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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 and their changes (cal ...
, two
sequence 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 and t ...

s of numbers, often
experimental data Experimental data in science Science (from the Latin word ''scientia'', meaning "knowledge") is a systematic enterprise that Scientific method, builds and Taxonomy (general), organizes knowledge in the form of Testability, testable explanatio ...
, are proportional or directly proportional if their corresponding elements have a
constant Constant or The Constant may refer to: Mathematics * Constant (mathematics) In mathematics, the word constant can have multiple meanings. As an adjective, it refers to non-variance (i.e. unchanging with respect to some other Value (mathematics ...
ratio In mathematics, a ratio indicates 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 ...

, which is called the coefficient of proportionality or proportionality constant. Two sequences are inversely proportional if corresponding elements have a constant product, also called the coefficient of proportionality. This definition is commonly extended to related varying quantities, which are often called ''variables''. This meaning of ''variable'' is not the common meaning of the term in mathematics (see
variable (mathematics) In , a variable (from ', "changeable") is a and placeholder for (historically) a that may change, or (nowadays) any . In particular, a variable may represent a , a , a , a , the , a , or an of a set. with variables as if they were explicit ...
); these two different concepts share the same name for historical reasons. Two
functions Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
$f\left(x\right)$ and $g\left(x\right)$ are ''proportional'' if their ratio $\frac$ 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 several pairs of variables share the same direct proportionality constant, the
equation 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 ge ...

expressing the equality of these ratios is called a proportion, e.g., (for details see
Ratio In mathematics, a ratio indicates 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 ...

). Proportionality is closely related to ''
linearity Linearity is the property of a mathematical relationship (''function (mathematics), function'') that can be graph of a function, graphically represented as a straight Line (geometry), line. Linearity is closely related to ''Proportionality (math ...

''.

# Direct proportionality

Given two variables ''x'' and ''y'', ''y'' is directly proportional to ''x'' if there is a non-zero constant ''k'' such that : $y = kx.$ The relation is often denoted using the symbols "∝" (not to be confused with the Greek letter
alpha Alpha (uppercase , lowercase ; grc, ἄλφα, ''álpha'', modern pronunciation ''álfa'') is the first letter Letter, letters, or literature may refer to: Characters typeface * Letter (alphabet) A letter is a segmental symbol A s ...

) or "~": : $y \propto x,$ or $y \sim x.$ For $x \ne 0$ the proportionality constant can be expressed as the ratio : $k = \frac.$ It is also called the constant of variation or constant of proportionality. A direct proportionality can also be viewed as a
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 ...

in two variables with a of and a
slope 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'', ...

of ''k''. This corresponds to
linear growth Linearity is the property of a mathematical relationship (''function (mathematics), function'') that can be graph of a function, graphically represented as a straight Line (geometry), line. Linearity is closely related to Proportionality (mathema ...
.

## Examples

* If an object travels at a constant
speed In everyday use and in kinematics Kinematics is a subfield of physics, developed in classical mechanics, that describes the Motion (physics), motion of points, bodies (objects), and systems of bodies (groups of objects) without considerin ...

, then the
distance Distance is a numerical measurement ' Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can be used to compare with other objects or eve ...

traveled is directly proportional to the
time Time is the continued sequence of existence and event (philosophy), events that occurs in an apparently irreversible process, irreversible succession from the past, through the present, into the future. It is a component quantity of various me ...

spent traveling, with the speed being the constant of proportionality. * The
circumference 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 a
circle A circle is a shape A shape or figure is the form of an object or its external boundary, outline, or external surface File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to preven ...

is directly proportional to its
diameter 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 ...

, with the constant of proportionality equal to

. * On a
map A map is a symbol A symbol is a mark, sign, or that indicates, signifies, or is understood as representing an , , or . Symbols allow people to go beyond what is n or seen by creating linkages between otherwise very different s and s. A ...

of a sufficiently small geographical area, drawn to
scale Scale or scales may refer to: Mathematics * Scale (descriptive set theory)In the mathematical discipline of descriptive set theory, a scale is a certain kind of object defined on a set (mathematics), set of point (mathematics), points in some Poli ...
distances, the distance between any two points on the map is directly proportional to the beeline distance between the two locations represented by those points; the constant of proportionality is the scale of the map. * The
force In physics, a force is an influence that can change the motion (physics), motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving from a Newton's first law, state of rest), i.e., to acce ...
, acting on a small object with small
mass Mass is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value ...
by a nearby large extended mass due to
gravity Gravity (), or gravitation, is a by which all things with or —including s, s, , and even —are attracted to (or ''gravitate'' toward) one another. , gravity gives to s, and the causes the s of the oceans. The gravitational attracti ...

, is directly proportional to the object's mass; the constant of proportionality between the force and the mass is known as
gravitational acceleration In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...
. * The net force acting on an object is proportional to the acceleration of that object with respect to an inertial frame of reference. The constant of proportionality in this,
Newton's second law Newton's laws of motion are three Scientific law, laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: ''Law 1''. A body continues ...
, is the classical mass of the object.

# Inverse proportionality

The concept of ''inverse proportionality'' can be contrasted with ''direct proportionality''. Consider two variables said to be "inversely proportional" to each other. If all other variables are held constant, the magnitude or absolute value of one inversely proportional variable decreases if the other variable increases, while their product (the constant of proportionality ''k'') is always the same. As an example, the time taken for a journey is inversely proportional to the speed of travel. Formally, two variables are inversely proportional (also called varying inversely, in inverse variation, in inverse proportion) if each of the variables is directly proportional to the
multiplicative inverse Image:Hyperbola one over x.svg, thumbnail, 300px, alt=Graph showing the diagrammatic representation of limits approaching infinity, The reciprocal function: . For every ''x'' except 0, ''y'' represents its multiplicative inverse. The graph forms a r ...

(reciprocal) of the other, or equivalently if their product is a constant.Weisstein, Eric W
"Inversely Proportional"
''MathWorld'' – A Wolfram Web Resource.
It follows that the variable ''y'' is inversely proportional to the variable ''x'' if there exists a non-zero constant ''k'' such that : $y = \frac,$ or equivalently, $xy = k.$ Hence the constant "''k''" is the product of ''x'' and ''y''. The graph of two variables varying inversely on the
Cartesian coordinate A Cartesian coordinate system (, ) in a plane is a coordinate system In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the o ...

plane is a
rectangular hyperbola 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 and ...

. The product of the ''x'' and ''y'' values of each point on the curve equals the constant of proportionality (''k''). Since neither ''x'' nor ''y'' can equal zero (because ''k'' is non-zero), the graph never crosses either axis.

# Hyperbolic coordinates

The concepts of ''direct'' and ''inverse'' proportion lead to the location of points in the Cartesian plane by
hyperbolic coordinatesImage:Hyperbolic coordinates.svg, 400px, Hyperbolic coordinates plotted on the Euclidean plane: all points on the same blue ray share the same coordinate value ''u'', and all points on the same red hyperbola share the same coordinate value ''v''. In ...

; the two coordinates correspond to the constant of direct proportionality that specifies a point as being on a particular
ray Ray may refer to: Science and mathematics * Ray (geometry), half of a line proceeding from an initial point * Ray (graph theory), an infinite sequence of vertices such that each vertex appears at most once in the sequence and each two consecutive ...

and the constant of inverse proportionality that specifies a point as being on a particular hyperbola.

*
Linear map 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 and ...

*
Correlation 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 m ...

*
Eudoxus of Cnidus Eudoxus of Cnidus (; grc, Εὔδοξος ὁ Κνίδιος, ''Eúdoxos ho Knídios''; ) was an ancient Greek Ancient Greek includes the forms of the Greek language used in ancient Greece and the classical antiquity, ancient world from ...
*
Golden ratio 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 ...

*
Inverse-square law 420px, S represents the light source, while r represents the measured points. The lines represent the flux emanating from the sources and fluxes. The total number of flux lines depends on the strength of the light source and is constant with in ...

*
Proportional font A typeface is the design of lettering that can include variations, such as extra bold, bold, regular, light, italic, condensed, extended, etc. Each of these variations of the typeface is a font. There are list of typefaces, thousands of differe ...
*
Ratio In mathematics, a ratio indicates 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 ...

*
Rule of three (mathematics) In mathematics, specifically in elementary arithmetic and elementary algebra, given an equation between two Fraction (mathematics), fractions or rational fraction, rational expressions, one can cross-multiply to simplify the equation or determine th ...
*
Sample size Sample size determination is the act of choosing the number of observations or replicates to include in a statistical sample In statistics and quantitative research methodology, a sample is a set of individuals or objects collected or selected ...
* Similarity * Basic proportionality theorem * the ''a'' is to ''b'' as ''c'' is to ''d'' symbol (U+2237 ''PROPORTION'')

## Growth

*
Linear growth Linearity is the property of a mathematical relationship (''function (mathematics), function'') that can be graph of a function, graphically represented as a straight Line (geometry), line. Linearity is closely related to Proportionality (mathema ...
*
Hyperbolic growth When a quantity grows towards a under a finite variation (a "") it is said to undergo hyperbolic growth. More precisely, the 1/x has a as a graph, and has a singularity at 0, meaning that the as x \to 0 is infinite: any similar graph is said ...

# References

* Ya. B. Zeldovich, I. M. Yaglom: ''Higher math for beginners''
p. 34–35
* Brian Burrell: ''Merriam-Webster's Guide to Everyday Math: A Home and Business Reference''. Merriam-Webster, 1998,
p. 85–101
* Lanius, Cynthia S.; Williams Susan E.
''PROPORTIONALITY: A Unifying Theme for the Middle Grades''
Mathematics Teaching in the Middle School 8.8 (2003), p. 392–396. * Seeley, Cathy; Schielack Jane F.
''A Look at the Development of Ratios, Rates, and Proportionality''
Mathematics Teaching in the Middle School, 13.3, 2007, p. 140–142. * Van Dooren, Wim; De Bock Dirk; Evers Marleen; Verschaffel Lieven
''Students' Overuse of Proportionality on Missing-Value Problems: How Numbers May Change Solutions''
Journal for Research in Mathematics Education, 40.2, 2009, p. 187–211. {{DEFAULTSORT:Proportionality (Mathematics) Mathematical terminology Ratios