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In mathematics, the slope or gradient of a
line Line most often refers to: * Line (geometry), object with zero thickness and curvature that stretches to infinity * Telephone line, a single-user circuit on a telephone communication system Line, lines, The Line, or LINE may also refer to: Arts ...
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 In mathematics, a ratio shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ...
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 A diagram is a symbolic representation of information using visualization techniques. Diagrams have been used since prehistoric times on walls of caves, but became more prevalent during the Enlightenment. Sometimes, the technique uses a three- ...
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, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
of the slope. A slope with a greater absolute value indicates a steeper line. The ''direction'' of a
line Line most often refers to: * Line (geometry), object with zero thickness and curvature that stretches to infinity * Telephone line, a single-user circuit on a telephone communication system Line, lines, The Line, or LINE may also refer to: Arts ...
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 In mathematics, a constant function is a function whose (output) value is the same for every input value. For example, the function is a constant function because the value of is 4 regardless of the input value (see image). Basic properties ...
. *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 Spherical Earth or Earth's curvature refers to the approximation of figure of the Earth as a sphere. The earliest documented mention of the concept dates from around the 5th century BC, when it appears in the writings of Greek philosophers. ...
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 most commonly refers to: * Grade (education), a measurement of a student's performance * Grade, the number of the year a student has reached in a given educational stage * Grade (slope), the steepness of a slope Grade or grading may also ref ...
s or
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gradi ...
s in
geography Geography (from Greek: , ''geographia''. Combination of Greek words ‘Geo’ (The Earth) and ‘Graphien’ (to describe), literally "earth description") is a field of science devoted to the study of the lands, features, inhabitants, and ...
and
civil engineering Civil engineering is a professional engineering discipline that deals with the design, construction, and maintenance of the physical and naturally built environment, including public works such as roads, bridges, canals, dams, airports, sewage ...
. Through
trigonometry Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. T ...
, the slope ''m'' of a line is related to its
angle In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle. Angles formed by two ...
of inclination ''θ'' by the
tangent function In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...
:m = \tan (\theta) 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, differential calculus is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve. ...
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 (ge ...
at a point as the slope of the
tangent line In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
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 In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
, perhaps as an
algebraic expression In mathematics, an algebraic expression is an expression built up from integer constants, variables, and the algebraic operations (addition, subtraction, multiplication, division and exponentiation by an exponent that is a rational number). For ex ...
, 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, at a river mouth * D (NATO phonetic alphabet: "Delta") * Delta Air Lines, US * Delta variant of SARS-CoV-2 that causes COVID-19 Delta may also re ...
'', Δ, is commonly used in mathematics to mean "difference" or "change".) Given two points (x_1,y_1) and (x_2,y_2), 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, division by zero is division (mathematics), division where the divisor (denominator) is 0, zero. Such a division can be formally expression (mathematics), expressed as \tfrac, where is the dividend (numerator). In ordinary ari ...
), 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 (group), a South Korean boy band *''Infinite'' (EP), debut EP of American m ...
, 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


Examples

For example, consider a line running through 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(x - 2) = 12x - 24. or: : y = 12x - 16. The angle θ between −90° and 90° that this line makes with the -axis is :\theta = \arctan(12) \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 (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
, the gradient of the least-squares regression
best-fitting line Line fitting is the process of constructing a straight line that has the best fit to a series of data points. Several methods exist, considering: *Vertical distance: Simple linear regression **Resistance to outliers: Robust simple linear regre ...
for a given
sample Sample or samples may refer to: Base meaning * Sample (statistics), a subset of a population – complete data set * Sample (signal), a digital discrete sample of a continuous analog signal * Sample (material), a specimen or small quantity of s ...
of data may be written as: :m = \frac, This quantity ''m'' is called as the ''
regression slope In statistics, linear regression is a linear approach for modelling the relationship between a scalar response and one or more explanatory variables (also known as dependent and independent variables). The case of one explanatory variable is cal ...
'' for the line y=mx+c. The quantity r is
Pearson's correlation coefficient In statistics, the Pearson correlation coefficient (PCC, pronounced ) ― also known as Pearson's ''r'', the Pearson product-moment correlation coefficient (PPMCC), the bivariate correlation, or colloquially simply as the correlation coefficient ...
, s_y is the
standard deviation In statistics, the standard deviation is a measure of the amount of variation or 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 value) of the set, while ...
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 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 value) of the set, while ...
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 les ...
s: :m = \frac


Slope of a road or railway

There are two common ways to describe the steepness of a
road A road is a linear way for the conveyance of traffic that mostly has an improved surface for use by vehicles (motorized and non-motorized) and pedestrians. Unlike streets, the main function of roads is transportation. There are many types of ...
or
railroad Rail transport (also known as train transport) is a means of transport that transfers passengers and goods on wheeled vehicles running on rails, which are incorporated in tracks. In contrast to road transport, where the vehicles run on a pre ...
. 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 trains are fitted with one or more cog wheels or pinions that mesh with ...
. The formulae for converting a slope given as a percentage into an angle in degrees and vice versa are: : \text = \arctan \left( \frac \right) (this is the inverse function of tangent; see
trigonometry Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. T ...
) and : \mbox = 100\% \times \tan( \mbox), where ''angle'' is in degrees and the trigonometric functions operate in degrees. For example, a slope of 100 % or 1000
Per mille (from Latin , "in each thousand") is an expression that means parts per thousand. Other recognised spellings include per mil, per mill, permil, permill, or permille. The associated sign is written , which looks like a percent sig ...
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 ) , anthem = ( en, "William of Nassau") , image_map = , map_caption = , subdivision_type = Sovereign state , subdivision_name = Kingdom of the Netherlands , established_title = Before independence , established_date = Spanish Netherl ...
File:PL road sign A-23.svg, Slope warning sign in
Poland Poland, officially the Republic of Poland, is a country in Central Europe. It is divided into 16 administrative provinces called voivodeships, covering an area of . Poland has a population of over 38 million and is the fifth-most populous ...
File: Skloník-klesání.jpg, A 1371-meter distance of a railroad with a 20
Per mille (from Latin , "in each thousand") is an expression that means parts per thousand. Other recognised spellings include per mil, per mill, permil, permill, or permille. The associated sign is written , which looks like a percent sig ...
slope.
Czech Republic The Czech Republic, or simply Czechia, is a landlocked country in Central Europe. Historically known as Bohemia, it is bordered by Austria to the south, Germany to the west, Poland to the northeast, and Slovakia to the southeast. The ...
File: Railway gradient post.jpg, Steam-age railway gradient post indicating a slope in both directions at
Meols railway station Meols railway station is a station serving the village of Meols, in Merseyside, England. It lies on the West Kirby branch of the Wirral Line, part of the Merseyrail network. History Meols was one of the original stations on the Hoylake Rail ...
,
United Kingdom The United Kingdom of Great Britain and Northern Ireland, commonly known as the United Kingdom (UK) or Britain, is a country in Europe, off the north-western coast of the continental mainland. It comprises England, Scotland, Wales and North ...


Calculus

The concept of a slope is central to
differential calculus In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve. ...
. For non-linear functions, the rate of change varies along the curve. The
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
of the function at a point is the slope of the line
tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
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 Secant is a term in mathematics derived from the Latin ''secare'' ("to cut"). It may refer to: * a secant line, in geometry * the secant variety, in algebraic geometry * secant (trigonometry) (Latin: secans), the multiplicative inverse (or reciproc ...
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, the mean value theorem (or Lagrange theorem) states, roughly, that for a given planar arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant through its endpoints. It i ...
.) 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, differential calculus is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve. ...
, we can determine the
limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
, or the value that Δ''y''/Δ''x'' approaches as Δ''y'' and Δ''x'' get closer to
zero 0 (zero) is a number representing an empty quantity. In place-value notation Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or ...
; 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, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
. :\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 . So the slope of the line tangent to ''y'' at (−2,4) is . The equation of this tangent line is: or .


Difference of slopes

An extension of the idea of angle follows from the difference of slopes. Consider the
shear mapping In plane geometry, a shear mapping is a linear map that displaces each point in a fixed direction, by an amount proportional to its signed distance from the line that is parallel to that direction and goes through the origin. This type of mappi ...
:(u,v) = (x,y) \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 :(1,y)\begin1 & v \\ 0 & 1\end = (1, y + v) 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 measure In mathematics, an invariant measure is a measure that is preserved by some function. The function may be a geometric transformation. For examples, circular angle is invariant under rotation, hyperbolic angle is invariant under squeeze mapping, an ...
, on a par with circular angle (invariant under rotation) and hyperbolic angle, with invariance group of
squeeze mapping In linear algebra, a squeeze mapping, also called a squeeze transformation, is a type of linear map that preserves Euclidean area of regions in the Cartesian plane, but is ''not'' a rotation or shear mapping. For a fixed positive real number , th ...
s.


See also

*
Euclidean distance In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefor ...
*
Grade Grade most commonly refers to: * Grade (education), a measurement of a student's performance * Grade, the number of the year a student has reached in a given educational stage * Grade (slope), the steepness of a slope Grade or grading may also ref ...
*
Inclined plane An inclined plane, also known as a ramp, is a flat supporting surface tilted at an angle from the vertical direction, 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 clas ...
*
Linear function In mathematics, the term linear function refers to two distinct but related notions: * In calculus and related areas, a linear function is a function (mathematics), function whose graph of a function, graph is a straight line, that is, a polynomia ...
*
Line of greatest slope Line most often refers to: * Line (geometry), object with zero thickness and curvature that stretches to infinity * Telephone line, a single-user circuit on a telephone communication system Line, lines, The Line, or LINE may also refer to: Arts ...
*
Mediant In music, the mediant (''Latin'': to be in the middle) is the third scale degree () of a diatonic scale, being the note halfway between the tonic and the dominant.Benward & Saker (2003), p.32. In the movable do solfège system, the mediant note i ...
* Slope definitions *
Theil–Sen estimator In non-parametric statistics, the Theil–Sen estimator is a method for 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. It has also bee ...
, a line with the
median In statistics and probability theory, the median is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. For a data set, it may be thought of as "the middle" value. The basic fe ...
slope among a set of sample points


References


External links

* interactive {{Calculus topics Elementary mathematics Analytic geometry Ratios