: Computing * Parallel algorithm In computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their a ...
if and only if they are not the same line (coincident) and either their slopes are equal or they both are vertical and therefore both have undefined slopes. Two lines are
perpendicular In elementary geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relativ ...

if the product of their slopes is −1 or one has a slope of 0 (a horizontal line) and the other has an undefined slope (a vertical line). *The angle θ between −90° and 90° that a line makes with the ''x''-axis is related to the slope ''m'' as follows: ::$m = \tan\left(\theta\right)$ :and ::$\theta = \arctan \left(m\right)$   (this is the inverse function of tangent; see
inverse trigonometric functions In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted Domain of a fun ...
).

## Examples

For example, consider a line running through the points (2,8) and (3,20). This line has a slope, , of :$\frac \; = 12.$ One can then write the line's equation, in point-slope form: :$y - 8 = 12\left(x - 2\right) = 12x - 24$ or: :$y = 12x - 16.$ The angle θ between −90° and 90° that this line makes with the -axis is :$\theta = \arctan\left(12\right) \approx 85.2^ .$ Consider the two lines: and . Both lines have slope . They are not the same line. So they are parallel lines. Consider the two lines and . The slope of the first line is . The slope of the second line is . The product of these two slopes is −1. So these two lines are perpendicular.

# Statistics

In
statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data Data (; ) are individual facts, statistics, or items of information, often numeric. In a more technical sens ...

, the gradient of the least-squares regression best-fitting line for a given sample of data may be written as: :$m = \frac$, This quantity ''m'' is called as the '' regression slope'' for the line $y=mx+c$. The quantity $r$ is Pearson's correlation coefficient, $s_y$ is the
standard deviation In statistics, the standard deviation is a measure of the amount of variation or statistical dispersion, dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected v ...

of the y-values and $s_x$ is the
standard deviation In statistics, the standard deviation is a measure of the amount of variation or statistical dispersion, dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected v ...

of the x-values. This may also be written as a ratio of
covariance In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the less ...

s: :$m = \frac$

# Slope of a road or railway

:''Main articles:
Grade (slope) The grade (also called slope, incline, gradient, mainfall, pitch or rise) of a physical feature, landform or constructed line refers to the tangent of the angle of that surface to the horizontal. It is a special case of the slope In mathemati ...
,
Grade separation In civil engineering (more specifically highway engineering), grade separation is a method of aligning a junction (traffic), junction of two or more surface transport axes at different heights (grades) so that they will not disrupt the traffic ...
'' There are two common ways to describe the steepness of a
road A road is a wide way leading from one place to another, typically one with a specially prepared surface which vehicles and bikes can use. Roads consist of one or two roadway A carriageway (British English British English (BrE) is the ...

or
railroad Rail transport (also known as train transport) is a means of transferring passengers and goods on wheeled vehicle A vehicle (from la, vehiculum) is a machine A machine is any physical system with ordered structural and functional p ...

. One is by the angle between 0° and 90° (in degrees), and the other is by the slope in a percentage. See also
steep grade railway A steep grade railway is a railway that ascends and descends a slope that has a steep grade. Such railways can use a number of different technologies to overcome the steepness of the grade. Usage Many steep grade railways are located in mountain ...
and
rack railway A rack railway (also rack-and-pinion railway, cog railway, or cogwheel railway) is a steep grade railway with a toothed rack rail, usually between the running rails. The train pulling passenger cars in Nevada Nevada (, ) is a U.S. ...

. The formulae for converting a slope given as a percentage into an angle in degrees and vice versa are: ::$\text = \arctan \left\left( \frac \right\right)$  , (this is the inverse function of tangent; see
trigonometry Trigonometry (from ', "triangle" and ', "measure") is a branch of that studies relationships between side lengths and s of s. The field emerged in the during the 3rd century BC from applications of to . The Greeks focused on the , while ...

) :and ::$\mbox = 100\% \cdot \tan\left( \mbox\right),$ where ''angle'' is in degrees and the trigonometric functions operate in degrees. For example, a slope of 100 % or 1000 is an angle of 45°. A third way is to give one unit of rise in say 10, 20, 50 or 100 horizontal units, e.g. 1:10. 1:20, 1:50 or 1:100 (or "1 ''in'' 10", "1 ''in'' 20" etc.) Note that 1:10 is steeper than 1:20. For example, steepness of 20% means 1:5 or an incline with angle 11,3°. Roads and railways have both longitudinal slopes and cross slopes. File:Nederlands verkeersbord J6.svg, Slope warning sign in the
Netherlands ) , national_anthem = ( en, "William of Nassau") , image_map = EU-Netherlands.svg , map_caption = , image_map2 = BES islands location map.svg , map_caption2 = , image_map3 ...

Poland Poland, officially the Republic of Poland, is a country located in Central Europe. It is divided into 16 Voivodeships of Poland, administrative provinces, covering an area of , and has a largely Temperate climate, temperate seasonal cli ...

File:Skloník-klesání.jpg, A 1371-meter distance of a railroad with a 20 slope.
Czech Republic The Czech Republic, also known by its short-form name Czechia and formerly known as Bohemia, is a landlocked country in Central Europe. It is bordered by Austria to the south, Germany to the west, Poland to the northeast, and Slovakia to ...
File:Railway gradient post.jpg, Steam-age railway gradient post indicating a slope in both directions at Meols railway station,
United Kingdom The United Kingdom of Great Britain and Northern Ireland, commonly known as the United Kingdom (UK) or Britain,Usage is mixed. The Guardian' and Telegraph' use Britain as a synonym for the United Kingdom. Some prefer to use Britain as shorth ...

# Calculus

The concept of a slope is central to
differential calculus In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
. For non-linear functions, the rate of change varies along the curve. The
derivative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

of the function at a point is the slope of the line
tangent In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

to the curve at the point, and is thus equal to the rate of change of the function at that point. If we let Δ''x'' and Δ''y'' be the distances (along the ''x'' and ''y'' axes, respectively) between two points on a curve, then the slope given by the above definition, :$m = \frac$, is the slope of a
secant line In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...
to the curve. For a line, the secant between any two points is the line itself, but this is not the case for any other type of curve. For example, the slope of the secant intersecting ''y'' = ''x''2 at (0,0) and (3,9) is 3. (The slope of the tangent at is also 3—''a'' consequence of the
mean value theorem In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

.) By moving the two points closer together so that Δ''y'' and Δ''x'' decrease, the secant line more closely approximates a tangent line to the curve, and as such the slope of the secant approaches that of the tangent. Using
differential calculus In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
, we can determine the
limit Limit or Limits may refer to: Arts and media * Limit (music), a way to characterize harmony * Limit (song), "Limit" (song), a 2016 single by Luna Sea * Limits (Paenda song), "Limits" (Paenda song), 2019 song that represented Austria in the Eurov ...

, or the value that Δ''y''/Δ''x'' approaches as Δ''y'' and Δ''x'' get closer to
zero 0 (zero) is a number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and ...

; it follows that this limit is the exact slope of the tangent. If ''y'' is dependent on ''x'', then it is sufficient to take the limit where only Δ''x'' approaches zero. Therefore, the slope of the tangent is the limit of Δ''y''/Δ''x'' as Δ''x'' approaches zero, or ''dy''/''dx''. We call this limit the
derivative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...
. :$\frac = \lim_\frac$ Its value at a point on the function gives us the slope of the tangent at that point. For example, let ''y'' = ''x''2. A point on this function is (−2,4). The derivative of this function is d''y''/d''x'' = 2''x''. So the slope of the line tangent to ''y'' at (−2,4) is 2 · (−2) = −4. The equation of this tangent line is: ''y''−4 = (−4)(''x''−(−2)) or ''y'' = −4''x'' − 4.

# Difference of slopes

An extension of the idea of angle follows from the difference of slopes. Consider the
shear mappingImage:VerticalShear m=1.25.svg, 175px, alt=Mesh Shear 5/4, A horizontal shearing of the plane with coefficient ''m'' = 1.25, illustrated by its effect (in green) on a rectangular grid and some figures (in blue). The black dot is the origin. In plane ...
:$\left(u,v\right) = \left(x,y\right) \begin1 & v \\ 0 & 1 \end.$ Then (1,0) is mapped to (1,''v''). The slope of (1,0) is zero and the slope of (1,''v'') is ''v''. The shear mapping added a slope of ''v''. For two points on with slopes ''m'' and ''n'', the image :$\left(1,y\right)\begin1 & v \\ 0 & 1\end = \left(1, y + v\right)$ has slope increased by ''v'', but the difference ''n'' − ''m'' of slopes is the same before and after the shear. This invariance of slope differences makes slope an angular
invariant measureIn mathematics, an invariant measure is a measure (mathematics), measure that is preserved by some function (mathematics), function. Ergodic theory is the study of invariant measures in dynamical systems. The Krylov–Bogolyubov theorem proves the ex ...
, on a par with circular angle (invariant under rotation) and hyperbolic angle, with invariance group of
squeeze mapping ''r'' = 3/2 squeeze mapping In linear algebra, a squeeze mapping is a type of linear map In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structu ...
s.

*
Euclidean distance In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
*
Grade Grade or grading may refer to: Arts and entertainment * Grade (band) Grade is a melodic hardcore band from Canada, often credited as pioneers in blending metallic hardcore with the hon and melody of emo, and - most notably - the alternating scr ...
*
Inclined plane An inclined plane, also known as a ramp, is a flat supporting surface tilted at an angle, with one end higher than the other, used as an aid for raising or lowering a load. The inclined plane is one of the six classical simple machines defin ...

*
Linear function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
*
Line of greatest slope Line, lines, The Line, or LINE may refer to: Arts, entertainment, and media Films * ''Lines'' (film), a 2016 Greek film * ''The Line'' (2017 film) * ''The Line'' (2009 film) * ''The Line'', a 2009 independent film by Nancy Schwartzman Nanc ...
*
Mediant In music Music is the of arranging s in time through the of melody, harmony, rhythm, and timbre. It is one of the aspects of all human societies. General include common elements such as (which governs and ), (and its associated concept ...

* *
Theil–Sen estimator In non-parametric statistics, the Theil–Sen estimator is a method for robust estimator, robustly fitting a line to sample points in the plane (simple linear regression) by choosing the median of the slopes of all lines through pairs of points. ...
, a line with the
median In statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin wi ...

slope among a set of sample points