The variabley is directly proportional to the variablex with proportionality constant ~0.6.The variabley is inversely proportional to the variablex with proportionality constant 1.
Inmathematics, twosequences of numbers, oftenexperimental data, areproportional ordirectly proportional if their corresponding elements have a constantratio. The ratio is calledcoefficient of proportionality (orproportionality constant) and its reciprocal is known asconstant of normalization (ornormalizing constant). Two sequences areinversely proportional if corresponding elements have a constantproduct.
If several pairs of variables share the same direct proportionality constant, theequation expressing the equality of these ratios is called aproportion, e.g.,a/b =x/y = ⋯ =k (for details seeRatio).Proportionality is closely related tolinearity.
Given anindependent variablex and a dependent variabley,y isdirectly proportional tox[1] if there is a positive constantk such that:
The relation is often denoted using the symbols∝ (not to be confused with the Greek letteralpha) or~, with exception of Japanese texts, where~ is reserved for intervals:
Forx ≠ 0 theproportionality constant can be expressed as the ratio:
It is also called theconstant of variation orconstant of proportionality. Given such a constantk, the proportionalityrelation∝ with proportionality constantk between two setsA andB is theequivalence relation defined by
If an object travels at a constantspeed, then thedistance traveled is directly proportional to thetime spent traveling, with the speed being the constant of proportionality.
Thecircumference of acircle is directly proportional to itsdiameter, with the constant of proportionality equal toπ.
On amap of a sufficiently small geographical area, drawn toscale 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.
Theforce, acting on a small object with smallmass by a nearby large extended mass due togravity, is directly proportional to the object's mass; the constant of proportionality between the force and the mass is known asgravitational acceleration.
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, is the classical mass of the object.
Two variables areinversely proportional (also calledvarying inversely, ininverse variation, ininverse proportion)[2] if each of the variables is directly proportional to themultiplicative inverse (reciprocal) of the other, or equivalently if theirproduct is a constant.[3] It follows that the variabley is inversely proportional to the variablex if there exists a non-zero constantk such thatHence the constantk is the product ofx andy.
The graph of two variables varying inversely on theCartesian coordinate plane is arectangular hyperbola. The product of thex andy values of each point on the curve equals the constant of proportionalityk. Since neitherx nory can equal zero (becausek is non-zero), the graph never crosses either axis.
Direct and inverse proportion contrast as follows: in direct proportion the variables increase or decrease together. With inverse proportion, an increase in one variable is associated with a decrease in the other. For instance, in travel, a constant speed dictates a direct proportion between distance and time travelled; in contrast, for a given distance (the constant), the time of travel is inversely proportional to speed:s ×t =d.
The concepts ofdirect andinverse proportion lead to the location of points in the Cartesian plane byhyperbolic coordinates; the two coordinates correspond to the constant of direct proportionality that specifies a point as being on a particularray and theconstant of inverse proportionality that specifies a point as being on a particularhyperbola.
