In mathematics Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions, two quantities 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. Being a fundamental term, quantity is used to refer to any type of quantitative properties or attributes of things are said to be proportional if they vary in such a way that one of the quantities is a constant Some mathematical constants, such as e and π, arise in many different contexts. Others, such as Graham's number or Skewes' number, only arise in a single specific context, but are notable because they are the earliest found, largest or smallest exemplar of a class of numbers. Many of the more interesting mathematical constants have a name, also multiple of the other, or equivalently if they have a constant ratio In mathematics, a ratio expresses the magnitude of quantities relative to each other. Specifically, the ratio of two quantities indicates how many times the first quantity is contained in the second and may be expressed algebraically as their quotient. Example: For every Spoon of sugar, you need 2 spoons of flour .
Proportion also refers to the equality of two ratios.
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Symbol
The mathematical symbol '∝' is used to indicate that two values are proportional. For example, A ∝ B.
In Unicode Unicode is a computing industry standard for the consistent representation and handling of text expressed in most of the world's writing systems. Developed in conjunction with the Universal Character Set standard and published in book form as The Unicode Standard, the latest version of Unicode consists of a repertoire of more than 107,000 this is symbol U+221D.
Direct proportionality
Given two variables A variable is a symbol that stands for a value that may vary; the term usually occurs in opposition to constant, which is a symbol for a non-varying value, i.e. completely fixed or fixed in the context of use. The concepts of constants and variables are fundamental to all modern mathematics, science, engineering, and computer programming 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
The relation is often denoted
or, alternatively,
and the constant ratio
is called the proportionality constant or constant of proportionality.
Examples
- If an object travels at a constant speed In kinematics, the instantaneous speed of an object is the magnitude of its instantaneous velocity (the rate of change of its position); it is thus the scalar equivalent of velocity. The average speed of an object in an interval of time is the distance traveled by the object divided by the duration of the interval; the instantaneous speed is the, then the distance Distance is a numerical description of how far apart objects are. In physics or everyday discussion, distance may refer to a physical length, or an estimation based on other criteria . In mathematics, a distance function or metric is a generalization of the concept of physical distance. A metric is a function that behaves according to a specific traveled is proportional to the time Time is a physical process and non-spatial dimension in which reality is macroscopically transformed in continuity from the past through the present and on to the future. Time has been defined as a one-dimensional quantity used to sequence events, to quantify the durations of events and the intervals between them, and to quantify and measure the spent traveling, with the speed being the constant of proportionality.
- The circumference where r is the radius and d is the diameter of the circle, and π is defined as the ratio of the circumference of the circle to its diameter (the numerical value of pi is 3.141 592 653 589 793...) of a circle A circle is a simple shape of Euclidean geometry consisting of those points in a plane which are equidistant from a given point called the center. The common distance of the points of a circle from its center is called its radius is proportional to its diameter In geometry, a diameter 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. The word "diameter" derives from Greek διάμετρος , "diagonal of a circle", from δια- (dia-), "across, through&, with the constant of proportionality equal to π π is a mathematical constant whose value is the ratio of any circle's circumference to its diameter in Euclidean space; this is the same value as the ratio of a circle's area to the square of its radius. It is approximately equal to 3.141593 in the usual decimal notation. Many formulae from mathematics, science, and engineering involve π, which.
- On a map A map is a visual representation of an area—a symbolic depiction highlighting relationships between elements of that space such as objects, regions, and themes drawn to scale The scale of a map is defined as the ratio of a distance on the map to the corresponding distance on the ground. If the region of the map is small enough for the curvature of the Earth to be neglected, then the scale may be taken as a constant ratio over the whole map. . For maps covering larger areas, or the whole Earth, it is essential to use a, 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 In physics, a force is any influence that causes a free body to undergo an acceleration. Force can also be described by intuitive concepts such as a push or pull that can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform. A force has both magnitude and direction, making it a acting on a certain object due to gravity Gravitation, or gravity, is one of the four fundamental interactions of nature , in which objects with mass attract one another. In everyday life, gravitation is most familiar as the agent that gives weight to objects with mass and causes them to fall to the ground when dropped. Gravitation causes dispersed matter to coalesce, thus accounting for is proportional to the object's mass In physics, mass commonly refers to any of three properties of matter, which have been shown experimentally to be equivalent: Inertial mass, active gravitational mass and passive gravitational mass. In everyday usage, mass is often taken to mean weight, but in scientific use, they refer to different properties; the constant of proportionality between the mass and the force is known as gravitational acceleration In physics, gravitational acceleration is the specific force or acceleration on an object caused by gravity. In a vacuum, all small bodies accelerate in a gravitational field at the same rate relative to the center of mass. This is true regardless of the mass or composition of the body. On the surface of the Earth, all objects fall with an.
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 The mathematical concept of a function expresses the intuitive idea that one quantity completely determines another quantity (the value, or the output). A function assigns a unique value to each input of a specified type. The argument and the value may be real numbers, but they can also be elements from any given sets: the domain and the codomain of x will be a straight line In Euclidean geometry, a line is a straight curve. When geometry is used to model the real world, lines are used to represent straight objects with negligible width and height. Lines are an idealisation of such objects and have no width or height at all and are usually considered to be infinitely long. Lines are a fundamental concept in some passing through the origin In mathematics, the 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. In Euclidean geometry, the origin may be chosen freely as any convenient with the slope In mathematics, the slope or gradient of a line describes its steepness, incline, or grade. A higher slope value indicates a steeper incline of the line equal to the constant of proportionality: it corresponds to linear growth In analytic geometry, the term linear function is sometimes used to mean a first-degree polynomial function of one variable. These functions are known as "linear" because they are precisely the functions whose graph in the Cartesian coordinate plane is a straight line.
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 In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1⁄x or x −1, is a number which when multiplied by x yields the multiplicative identity, 1. The multiplicative inverse of a fraction a⁄b is b⁄a. For the multiplicative inverse of a real number, divide 1 by the number. For example, the reciprocal of 5 is one (reciprocal) of the other, or equivalently if their product In mathematics, a product is the result of multiplying, or an expression that identifies factors to be multiplied. The order in real or complex numbers are multiplied has no bearing on the product; this is known as the commutative law of multiplication. When matrices or members of various other associative algebras are multiplied the 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
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 In mathematics, the absolute value |a| of a real number a is a's numerical value without regard to its sign. So, for example, 3 is the absolute value of both 3 and −3 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 A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length plane is a hyperbola In mathematics a hyperbola is a curve, specifically a smooth curve that lies in a plane, which can be defined either by its geometric properties or by the kinds of equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, which are mirror images of each other and resembling two infinite bows. 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
Main article: Hyperbolic coordinates affords the hyperbolic geometry structure to Q that is erected on HP by hyperbolic motions. The hyperbolic lines in Q are rays from the origin or petal-shaped curves leaving and re-entering the origin. The left-right shift in HP corresponds to a squeeze mapping applied to QThe concepts of direct and inverse proportion lead to the location of points in the Cartesian plane by hyperbolic coordinates affords the hyperbolic geometry structure to Q that is erected on HP by hyperbolic motions. The hyperbolic lines in Q are rays from the origin or petal-shaped curves leaving and re-entering the origin. The left-right shift in HP corresponds to a squeeze mapping applied to Q; the two coordinates correspond to the constant of direct proportionality that locates a point on a ray In Euclidean geometry, a line is a straight curve. When geometry is used to model the real world, lines are used to represent straight objects with negligible width and height. Lines are an idealisation of such objects and have no width or height at all and are usually considered to be infinitely long. Lines are a fundamental concept in some 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 In mathematics, the exponential function is the function ex, where e is the number such that the function ex equals its own derivative. The exponential function is used to model phenomena when a constant change in the independent variable gives the same proportional change (increase or decrease) in the dependent variable. The exponential function of x, that is if there exists a non-zero constant k
Likewise, a variable y is logarithmically proportional to a variable x, if y is directly proportional to the logarithm In mathematics, the logarithm of a number to a given base is the power or exponent to which the base must be raised in order to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 3 is the power to which ten must be raised to produce 1000: 103 = 1000, so log101000 = 3. Only positive real numbers have real number of x, that is if there exists a non-zero constant k
Experimental determination
To determine experimentally whether two physical Physics is a natural science that involves the study of matter and its motion through space-time, as well as all applicable concepts, such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves quantities are directly proportional, one performs several measurements and plots the resulting data points in a Cartesian coordinate system A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length. 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 In mathematics, the slope or gradient of a line describes its steepness, incline, or grade. A higher slope value indicates a steeper incline. TWAT
See also
| Wikiversity has learning materials about Proportions |
- Correlation In statistics, correlation and dependence are any of a broad class of statistical relationships between two or more random variables or observed data values
- Eudoxus of Cnidus Eudoxus of Cnidus was a Greek astronomer, mathematician, scholar and student of Plato. Since all his own works are lost, our knowledge of him is obtained from secondary sources, such as Aratus's poem on astronomy. Theodosius of Bithynia's Sphaerics may be based on a work of Eudoxus
- Golden ratio In mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The golden ratio is an irrational mathematical constant, approximately 1.6180339887. Other names frequently used for the golden ratio are the golden
- Proportional font In typography, a typeface is a set of one or more fonts, in one or more sizes, designed with stylistic unity, each comprising a coordinated set of glyphs. A typeface usually comprises an alphabet of letters, numerals, and punctuation marks; it may also include ideograms and symbols, or consist entirely of them, for example, mathematical or map-
- Rule of three (mathematics) In elementary arithmetic, given an equation between two fractions or rational expressions, one can cross-multiply to simplify the equation or determine the value of a variable
- Sample size The sample size of a statistical sample is the number of observations that constitute it. It is typically denoted n, a positive integer
- Similarity Two geometrical objects are called similar if they both have the same shape. More precisely, one is congruent to the result of a uniform scaling of the other. Corresponding sides of similar polygons are in proportion, and corresponding angles of similar polygons have the same measure. One can be obtained from the other by uniformly "
Growth
- Linear growth In analytic geometry, the term linear function is sometimes used to mean a first-degree polynomial function of one variable. These functions are known as "linear" because they are precisely the functions whose graph in the Cartesian coordinate plane is a straight line
- Hyperbolic growth
Categories: Mathematical terminology Domain-specific terms must be recategorized into the corresponding mathematical domain. If the domain is unclear, but reasonably believed to exist, it is better to put the page into the root category:mathematics, where it will have a better chance of spotting and classification | Ratios
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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
bsaxberg
ue, 18 Mar 2008 21:53:41 GM
My last blog celebrated the release of the National . Math. Advisory Panel's (NMAP's) final report last week. This week I begin a series of blogs about many of the subsidiary reports each of them offers insights into America's current ...
