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Information about Proportionality (mathematics)

This article is about proportionality, the mathematical relation. For other uses of the term proportionality, see Proportionality (disambiguation).


In mathematics, two quantities are called proportional if they vary in such a way that one of the quantities is a constant multiple of the other, or equivalently if they have a constant ratio.

Definition

More formally, the variable y is said to be proportional (or sometimes directly proportional) to the variable x, if there exists a constant non-zero number k such that



The relation is often denoted



and the constant ratio


is called the proportionality constant or constant of proportionality of the proportionality relation.

Examples

  • 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 π.
  • On a map drawn to scale, the distance between any two points on the map is proportional to the distance between the two locations the points represent, with the constant of proportionality being the scale of the map.
  • The amount of force acting on a certain object from the gravity of the Earth at sea level is proportional to the object's mass, with the gravitational acceleration being the constant of proportionality on the object.

Properties

Since



is equivalent to



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.

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 proportionality with inverse proportionality.

Two variables are inversely proportional (or varying inversely) if one of the variables is directly proportional with the multiplicative inverse 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



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.

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 exists a non-zero constant k such that



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 exists a non-zero constant k such that

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.

See also

Proportionality may refer to:
  • Proportionality (mathematics), a special mathematical relationship between two quantities
  • Proportionality (physics)
  • Proportionality (law), a legal principle under municipal law in which the punishment of a certain crime should be in

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Mathematics (colloquially, maths or math) is the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies them. Benjamin Peirce called it "the science that draws necessary conclusions".
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Quantity is a kind of property which exists as magnitude or multitude. It is among the basic classes of things along with quality, substance, change, and relation. Quantity was first introduced as quantum, an entity having quantity.
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In mathematics and the mathematical sciences, a constant is a fixed, but possibly unspecified, value. This is in contrast to a variable, which is not fixed.

Unspecified constants

The most widely mentioned sort of constant
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The word multiple can refer to:
  • multiples of numbers
  • dissociative identity disorder, for people with multiple personalities, sometimes called "multiples".
  • multiple birth, because having twins is sometimes called having "multiples".

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variable (IPA pronunciation: [ˈvæɹiəbl]) (sometimes called a pronumeral) is a symbolic representation denoting a quantity or expression.
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In mathematics, a binary relation (or a dyadic or 2-place relation) is an arbitrary association of elements within a set or with elements of another set.

An example is the "divides" relation between the set of prime numbers P and the set of integers
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Speed is the rate of motion, or equivalently the rate of change in position, many times expressed as distance d traveled per unit of time t.

Speed is a scalar quantity with dimensions distance/time; the equivalent vector quantity to speed is known as
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Distance is a numerical description of how far apart objects are at any given moment in time. In physics or everyday discussion, distance may refer to a physical length, a period of time, or an estimation based on other criteria (e.g. "two counties over").
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time.

One view is that time is part of the fundamental structure of the universe, a dimension in which events occur in sequence, and time itself is something that can be measured.
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The circumference is the distance around a closed curve. Circumference is a kind of perimeter.

Circle

The circumference of a circle can be calculated from its diameter using the formula:


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circle is the set of all points in a plane at a fixed distance, called the radius, from a given point, the centre.

Circles are simple closed curves which divide the plane into an interior and exterior.
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diameter (Greek words diairo = divide and metro = measure) of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle. The diameters are the longest chords of the circle.
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MAP may refer to:
  • MAP, the ISO 639 alpha-3 for Austronesian languages
  • MAP (band), an indie band from Riverside, California
  • Maghreb Arab Press, the official Moroccan news agency
  • Malawi Against Polio

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The scale of a map is the ratio of a single unit of distance on the map to the equivalent distance on the ground. The scale can be expressed in four ways: as a ratio, a fraction, in words and as a graphical (bar) scale.
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In physics, force is an action or agency that causes a body of mass m to accelerate. It may be experienced as a lift, a push, or a pull. The acceleration of the body is proportional to the vector sum of all forces acting on it (known as net force or resultant force).
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Gravitation is a natural phenomenon by which all objects with mass attract each other. In everyday life, gravitation is most familiar as the agency that endows objects with weight.
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EARTH was a short-lived Japanese vocal trio which released 6 singles and 1 album between 2000 and 2001. Their greatest hit, their debut single "time after time", peaked at #13 in the Oricon singles chart.
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Mass is a fundamental concept in physics, roughly corresponding to the intuitive idea of "how much matter there is in an object". Mass is a central concept of classical mechanics and related subjects, and there are several definitions of mass within the framework of relativistic
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In physics, gravitational acceleration is the acceleration of an object caused by the force of gravity from another object. An interesting fact is that any object will accelerate towards a large object at the same rate, regardless of the mass of the object.
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function expresses dependence between two quantities, one of which is given (the independent variable, argument of the function, or its "input") and the other produced (the dependent variable, value of the function, or "output").
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line can be described as an ideal zero-width, infinitely long, perfectly straight curve (the term curve in mathematics includes "straight curves") containing an infinite number of points. In Euclidean geometry, exactly one line can be found that passes through any two points.
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origin of a Euclidean space is a special point, usually denoted by the letter O, used as a fixed point of reference for the geometry of the surrounding space. In a Cartesian coordinate system, the origin is the point where the axes of the system intersect.
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Slope is often used to describe the measurement of the steepness, incline, gradient, or grade of a straight line. A higher slope value indicates a steeper incline. The slope is defined as the ratio of the "rise" divided by the "run
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multiplicative inverse for a number x, denoted by 1⁄x or x −1, is a number which when multiplied by x yields the multiplicative identity, 1.
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For other senses of this word, see product.
In mathematics, a product is the result of multiplying, or an expression that identifies factors to be multiplied.
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In mathematics, the absolute value (or modulus[1]) of a real number is its numerical value without regard to its sign. So, for example, 3 is the absolute value of both 3 and −3.
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hyperbola (Greek ὑπερβολή literally 'overshooting' or 'excess') is a type of conic section defined as the intersection between a right circular conical surface and a plane which cuts through both halves
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