Proportionality vs. Linearity

Propotionality

The variable y is directly proportional to the variable x with proportionality constant ~0.6.

The variable y is inversely proportional to the variable x with proportionality constant 1.

In mathematics, two varying quantities are said to be in a relation of proportionality, multiplicatively connected to a constant; that is, when either their ratio or their product yields a constant. The value of this constant is called the coefficient of proportionality or proportionality constant.

• If the ratio (${\frac {y}{x}}$) of two variables (x and y) is equal to a constant (k = ${\frac {y}{x}}$), then the variable in the numerator of the ratio (y) can be product of the other variable and the constant (y = k ⋅ x). In this case y is said to be directly proportional to x with proportionality constant k. Equivalently one may write $x={\frac {1}{k}}$⋅ y; that is, x is directly proportional to y with proportionality constant ${\frac {1}{k}}=({\frac {x}{y}})$. If the term proportional is connected to two variables without further qualification, generally direct proportionality can be assumed.
• If the product of two variables (x ⋅ y) is equal to a constant (k = x ⋅ y), then the two are said to be inversely proportional to each other with the proportionality constant k. Equivalently, both variables are directly proportional to the reciprocal of the respective other with proportionality constant k ($x$ = k ⋅ ${\frac {1}{y}}$ and y = k ⋅ ${\frac {1}{x}}$).

If several pairs of variables share the same direct proportionality constant, the equation expressing the equality of these ratios is called a proportion, e.g., ${\frac {a}{b}}={\frac {x}{y}}$ = ⋯ = k (for details see Ratio).

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) or "~":

$y\propto x,$ or $y\sim x.$

For $x\neq 0$ the proportionality constant can be expressed as the ratio

$k={\frac {y}{x}}.$

It is also called the constant of variation or constant of proportionality.

A direct proportionality can also be viewed as a linear equation in two variables with a y-intercept of 0 and a slope of k. This corresponds to linear growth.

Examples

• If an object travels at a constant speed, then the distance traveled is directly proportional to the time spent traveling, with the speed being the constant of proportionality.
• The circumference of a circle is directly proportional to its diameter, with the constant of proportionality equal to π.
• On a map of a sufficiently small geographical area, drawn to scale 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, acting on a small object with small mass by a nearby large extended mass due to gravity, is directly proportional to the object's mass; the constant of proportionality between the force and the mass is known as gravitational 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.

Inverse proportionality

Inverse proportionality with a function of y = 1/x

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, in reciprocal proportion) if each of the variables is directly proportional to 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

$y={\frac {k}{x}},$

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 plane is a rectangular hyperbola. 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 coordinates; the two coordinates correspond to the constant of direct proportionality that specifies a point as being on a particular ray and the constant of inverse proportionality that specifies a point as being on a particular hyperbola.

Linearity

Linearity is the property of a mathematical relationship (function) that can be graphically represented as a straight line. Linearity is closely related to proportionality. Examples in physics include the linear relationship of voltage and current in an electrical conductor (Ohm's law), and the relationship of mass and weight. By contrast, more complicated relationships are nonlinear.

Generalized for functions in more than one dimension, linearity means the property of a function of being compatible with addition and scaling, also known as the superposition principle.

The word linear comes from Latin linearis, "pertaining to or resembling a line".

In mathematics

In mathematics, a linear map or linear function f(x) is a function that satisfies the two properties:

• Additivity: f(x + y) = f(x) + f(y).
• Homogeneity of degree 1: f(αx) = α f(x) for all α.

These properties are known as the superposition principle. In this definition, x is not necessarily a real number, but can in general be an element of any vector space. A more special definition of linear function, not coinciding with the definition of linear map, is used in elementary mathematics (see below).

Additivity alone implies homogeneity for rational α, since $f(x+x)=f(x)+f(x)$ implies $f(nx)=nf(x)$ for any natural number n by mathematical induction, and then $nf(x)=f(nx)=f(m{\tfrac {n}{m}}x)=mf({\tfrac {n}{m}}x)$ implies $f({\tfrac {n}{m}}x)={\tfrac {n}{m}}f(x)$. The density of the rational numbers in the reals implies that any additive continuous function is homogeneous for any real number α, and is therefore linear.

The concept of linearity can be extended to linear operators. Important examples of linear operators include the derivative considered as a differential operator, and other operators constructed from it, such as del and the Laplacian. When a differential equation can be expressed in linear form, it can generally be solved by breaking the equation up into smaller pieces, solving each of those pieces, and summing the solutions.

Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called 'linear spaces'), linear transformations (also called 'linear maps'), and systems of linear equations.

For a description of linear and nonlinear equations, see linear equation.

Linear polynomials

In a different usage to the above definition, a polynomial of degree 1 is said to be linear, because the graph of a function of that form is a straight line.

Over the reals, a linear equation is one of the forms:

$f(x)=mx+b\$

where m is often called the slope or gradient; b the y-intercept, which gives the point of intersection between the graph of the function and the y-axis.

Note that this usage of the term linear is not the same as in the section above, because linear polynomials over the real numbers do not in general satisfy either additivity or homogeneity. In fact, they do so if and only if b = 0. Hence, if b ≠ 0, the function is often called an affine function (see in greater generality affine transformation).

Boolean functions

Hasse diagram of a linear Boolean function

In Boolean algebra, a linear function is a function $f$ for which there exist $a_{0},a_{1},\ldots ,a_{n}\in \{0,1\}$ such that

$f(b_{1},\ldots ,b_{n})=a_{0}\oplus (a_{1}\land b_{1})\oplus \cdots \oplus (a_{n}\land b_{n})$, where $b_{1},\ldots ,b_{n}\in \{0,1\}.$

Note that if $a_{0}=1$, the above function is considered affine in linear algebra (i.e. not linear).

A Boolean function is linear if one of the following holds for the function's truth table:

1. In every row in which the truth value of the function is T, there are an odd number of Ts assigned to the arguments, and in every row in which the function is F there is an even number of Ts assigned to arguments. Specifically, f(F, F, ..., F) = F, and these functions correspond to linear maps over the Boolean vector space.
2. In every row in which the value of the function is T, there is an even number of Ts assigned to the arguments of the function; and in every row in which the truth value of the function is F, there are an odd number of Ts assigned to arguments. In this case, f(F, F, ..., F) = T.

Another way to express this is that each variable always makes a difference in the truth value of the operation or it never makes a difference.

Negation, Logical biconditional, exclusive or, tautology, and contradiction are linear functions.

Licensing

Content obtained and/or adapted from:

• Proportionality (mathematics), Wikipedia under a CC BY-SA license
• Linearity, Wikipedia under a CC BY-SA license