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Proportionality (mathematics)

In mathematics, two quantities are proportional if they vary in such a way that one of them is a constantmultiple of the other.


The mathematical symbol '�' is used to indicate that two values are proportional. For example, A � B.

In Unicode this is symbol U+221D.

Direct proportionality

Given two variables x and y, y is '(directly) proportional to x (x and y vary directly, or x and y are in direct variation) if there is a non-zero constant k such that

y = kx.\,

The relation is often denoted

y \propto x

and the constant ratio

k = y/x\,

is called the proportionality constant or constant of proportionality.


  • If an object travels at a constant speed, then the distance traveled is proportional to the time spent traveling, with the speed being the constant of proportionality.
  • The circumference of a circle is proportional to its diameter, with the constant of proportionality equal to Ï€.
  • On a map drawn to scale, the distance between any two points on the map is proportional to the distance between the two locations that the points represent, with the constant of proportionality being the scale of the map.
  • The force acting on a certain object due to gravity is proportional to the object's mass; the constant of proportionality between the mass and the force is known as gravitational acceleration.



y = kx\,

is equivalent to

x = \left(\frac{1}{k}\right)y,

it follows that if y is proportional to x, with (nonzero) proportionality constant k, then x is also proportional to y with proportionality constant 1/k.

If y is proportional to x, then the graph of y as a function of x will be a straight line passing through the origin with the slope of the line equal to the constant of proportionality: it corresponds to linear growth.

Inverse proportionality

As noted in the definition above, two proportional variables are sometimes said to be directly proportional. This is done so as to contrast direct proportionality with inverse proportionality.

Two variables are inversely proportional (or varying inversely, or in inverse variation, or in inverse proportion or reciprocal proportion) if one of the variables is directly proportional with the multiplicative inverse (reciprocal) of the other, or equivalently if their product is a constant. It follows that the variable y is inversely proportional to the variable x if there exists a non-zero constant k such that

y = {k \over x}

The constant can be found by multiplying the original x variable and the original y variable.

Basically, the concept of inverse proportion means that as the absolute value or magnitude of one variable gets bigger, the absolute value or magnitude of another gets smaller, such that their product (the constant of proportionality) is always the same.

For example, the time taken for a journey is inversely proportional to the speed of travel; the time needed to dig a hole is (approximately) inversely proportional to the number of people digging.

The graph of two variables varying inversely on the Cartesian coordinate plane is a hyperbola. The product of the X and Y values of each point on the curve will equal the constant of proportionality (k). Since k can never equal zero, the graph will never cross either axis.

Hyperbolic coordinates

The concepts of direct and inverse proportion lead to the location of points in the Cartesian plane by hyperbolic coordinates; the two coordinates correspond to the constant of direct proportionality that locates a point on a ray and the constant of inverse proportionality that locates a point on a hyperbola.

Exponential and logarithmic proportionality

A variable y is exponentially proportional to a variable x, if y is directly proportional to the exponential function of x, that is if there exist non-zero constants k and a

y = k a^x.\,

Likewise, a variable y is logarithmically proportional to a variable x, if y is directly proportional to the logarithm of x, that is if there exist non-zero constants k and a

y = k \log_a (x).\,

Experimental determination

To determine experimentally whether two physical quantities are directly proportional, one performs several measurements and plots the resulting data points in a Cartesian coordinate system. If the points lie on or close to a straight line that passes through the origin (0, 0), then the two variables are probably proportional, with the proportionality constant given by the line's slope.

Unrelated proportionality

Given two variables x and y, y

Power law

A power law is a special kind of mathematical relationship between two quantities. When the frequency of an event varies as a power of some attribute of that event (e.g. its size), the frequency is said to follow a power law. For instance, the number of cities having a certain population size is found to vary as a power of the size of the population, and hence follows a power law. The distribution of a wide variety of natural and man-made phenomena follow a power law, including frequencies of words in most languages, frequencies of family names, sizes of craters on the moon and of solar flares, the sizes of power outages, earthquakes, and wars, the popularity of books and music, and many other quantities.

Technical definition

A power law is any polynomial relationship that exhibits the property of scale invariance. The most common power laws relate two variables and have the form

f(x) = ax^k\! +o(x^k),

where a and k are constants, and o(x^k) is an asymptotically small function of x^k. Here, k is typically called the scalingexponent, where the word "scaling" denotes the fact that a power-law function satisfies f(c x) \propto f(x) where c is a constant. Thus, a rescaling of the function's argument changes the constant of proportionality but preserves the shape of the function itself. This point becomes clearer if we take the logarithm of both sides:

\log\left(f(x)\right) = k \log x + \log a.

Notice that this expression has the form of a linear relationship with slope k. Rescaling the argument produces a linear shift of the function up or down but leaves both the basic form and the slope k unchanged.

Power-law relations characterize a staggering number of naturally occurring phenomena, and this is one of the principal reasons why they have attracted such wide interest. For instance, inverse-square laws, such as gravitation and the Coulomb force, are power laws, as are many common mathematical formulae such as the quadratic law of area of the circle. However much of the recent interest in power laws comes from the study of probability distributions: it's now known that the distributions of a wide variety of quantities seem to follow the power-law form, at least in their upper tail (large events). The behavior of these large events connects these quantities to the study of theory of large deviations (also called extreme value theory), which considers the frequency of extremely rare events like stock market crashes and large natural disasters. It is primarily in the study of statistical distributions that the name "power law" is used; in other areas the power-law functional form is more often referred to simply as a polynomial form or polynomial function.

Scientific interest in power law relations stems partly from the ease with which certain general classes of mechanisms generate them (see the Sornette reference below). The demonstration of a power-law relation in some data can point to specific kinds of mechanisms that might underlie the natural phenomenon in question, and can indicate a deep connection with other, seemingly unrelated systems (see the reference by Simon and the subsection on universality below). The ubiquity of power-law relations in physics is partly due to dimensional constraints, while in complex systems, power laws are often thought to be signatures of hierarchy or of specific stochastic processes. A few notable examples of power laws are the Gutenberg-Richter law for earthquake sizes, Pareto's law of income distribution, structural self-similarity of fractals, and scaling laws in biological systems. Research on the origins of power-law relations, and efforts to observe and validate them in the real world, is an active topic of research in many fields of science, including physics, computer science, linguistics, geophysics, sociology, economics and more.

Properties of power laws

Scale invariance

The main property of power laws that makes them interesting is their scale invariance. Given a relation f(x) = ax^k, scaling the argument x by a constant factor causes only a proportionate scaling of the function itself. That is,

f(c x) = a(c x)^k = c^{k}f(x) \propto f(x).\!

That is, scaling by a constant simply multiplies the original power-law relation by the constant c^k. Thus, it follows that all power laws with a particular scaling exponent are equivalent up to constant factors, since each is simply a scaled version of the others. This behavior is what produces the linear relationship when logarithms are taken of both f(x) and x, and the straight-line on the log-log plot is often called the signature of a power law. Notably, however, with real data, such straightness is necessary, but not a sufficient condition for the data following a power-law relation. In fact, there are many wa

Boyle's law

Boyle's law (sometimes referred to as the Boyle-Mariotte law) is one of many gas laws and a special case of the ideal gas law. Boyle's law describes the inversely proportional relationship between the absolute pressure and volume of a gas, if the temperature is kept constant within a closed system. The law was named after chemist and physicistRobert Boyle, who published the original law in 1662. The law itself can be stated as follows:

For a fixed amount of an ideal gas kept at a fixed temperature, P [pressure] and V [volume] are inversely proportional (while one doubles, the other halves).|


This relationship between pressure and volume was first noted by two amateur scientists, Richard Towneley and Henry Power. Boyle confirmed their discovery through experiments and published the results. According to Robert Gunther and other authorities, it was Boyle's assistant, Robert Hooke, who built the experimental apparatus. Boyle's law is based on experiments with air, which he considered to be a fluid of particles at rest in between small invisible springs. At that time, air was still seen as one of the four elements, but Boyle disagreed. Boyle's interest was probably to understand air as an essential element of life; he published e.g. the growth of plants without air. The French physicist Edme Mariotte (1620–1684) discovered the same law independently of Boyle in 1676, but Boyle had already published it in 1662. Thus this law may, improperly, be referred to as Mariotte's or the Boyle-Mariotte law. Later, in 1687 in the Philosophiæ Naturalis Principia Mathematica, Newton showed mathematically that if an elastic fluid consisting of particles at rest, between which are repulsive forces inversely proportional to their distance, the density would be directly proportional to the pressure, but this mathematical treatise is not the physical explanation for the observed relationship. Instead of a static theory a kinetic theory is needed, which was provided two centuries later by Maxwell and Boltzmann.


Relation to Kinetic Theory and Ideal Gases

Boyle’s law states that at constant temperature for a fixed mass, the absolute pressure and the volume of a gas are inversely proportional. The law can also be stated in a slightly different manner, that the product of absolute pressure and volume is always constant.

Most gases behave like ideal gases at moderate pressures and temperatures. The technology of the 17th century could not produce high pressures or low temperatures. Hence, the law was not likely to have deviations at the time of publication. As improvements in technology permitted higher pressures and lower temperatures, deviations from the ideal gas behavior would become noticeable, and the relationship between pressure and volume can only be accurately described employing real gas theory. The deviation is expressed as the compressibility factor.

Robert Boyle (and Edme Mariotte) derived the law solely on experimental grounds. The law can also be derived theoretically based on the presumed existence of atoms and molecules and assumptions about motion and perfectly elastic collisions (see kinetic theory of gases). These assumptions were met with enormous resistance in the positivist scientific community at the time however, as they were seen as purely theoretical constructs for which there was not the slightest observational evidence.

Daniel Bernoulli in 1738 derived Boyle's law using Newton's laws of motion with application on a molecular level. It remained ignored until around 1845, when John Waterston published a paper building the main precepts of kinetic theory; this was rejected by the Royal Society of England. Later works of James Prescott Joule, Rudolf Clausius and in particular Ludwig Boltzmann firmly established the kinetic theory of gases and brought attention to both the theories of Bernoulli and Waterston.

The debate between proponents of Energetics and Atomism led Boltzmann to write a book in 1898, which endured criticism up to his suicide in 1906. Albert Einstein in 1905 showed how kinetic theory applies to the Brownian motion of a fluid-suspended particle, which was confirmed in 1908 by Jean Perrin.


The mathematical equation for Boyle's law is:

\qquad\qquad pV = k


p denotes the pressure of the system.
V denotes the volume of the gas.
k is a constant value representative of the pressure and volume of the system.

So long as temperature remains constant the same amount of energy given to the system persists throughout its operation and therefore, theoretically, the value of k will remain constant. However, due to the derivation of pressure as perpendicular applied force and the probabilistic likelihood of collisions with other particles through collision theory, the application of force to a surfa

Laws of war

The law of war is a body of law concerning acceptable justifications to engage in war (jus ad bellum) and the limits to acceptable wartime conduct (jus in bello). The law of war is considered an aspect ofpublic international law (the law of nations) and is distinguished from other bodies of law, such as the domestic law of a particular belligerent to a conflict, that may also provide legal limits to the conduct or justification of war.

Among other issues, modern laws of war address declarations of war, acceptance of surrender and the treatment of prisoners of war; military necessity, along with distinction and proportionality; and the prohibition of certain weapons that may cause unnecessary suffering.

Early sources and history

Attempts to define and regulate the conduct of individuals, nations, and other agents in war and to mitigate the worst effects of war have a long history. The earliest known instances are found in the Hebrew Bible (Old Testament). For example, Deuteronomy 20:19-20 limits the amount of acceptable collateral and environmental damage:

When thou shalt besiege a city a long time, in making war against it to take it, thou shalt not destroy the trees thereof by forcing an axe against them: for thou mayest eat of them, and thou shalt not cut them down (for the tree of the field is man's life) to employ them in the siege: Only the trees which thou knowest that they be not trees for meat, thou shalt destroy and cut them down; and thou shalt build bulwarks against the city that maketh war with thee, until it be subdued.

Similarly, Deuteronomy 21:10-14 requires that female captives who were forced to marry the victors of a war could not be sold as slaves.

In the early 7th century, the first Caliph, Abu Bakr, whilst instructing his Muslim army, laid down the following rules concerning warfare:

Stop, O people, that I may give you ten rules for your guidance in the battlefield. Do not commit treachery or deviate from the right path. You must not mutilate dead bodies. Neither kill a child, nor a woman, nor an aged man. Bring no harm to the trees, nor burn them with fire, especially those which are fruitful. Slay not any of the enemy's flock, save for your food. You are likely to pass by people who have devoted their lives to monastic services; leave them alone.

These rules were put into practice during the early Muslim conquests of the 7th and 8th centuries. After the expansion of the Caliphate, Islamic legal treatises on international law from the 9th century onwards covered the application of Islamic military jurisprudence to international law, including the law of treaties; the treatment of diplomats, hostages, refugees and prisoners of war in Islam; the right of asylum; conduct on the battlefield; protection of women, children and non-combatantcivilians; contracts across the lines of battle; the use of poisonous weapons; and devastation of enemy territory. These laws were put into practice by Muslim armies during the Crusades, most notably by Saladin and Sultan al-Kamil. For example, after al-Kamil defeated the Franks, Oliverus Scholasticus praised the Islamic laws of war, commenting on how al-Kamil supplied the defeated Frankish army with food:

Who could doubt that such goodness, friendship and charity come from God? Men whose parents, sons and daughters, brothers and sisters, had died in agony at our hands, whose lands we took, whom we drove naked from their homes, revived us with their own food when we were dying of hunger and showered us with kindness even when we were in their power.{{Verify source|date=September 2010

In medieval Europe, the Roman Catholic Church also began promulgating teachings on just war, reflected to some extent in movements such as the Peace and Truce of God. The impulse to restrict the extent of warfare, and especially protect the lives and property of non-combatants continued with Hugo Grotius and his attempts to write laws of war.

Modern sources

The modern law of war is derived from two principal sources:


From Encyclopedia

distributive law

distributive law In mathematics, given any two operations, symbolized by * and [symbol], the first operation, *, is distributive over the second, [symbol], if a *( b [symbol] c )=( a * b )[symbol]( a * c ) for all possible choices of a, b, and c. Multiplication, ×, is distributive over addition, +, since for any numbers a, b, and c, a ×( b + c )=( a × b )+( a × c ). For example, for the numbers 2, 3, and 4, 2×(3+4)=14 and (2×3)+(2×4)=14, meaning that 2×(3+4)=(2×3)+(2×4). Strictly speaking, this law expresses only left distributivity, i.e., a is distributed from the left side of ( b + c ); the corresponding definition for right distributivity is ( a + b )× c =( a × c )+( b × c ).

From Yahoo Answers

Question:in the definition of law of reciprocal proportions: "when two (or more) elements A and B combine sepatately with the fixed mass of the third element E, then the ratio in which A and B combine with E is the same or simple multiple of the ratio in which A and B combine with each other" what does "simple multiple of it mean" ???? and please explain this law thoroughly and its applications and use... thnx :)

Answers:Okay, so the law of reciprocal proportions says that the masses of two elements that react with a third element can also react with each other, and also with some other element with also a certain mass. This is pretty hard to understand so let's try an example: If 2 gram of hydrogen reacts with 3 grams of carbon, to form methane, and the same 1 gram of hydrogen reach with 4 grams of oxygen to form water, then according to the law of reciprocal proportions, 3 grams of carbon can react with 4 grams of oxygen, since they are both able to reach with 2 gram of hydrogen. In that same sense the 3 grams of carbon and 4 grams of oxygen can react with other elements, not just hydrogen. For example, they can both reach with Chloride to form carbon tetrachloride for carbon, and dichlorine monoxide for oxygen. I remember I had trouble with this too. Here, check out this website, it's got a good example:

Question:if you know the answer to this please help me out: I've been thinking about it for a long time so a little help would be great :D thank you ! ohmygosh yeah i am from smith !!

Answers:well first u must understand the definition of law of constant propotion. Im trying to solve the same question for a lab due tomorrow and the definition is unclear to me but i understand how it coincides or supports his theory. Dalton believes that atoms combine into compunds in whole number ratios. the law of constant proportion is the law staing that every pure substance always contains the same elements combined in the same proportions by weight. (if u understnad the defintion plz explain...) well anyhu the law of constant proportion backs up his theory ecause Dalton believed that compounds are made of certain amounts of atoms and when the compounds are broke apart, you get atoms in whole number ratios

Question:examples of 3 laws of motion

Answers:Newton first law of motion:every body continues to remain in its state of rest or uniform motion along a straight line unless it is compelled by external force to change that state. First law of newton is also called the law of inertia. ex : A person sitting in a vehicle at rest has his whole body at rest.when the vehicle suddenly starts moving forward,the lower part of in contact with the vehicle moves forward.But the upper part of the body continues too remain at rest due to inertia.As a result,the person has a tendency to fall back. Newton second law of motion:The acceleration given to a body is directly proportional to the force and inversely proportional to the mass and it takes place in the direction of force. ex : A force of 0.12N acts for 3 seconds on a body of mass 0.4 kg at rest.Find the velocity gained by the body. solution : F=0.12 N, m =0.4 kg and t = 3 s. using the equation F = ma, 0.12 = 0.41 therefore a = 0.12/0.4 = 0.3 meter per second square Initial velocity of the body U = 0 so final velocity of the body is given by v = u + at 0 + 0.3 * 3 = 0.9 meter per second square Third law of newton states that to every action there is any equal and opposite reaction. ex:When a body reaming at rest on a table.The body exerts a downward force on the table equal to its weight (action).The table in turn,exerts on the body an equal force in the opposite direction (reaction).

Question:1. reactant- Example: 2. product- Example: 3. chemical energy- Example: 4. exothermic reaction- Example: 5. endothermic reaction- Example: 6. chemical reaction- Example:. 7. Law of Conservation of Mass-

Answers:1. reactant- a thing you need to get a product! Example:money to get food! 2. product- result if something! Example:2+2=4 4 is the product 3. chemical energy- energy in a chemical reaction? Example:mass of hydrogen is 1.009 4. exothermic reaction-when a chemical reaction gives off energy! exo? exit? Example: 5. endothermic reaction-a chemical reaction where engery is entering the reaction. endo enter? Example: 6. chemical reaction- reaction between chemicals? Example:.H20 and salt?

From Youtube

Beg Algebra: Direct Proportion In order to explain direct proportionality, Professor Burger uses a real-world example of a spring and Hooke's Law. Hooke's law states that the distance a spring stretches varies directly to the force applied. If force, f, is directly proportional to distance, d, then d~f or d=kf. This equation allows us to find the constant, k, of how much the spring stretches when force is applied. After we have found this number, we can determine the distance the spring will stretch with varying forces applied. A lesson on inverse proportions can be found here: www.mindbites.comTaught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, Beginning Algebra. This course and others are available from Thinkwell, Inc. The full course can be found at The full course covers linear equations, inequalities, polynomials, rational expressions, relations and functions, roots and radicals, quadratic equations and systems of equations. Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College. He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the ...

Avogadro's Law :Check us out at Avogadro's law (sometimes referred to as Avogadro's hypothesis or Avogadro's principle) is a gas law named after Amedeo Avogadro who, in 1811,[1] hypothesized that "Equal volumes of ideal or perfect gases, at the same temperature and pressure, contain the same number of molecules." Thus, the number of molecules in a specific volume of gas is independent of the size or mass of the gas molecules. As an example, equal volumes of molecular hydrogen and nitrogen would contain the same number of molecules, as long as they are at the same temperature and pressure and observe ideal or perfect gas behavior. In practice, for real gases, the law only holds approximately, but the agreement is close enough for the approximation to be useful. The law can be stated mathematically as: . Where: V is the volume of the gas. n is the amount of substance of the gas. k is a proportionality constant. The most significant consequence of Avogadro's law is that the ideal gas constant has the same value for all gases. This means that the constant Where: p is the pressure of the gas T is the temperature in kelvin of the gas has the same value for all gases, independent of the size or mass of the gas molecules. One mole of an ideal gas occupies 22.414 litres (dm ) at STP, and occupies 24.45 litres at SATP (Standard Ambient Temperature and Pressure = 273K and 1 atm). This volume is often referred to as the molar volume of an ideal gas. Real gases may deviate from ...