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Dimensional analysis

In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantities (such as length, mass, time, and electric current) and units of measurement (such as metres and grams) and tracking these dimensions as calculations or comparisons are performed. The term dimensional analysis is also used to refer to conversion of units from one dimensional unit to another, which can be used to evaluate scientific formulae.

Commensurable physical quantities are of the same kind and have the same dimension, and can be directly compared to each other, even if they are expressed in differing units of measurement; e.g., metres and feet, grams and pounds, seconds and years. Incommensurable physical quantities are of different kinds and have different dimensions, and can not be directly compared to each other, no matter what units they are expressed in, e.g. metres and grams, seconds and grams, metres and seconds. For example, asking whether a gram is larger than an hour is meaningless.

Any physically meaningful equation, or inequality, must have the same dimensions on its left and right sides, a property known as dimensional homogeneity. Checking for dimensional homogeneity is a common application of dimensional analysis, serving as a plausibility check on derived equations and computations. It also serves as a guide and constraint in deriving equations that may describe a physical system in the absence of a more rigorous derivation.

The concept of physical dimension, and of dimensional analysis, was introduced by Joseph Fourier in 1822.

Formulation
The Buckingham π theorem describes how every physically meaningful equation involving $n$ variables can be equivalently rewritten as an equation of $n − m$ dimensionless parameters, where m is the rank of the dimensional matrix. Furthermore, and most importantly, it provides a method for computing these dimensionless parameters from the given variables.

A dimensional equation can have the dimensions reduced or eliminated through nondimensionalization, which begins with dimensional analysis, and involves scaling quantities by characteristic units of a system or physical constants of nature. This may give insight into the fundamental properties of the system, as illustrated in the examples below.

The dimension of a physical quantity can be expressed as a product of the base physical dimensions such as length, mass and time, each raised to an integer (and occasionally rational) power. The dimension of a physical quantity is more fundamental than some scale or unit used to express the amount of that physical quantity. For example, mass is a dimension, while the kilogram is a particular reference quantity chosen to express a quantity of mass. The choice of unit is arbitrary, and its choice is often based on historical precedent. Natural units, being based on only universal constants, may be thought of as being "less arbitrary".

There are many possible choices of base physical dimensions. The SI standard selects the following dimensions and corresponding dimension symbols:
 * time (T), length (L), mass (M), electric current (I), absolute temperature (Θ), amount of substance (N) and luminous intensity (J).

The symbols are by convention usually written in roman sans serif typeface. Mathematically, the dimension of the quantity $Q$ is given by
 * $$\operatorname{dim}Q = \mathsf{T}^a\mathsf{L}^b\mathsf{M}^c\mathsf{I}^d\mathsf{\Theta}^e\mathsf{N}^f\mathsf{J}^g$$

where $a$, $b$, $c$, $d$, $e$, $f$, $g$ are the dimensional exponents. Other physical quantities could be defined as the base quantities, as long as they form a linearly independent basis – for instance, one could replace the dimension (I) of electric current of the SI basis with a dimension (Q) of electric charge, since Q = TI.

A quantity that has only $b ≠ 0$ (with all other exponents zero) is known as a geometric quantity. A quantity that has only both $a ≠ 0$ and $b ≠ 0$ is known as a kinematic quantity. A quantity that has only all of $a ≠ 0$, $b ≠ 0$, and $c ≠ 0$ is known as a dynamic quantity. A quantity that has all exponents null is said to have dimension one.

The unit chosen to express a physical quantity and its dimension are related, but not identical concepts. The units of a physical quantity are defined by convention and related to some standard; e.g., length may have units of metres, feet, inches, miles or micrometres; but any length always has a dimension of L, no matter what units of length are chosen to express it. Two different units of the same physical quantity have conversion factors that relate them. For example, 1 in = 2.54 cm; in this case 2.54 cm/in is the conversion factor, which is itself dimensionless. Therefore, multiplying by that conversion factor does not change the dimensions of a physical quantity.

There are also physicists who have cast doubt on the very existence of incompatible fundamental dimensions of physical quantity, although this does not invalidate the usefulness of dimensional analysis.

Simple cases
As examples, the dimension of the physical quantity speed $v$ is
 * $$\operatorname{dim}v

= \frac{\text{length}}{\text{time}} = \frac{\mathsf{L}}{\mathsf{T}} = \mathsf{T}^{-1}\mathsf{L} .$$

The dimension of the physical quantity acceleration $a$ is
 * $$\operatorname{dim}a

= \frac{\text{speed}}{\text{time}} = \frac{\mathsf{T}^{-1}\mathsf{L}}{\mathsf{T}} = \mathsf{T}^{-2}\mathsf{L} .$$

The dimension of the physical quantity force $F$ is
 * $$\operatorname{dim}F

= \text{mass} \times \text{acceleration} = \mathsf{M} \times \mathsf{T}^{-2}\mathsf{L} = \mathsf{T}^{-2}\mathsf{L}\mathsf{M} .$$

The dimension of the physical quantity pressure $P$ is
 * $$\operatorname{dim}P

= \frac{\text{force}}{\text{area}} = \frac{\mathsf{T}^{-2}\mathsf{L}\mathsf{M}}{\mathsf{L}^2} = \mathsf{T}^{-2}\mathsf{L}^{-1}\mathsf{M} .$$

The dimension of the physical quantity energy $E$ is
 * $$\operatorname{dim}E

= \text{force} \times \text{displacement} = \mathsf{T}^{-2}\mathsf{L}\mathsf{M} \times \mathsf{L} = \mathsf{T}^{-2}\mathsf{L}^2\mathsf{M} .$$

The dimension of the physical quantity power $P$ is
 * $$\operatorname{dim}P

= \frac{\text{energy}}{\text{time}} = \frac{\mathsf{T}^{-2}\mathsf{L}^2\mathsf{M}}{\mathsf{T}} = \mathsf{T}^{-3}\mathsf{L}^2\mathsf{M} .$$

The dimension of the physical quantity electric charge $Q$ is
 * $$\operatorname{dim}Q

= \text{current} \times \text{time} = \mathsf{T}\mathsf{I} .$$

The dimension of the physical quantity voltage $V$ is
 * $$\operatorname{dim}V

= \frac{\text{power}}{\text{current}} = \frac{\mathsf{T}^{-3}\mathsf{L}^2\mathsf{M}}{\mathsf{I}} = \mathsf{T^{-3}}\mathsf{L}^2\mathsf{M} \mathsf{I}^{-1} .$$

The dimension of the physical quantity capacitance $C$ is
 * $$\operatorname{dim}C

= \frac{\text{electric charge}}{\text{electric potential difference}} = \frac {\mathsf{T}\mathsf{I}}{\mathsf{T}^{-3}\mathsf{L}^2\mathsf{M}\mathsf{I}^{-1}} = \mathsf{T^4}\mathsf{L^{-2}}\mathsf{M^{-1}}\mathsf{I^2} .$$

Rayleigh's method
In dimensional analysis, Rayleigh's method is a conceptual tool used in physics, chemistry, and engineering. It expresses a functional relationship of some variables in the form of an exponential equation. It was named after Lord Rayleigh.

The method involves the following steps:
 * 1) Gather all the independent variables that are likely to influence the dependent variable.
 * 2) If $R$ is a variable that depends upon independent variables $R_{1}$, $R_{2}$, $R_{3}$, ..., $R_{n}$, then the functional equation can be written as $R = F(R_{1}, R_{2}, R_{3}, ..., R_{n})$.
 * 3) Write the above equation in the form $R = C R_{1}^{a} R_{2}^{b} R_{3}^{c} ... R_{n}^{m}$, where $C$ is a dimensionless constant and $a$, $b$, $c$, ..., $m$ are arbitrary exponents.
 * 4) Express each of the quantities in the equation in some base units in which the solution is required.
 * 5) By using dimensional homogeneity, obtain a set of simultaneous equations involving the exponents $a$, $b$, $c$, ..., $m$.
 * 6) Solve these equations to obtain the values of the exponents $a$, $b$, $c$, ..., $m$.
 * 7) Substitute the values of exponents in the main equation, and form the non-dimensional parameters by grouping the variables with like exponents.

As a drawback, Rayleigh's method does not provide any information regarding number of dimensionless groups to be obtained as a result of dimensional analysis.

Concrete numbers and base units
Many parameters and measurements in the physical sciences and engineering are expressed as a concrete number—a numerical quantity and a corresponding dimensional unit. Often a quantity is expressed in terms of several other quantities; for example, speed is a combination of length and time, e.g. 60 kilometres per hour or 1.4 kilometres per second. Compound relations with "per" are expressed with division, e.g. 60 km/h. Other relations can involve multiplication (often shown with a centered dot or juxtaposition), powers (like m2 for square metres), or combinations thereof.

A set of base units for a system of measurement is a conventionally chosen set of units, none of which can be expressed as a combination of the others and in terms of which all the remaining units of the system can be expressed. For example, units for length and time are normally chosen as base units. Units for volume, however, can be factored into the base units of length (m3), thus they are considered derived or compound units.

Sometimes the names of units obscure the fact that they are derived units. For example, a newton (N) is a unit of force, which may be expressed as the product of mass (with unit kg) and acceleration (with unit m⋅s−2). The newton is defined as 1 N = 1 kg⋅m⋅s−2.

Percentages, derivatives and integrals
Percentages are dimensionless quantities, since they are ratios of two quantities with the same dimensions. In other words, the % sign can be read as "hundredths", since 1% = 1/100.

Taking a derivative with respect to a quantity divides the dimension by the dimension of the variable that is differentiated with respect to. Thus: Likewise, taking an integral adds the dimension of the variable one is integrating with respect to, but in the numerator.
 * position ($x$) has the dimension L (length);
 * derivative of position with respect to time ($dx/dt$, velocity) has dimension T−1L—length from position, time due to the gradient;
 * the second derivative $(dx/dt = d(dx/dt) / dt$, acceleration) has dimension.
 * force has the dimension (mass multiplied by acceleration);
 * the integral of force with respect to the distance ($s$) the object has travelled ($\textstyle\int F\ ds$, work) has dimension.

In economics, one distinguishes between stocks and flows: a stock has a unit (say, widgets or dollars), while a flow is a derivative of a stock, and has a unit of the form of this unit divided by one of time (say, dollars/year).

In some contexts, dimensional quantities are expressed as dimensionless quantities or percentages by omitting some dimensions. For example, debt-to-GDP ratios are generally expressed as percentages: total debt outstanding (dimension of currency) divided by annual GDP (dimension of currency)—but one may argue that, in comparing a stock to a flow, annual GDP should have dimensions of currency/time (dollars/year, for instance) and thus debt-to-GDP should have the unit year, which indicates that debt-to-GDP is the number of years needed for a constant GDP to pay the debt, if all GDP is spent on the debt and the debt is otherwise unchanged.

Dimensional homogeneity (commensurability)
The most basic rule of dimensional analysis is that of dimensional homogeneity.

However, the dimensions form an abelian group under multiplication, so:

For example, it makes no sense to ask whether 1 hour is more, the same, or less than 1 kilometre, as these have different dimensions, nor to add 1 hour to 1 kilometre. However, it makes sense to ask whether 1 mile is more, the same, or less than 1 kilometre, being the same dimension of physical quantity even though the units are different. On the other hand, if an object travels 100 km in 2 hours, one may divide these and conclude that the object's average speed was 50 km/h.

The rule implies that in a physically meaningful expression only quantities of the same dimension can be added, subtracted, or compared. For example, if $m_{man}$, $m_{rat}$ and $L_{man}$ denote, respectively, the mass of some man, the mass of a rat and the length of that man, the dimensionally homogeneous expression $m_{man} + m_{rat}$ is meaningful, but the heterogeneous expression $m_{man} + L_{man}$ is meaningless. However, $m_{man}/L^{2}_{man}$ is fine. Thus, dimensional analysis may be used as a sanity check of physical equations: the two sides of any equation must be commensurable or have the same dimensions.

Even when two physical quantities have identical dimensions, it may nevertheless be meaningless to compare or add them. For example, although torque and energy share the dimension, they are fundamentally different physical quantities.

To compare, add, or subtract quantities with the same dimensions but expressed in different units, the standard procedure is first to convert them all to the same unit. For example, to compare 32 metres with 35 yards, use 1 yard = 0.9144 m to convert 35 yards to 32.004 m.

A related principle is that any physical law that accurately describes the real world must be independent of the units used to measure the physical variables. For example, Newton's laws of motion must hold true whether distance is measured in miles or kilometres. This principle gives rise to the form that a conversion factor between two units that measure the same dimension must take multiplication by a simple constant. It also ensures equivalence; for example, if two buildings are the same height in feet, then they must be the same height in metres.

Conversion factor
In dimensional analysis, a ratio which converts one unit of measure into another without changing the quantity is called a conversion factor. For example, kPa and bar are both units of pressure, and 100 kPa = 1 bar. The rules of algebra allow both sides of an equation to be divided by the same expression, so this is equivalent to 100 kPa / 1 bar = 1. Since any quantity can be multiplied by 1 without changing it, the expression "100 kPa / 1 bar" can be used to convert from bars to kPa by multiplying it with the quantity to be converted, including the unit. For example, 5 bar × 100 kPa / 1 bar = 500 kPa because 5 × 100 / 1 = 500, and bar/bar cancels out, so 5 bar = 500 kPa.

Applications
Dimensional analysis is most often used in physics and chemistry – and in the mathematics thereof – but finds some applications outside of those fields as well.

Mathematics
A simple application of dimensional analysis to mathematics is in computing the form of the volume of an $n$-ball (the solid ball in n dimensions), or the area of its surface, the $n$-sphere: being an $n$-dimensional figure, the volume scales as $xn$, while the surface area, being $(n − 1)$-dimensional, scales as $xn−1$. Thus the volume of the $n$-ball in terms of the radius is $Cnr$, for some constant $Cn$. Determining the constant takes more involved mathematics, but the form can be deduced and checked by dimensional analysis alone.

Finance, economics, and accounting
In finance, economics, and accounting, dimensional analysis is most commonly referred to in terms of the distinction between stocks and flows. More generally, dimensional analysis is used in interpreting various financial ratios, economics ratios, and accounting ratios.
 * For example, the P/E ratio has dimensions of time (unit: year), and can be interpreted as "years of earnings to earn the price paid".
 * In economics, debt-to-GDP ratio also has the unit year (debt has a unit of currency, GDP has a unit of currency/year).
 * Velocity of money has a unit of 1/years (GDP/money supply has a unit of currency/year over currency): how often a unit of currency circulates per year.
 * Annual continuously compounded interest rates and simple interest rates are often expressed as a percentage (adimensional quantity) while time is expressed as an adimensional quantity consisting of the number of years. However, if the time includes year as the unit of measure, the dimension of the rate is 1/year. Of course, there is nothing special (apart from the usual convention) about using year as a unit of time: any other time unit can be used. Furthermore, if rate and time include their units of measure, the use of different units for each is not problematic. In contrast, rate and time need to refer to a common period if they are adimensional. (Note that effective interest rates can only be defined as adimensional quantities.)
 * In financial analysis, bond duration can be defined as $(dV/dr)/V$, where $V$ is the value of a bond (or portfolio), $r$ is the continuously compounded interest rate and $dV/dr$ is a derivative. From the previous point, the dimension of $r$ is 1/time. Therefore, the dimension of duration is time (usually expressed in years) because $dr$ is in the "denominator" of the derivative.

Fluid mechanics
In fluid mechanics, dimensional analysis is performed to obtain dimensionless pi terms or groups. According to the principles of dimensional analysis, any prototype can be described by a series of these terms or groups that describe the behaviour of the system. Using suitable pi terms or groups, it is possible to develop a similar set of pi terms for a model that has the same dimensional relationships. In other words, pi terms provide a shortcut to developing a model representing a certain prototype. Common dimensionless groups in fluid mechanics include:
 * Reynolds number ($Re$), generally important in all types of fluid problems: $$\mathrm{Re} = \frac{\rho\,ud}{\mu}.$$
 * Froude number ($Fr$), modeling flow with a free surface: $$\mathrm{Fr} = \frac{u}{\sqrt{g\,L}}.$$
 * Euler number ($Eu$), used in problems in which pressure is of interest: $$\mathrm{Eu} = \frac{\Delta p}{\rho u^2}.$$
 * Mach number ($Ma$), important in high speed flows where the velocity approaches or exceeds the local speed of sound: $$\mathrm{Ma} = \frac{u}{c},$$ where $c$ is the local speed of sound.

History
The origins of dimensional analysis have been disputed by historians. The first written application of dimensional analysis has been credited to François Daviet, a student of Lagrange, in a 1799 article at the Turin Academy of Science.

This led to the conclusion that meaningful laws must be homogeneous equations in their various units of measurement, a result which was eventually later formalized in the Buckingham π theorem. Simeon Poisson also treated the same problem of the parallelogram law by Daviet, in his treatise of 1811 and 1833 (vol I, p. 39). In the second edition of 1833, Poisson explicitly introduces the term dimension instead of the Daviet homogeneity.

In 1822, the important Napoleonic scientist Joseph Fourier made the first credited important contributions based on the idea that physical laws like F = ma should be independent of the units employed to measure the physical variables.

James Clerk Maxwell played a major role in establishing modern use of dimensional analysis by distinguishing mass, length, and time as fundamental units, while referring to other units as derived. Although Maxwell defined length, time and mass to be "the three fundamental units", he also noted that gravitational mass can be derived from length and time by assuming a form of Newton's law of universal gravitation in which the gravitational constant $G$ is taken as 1|unity, thereby defining M = T−2L3. By assuming a form of Coulomb's law in which the Coulomb constant ke is taken as unity, Maxwell then determined that the dimensions of an electrostatic unit of charge were Q = T−1L3/2M1/2, which, after substituting his M = T−2L3 equation for mass, results in charge having the same dimensions as mass, viz. Q = T−2L3.

Dimensional analysis is also used to derive relationships between the physical quantities that are involved in a particular phenomenon that one wishes to understand and characterize. It was used for the first time in this way in 1872 by Lord Rayleigh, who was trying to understand why the sky is blue. Rayleigh first published the technique in his 1877 book The Theory of Sound.

The original meaning of the word dimension, in Fourier's Theorie de la Chaleur, was the numerical value of the exponents of the base units. For example, acceleration was considered to have the dimension 1 with respect to the unit of length, and the dimension −2 with respect to the unit of time. This was slightly changed by Maxwell, who said the dimensions of acceleration are T−2L, instead of just the exponents.

A simple example: period of a harmonic oscillator
What is the period of oscillation $T$ of a mass $m$ attached to an ideal linear spring with spring constant $k$ suspended in gravity of strength $g$? That period is the solution for $T$ of some dimensionless equation in the variables $T$, $m$, $k$, and $g$. The four quantities have the following dimensions: $T$  [T];  $m$  [M]; $k$ [M/T2]; and  $g$ [L/T2]. From these we can form only one dimensionless product of powers of our chosen variables, $G1 = Tk/m$ [T2 · M/T2 / M = 1], and putting $G1 = C$ for some dimensionless constant $C$ gives the dimensionless equation sought. The dimensionless product of powers of variables is sometimes referred to as a dimensionless group of variables; here the term "group" means "collection" rather than mathematical group. They are often called dimensionless numbers as well.

The variable $g$ does not occur in the group. It is easy to see that it is impossible to form a dimensionless product of powers that combines $g$ with $k$, $m$, and $T$, because $g$ is the only quantity that involves the dimension L. This implies that in this problem the $g$ is irrelevant. Dimensional analysis can sometimes yield strong statements about the irrelevance of some quantities in a problem, or the need for additional parameters. If we have chosen enough variables to properly describe the problem, then from this argument we can conclude that the period of the mass on the spring is independent of $g$: it is the same on the earth or the moon. The equation demonstrating the existence of a product of powers for our problem can be written in an entirely equivalent way: $T = \kappa \sqrt\tfrac{m}{k}$, for some dimensionless constant $κ$ (equal to $$\sqrt{C}$$ from the original dimensionless equation).

When faced with a case where dimensional analysis rejects a variable ($g$, here) that one intuitively expects to belong in a physical description of the situation, another possibility is that the rejected variable is in fact relevant, but that some other relevant variable has been omitted, which might combine with the rejected variable to form a dimensionless quantity. That is, however, not the case here.

When dimensional analysis yields only one dimensionless group, as here, there are no unknown functions, and the solution is said to be "complete" – although it still may involve unknown dimensionless constants, such as $κ$.

A more complex example: energy of a vibrating wire
Consider the case of a vibrating wire of length $ℓ$ (L) vibrating with an amplitude $A$ (L). The wire has a linear density $ρ$ (M/L) and is under tension $s$ (LM/T2), and we want to know the energy $E$ (L2M/T2) in the wire. Let $π_{1}$ and $π_{2}$ be two dimensionless products of powers of the variables chosen, given by
 * $$\begin{align}

\pi_1 &= \frac{E}{As} \\ \pi_2 &= \frac{\ell}{A}. \end{align}$$

The linear density of the wire is not involved. The two groups found can be combined into an equivalent form as an equation
 * $$F\left(\frac{E}{As}, \frac{\ell}{A}\right) = 0 ,$$

where $F$ is some unknown function, or, equivalently as
 * $$E = As f\left(\frac{\ell}{A}\right) ,$$

where $f$ is some other unknown function. Here the unknown function implies that our solution is now incomplete, but dimensional analysis has given us something that may not have been obvious: the energy is proportional to the first power of the tension. Barring further analytical analysis, we might proceed to experiments to discover the form for the unknown function $f$. But our experiments are simpler than in the absence of dimensional analysis. We'd perform none to verify that the energy is proportional to the tension. Or perhaps we might guess that the energy is proportional to $ℓ$, and so infer that $E = ℓs$. The power of dimensional analysis as an aid to experiment and forming hypotheses becomes evident.

The power of dimensional analysis really becomes apparent when it is applied to situations, unlike those given above, that are more complicated, the set of variables involved are not apparent, and the underlying equations hopelessly complex. Consider, for example, a small pebble sitting on the bed of a river. If the river flows fast enough, it will actually raise the pebble and cause it to flow along with the water. At what critical velocity will this occur? Sorting out the guessed variables is not so easy as before. But dimensional analysis can be a powerful aid in understanding problems like this, and is usually the very first tool to be applied to complex problems where the underlying equations and constraints are poorly understood. In such cases, the answer may depend on a dimensionless number such as the Reynolds number, which may be interpreted by dimensional analysis.

A third example: demand versus capacity for a rotating disc
Consider the case of a thin, solid, parallel-sided rotating disc of axial thickness $t$ (L) and radius $R$ (L). The disc has a density $ρ$ (M/L3), rotates at an angular velocity $ω$ (T−1) and this leads to a stress $S$ (T−2L−1M) in the material. There is a theoretical linear elastic solution, given by Lame, to this problem when the disc is thin relative to its radius, the faces of the disc are free to move axially, and the plane stress constitutive relations can be assumed to be valid. As the disc becomes thicker relative to the radius then the plane stress solution breaks down. If the disc is restrained axially on its free faces then a state of plane strain will occur. However, if this is not the case then the state of stress may only be determined though consideration of three-dimensional elasticity and there is no known theoretical solution for this case. An engineer might, therefore, be interested in establishing a relationship between the five variables. Dimensional analysis for this case leads to the following (5 − 3 = 2) non-dimensional groups:

 * demand/capacity = $ρRω/S$
 * thickness/radius or aspect ratio = $t/R$

Through the use of numerical experiments using, for example, the finite element method, the nature of the relationship between the two non-dimensional groups can be obtained as shown in the figure. As this problem only involves two non-dimensional groups, the complete picture is provided in a single plot and this can be used as a design/assessment chart for rotating discs.

Mathematical properties
The dimensions that can be formed from a given collection of basic physical dimensions, such as T, L, and M, form an abelian group: The identity is written as 1; L0 = 1, and the inverse of L is 1/L or L−1. L raised to any integer power $p$ is a member of the group, having an inverse of L$−p$ or 1/L$p$. The operation of the group is multiplication, having the usual rules for handling exponents (L$n$ × L$m$ = L$n+m$). Physically, 1/L can be interpreted as reciprocal length, and 1/T as reciprocal time (see reciprocal second).

An abelian group is equivalent to a module over the integers, with the dimensional symbol T$i$ L$j$  M$k$ corresponding to the tuple $(i, j, k)$. When physical measured quantities (be they like-dimensioned or unlike-dimensioned) are multiplied or divided by one other, their dimensional units are likewise multiplied or divided; this corresponds to addition or subtraction in the module. When measurable quantities are raised to an integer power, the same is done to the dimensional symbols attached to those quantities; this corresponds to scalar multiplication in the module.

A basis for such a module of dimensional symbols is called a set of base quantities, and all other vectors are called derived units. As in any module, one may choose different bases, which yields different systems of units (e.g., choosing whether the unit for charge is derived from the unit for current, or vice versa).

The group identity, the dimension of dimensionless quantities, corresponds to the origin in this module, $(0, 0, 0)$.

In certain cases, one can define fractional dimensions, specifically by formally defining fractional powers of one-dimensional vector spaces, like $V$. However, it is not possible to take arbitrary fractional powers of units, due to representation-theoretic obstructions.

One can work with vector spaces with given dimensions without needing to use units (corresponding to coordinate systems of the vector spaces). For example, given dimensions $M$ and $L$, one has the vector spaces $V$ and $V$, and can define $V := V ⊗ V$ as the tensor product. Similarly, the dual space can be interpreted as having "negative" dimensions. This corresponds to the fact that under the natural pairing between a vector space and its dual, the dimensions cancel, leaving a dimensionless scalar.

The set of units of the physical quantities involved in a problem correspond to a set of vectors (or a matrix). The nullity describes some number (e.g., $V$) of ways in which these vectors can be combined to produce a zero vector. These correspond to producing (from the measurements) a number of dimensionless quantities, $V$. (In fact these ways completely span the null subspace of another different space, of powers of the measurements.) Every possible way of multiplying (and exponentiating) together the measured quantities to produce something with the same unit as some derived quantity $m$ can be expressed in the general form
 * $$X = \prod_{i=1}^m (\pi_i)^{k_i}\,.$$

Consequently, every possible commensurate equation for the physics of the system can be rewritten in the form
 * $$f(\pi_1,\pi_2, ..., \pi_m)=0\,.$$

Knowing this restriction can be a powerful tool for obtaining new insight into the system.

Mechanics
The dimension of physical quantities of interest in mechanics can be expressed in terms of base dimensions T, L, and M – these form a 3-dimensional vector space. This is not the only valid choice of base dimensions, but it is the one most commonly used. For example, one might choose force, length and mass as the base dimensions (as some have done), with associated dimensions F, L, M; this corresponds to a different basis, and one may convert between these representations by a change of basis. The choice of the base set of dimensions is thus a convention, with the benefit of increased utility and familiarity. The choice of base dimensions is not entirely arbitrary, because they must form a basis: they must span the space, and be linearly independent.

For example, F, L, M form a set of fundamental dimensions because they form a basis that is equivalent to T, L, M: the former can be expressed as [F = LM/T2], L, M, while the latter can be expressed as [T = (LM/F)1/2], L, M.

On the other hand, length, velocity and time (T, L, V) do not form a set of base dimensions for mechanics, for two reasons:
 * There is no way to obtain mass – or anything derived from it, such as force – without introducing another base dimension (thus, they do not span the space).
 * Velocity, being expressible in terms of length and time (V = L/T), is redundant (the set is not linearly independent).

Other fields of physics and chemistry
Depending on the field of physics, it may be advantageous to choose one or another extended set of dimensional symbols. In electromagnetism, for example, it may be useful to use dimensions of T, L, M and Q, where Q represents the dimension of electric charge. In thermodynamics, the base set of dimensions is often extended to include a dimension for temperature, Θ. In chemistry, the amount of substance (the number of molecules divided by the Avogadro constant, ≈ $6.02 mol-1$) is also defined as a base dimension, N. In the interaction of relativistic plasma with strong laser pulses, a dimensionless relativistic similarity parameter, connected with the symmetry properties of the collisionless Vlasov equation, is constructed from the plasma-, electron- and critical-densities in addition to the electromagnetic vector potential. The choice of the dimensions or even the number of dimensions to be used in different fields of physics is to some extent arbitrary, but consistency in use and ease of communications are common and necessary features.

Polynomials and transcendental functions
Bridgman’s theorem restricts the type of function that can be used to define a physical quantity from general (dimensionally compounded) quantities to only products of powers of the quantities, unless some of the independent quantities are algebraically combined to yield dimensionless groups, whose functions are grouped together in the dimensionless numeric multiplying factor. This excludes polynomials of more than one term or transcendental functions not of that form.

Scalar arguments to transcendental functions such as exponential, trigonometric and logarithmic functions, or to inhomogeneous polynomials, must be dimensionless quantities. (Note: this requirement is somewhat relaxed in Siano's orientational analysis described below, in which the square of certain dimensioned quantities are dimensionless.)

While most mathematical identities about dimensionless numbers translate in a straightforward manner to dimensional quantities, care must be taken with logarithms of ratios: the identity $\{π_{1}, ..., π_{m}\}$, where the logarithm is taken in any base, holds for dimensionless numbers $X$ and $log(a/b) = log a − log b$, but it does not hold if $a$ and $b$ are dimensional, because in this case the left-hand side is well-defined but the right-hand side is not.

Similarly, while one can evaluate monomials ($a$) of dimensional quantities, one cannot evaluate polynomials of mixed degree with dimensionless coefficients on dimensional quantities: for $b$, the expression (3 m)2 = 9 m2 makes sense (as an area), while for $x^{n}$, the expression (3 m)2 + 3 m = 9 m2 + 3 m does not make sense.

However, polynomials of mixed degree can make sense if the coefficients are suitably chosen physical quantities that are not dimensionless. For example,
 * $$ \tfrac{1}{2} \cdot (\mathrm{-9.8~m/s^2}) \cdot t^2 + (\mathrm{500~m/s}) \cdot t. $$

This is the height to which an object rises in time $x^{2}$ if the acceleration of gravity is 9.8 metres per second per second and the initial upward speed is 500 metres per second. It is not necessary for $x^{2} + x$ to be in seconds. For example, suppose $t$ = 0.01 minutes. Then the first term would be
 * $$\begin{align}

&\tfrac{1}{2} \cdot (\mathrm{-9.8~m/s^2}) \cdot (\mathrm{0.01~min})^2 \\[10pt] ={} &\tfrac{1}{2} \cdot -9.8 \cdot \left(0.01^2\right) (\mathrm{min/s})^2 \cdot \mathrm{m} \\[10pt] ={} &\tfrac{1}{2} \cdot -9.8 \cdot \left(0.01^2\right) \cdot 60^2 \cdot \mathrm{m}. \end{align}$$

Combining units and numerical values
The value of a dimensional physical quantity $t$ is written as the product of a unit [$t$] within the dimension and a dimensionless numerical value or numerical factor, $Z$.
 * $$Z = n \times [Z] = n [Z]$$

When like-dimensioned quantities are added or subtracted or compared, it is convenient to express them in the same unit so that the numerical values of these quantities may be directly added or subtracted. But, in concept, there is no problem adding quantities of the same dimension expressed in different units. For example, 1 metre added to 1 foot is a length, but one cannot derive that length by simply adding 1 and 1. A conversion factor, which is a ratio of like-dimensioned quantities and is equal to the dimensionless unity, is needed:
 * $$ \mathrm{1\,ft} = \mathrm{0.3048\,m}$$ is identical to $$ 1 = \frac{\mathrm{0.3048\,m}}{\mathrm{1\,ft}}.$$

The factor 0.3048 m/ft is identical to the dimensionless 1, so multiplying by this conversion factor changes nothing. Then when adding two quantities of like dimension, but expressed in different units, the appropriate conversion factor, which is essentially the dimensionless 1, is used to convert the quantities to the same unit so that their numerical values can be added or subtracted.

Only in this manner is it meaningful to speak of adding like-dimensioned quantities of differing units.

Quantity equations
A quantity equation, also sometimes called a complete equation, is an equation that remains valid independently of the unit of measurement used when expressing the physical quantities.

In contrast, in a numerical-value equation, just the numerical values of the quantities occur, without units. Therefore, it is only valid when each numerical values is referenced to a specific unit.

For example, a quantity equation for displacement $Z$ as speed $n$ multiplied by time difference $d$ would be:

for $s$ = 5 m/s, where $t$ and $d = s t$ may be expressed in any units, converted if necessary. In contrast, a corresponding numerical-value equation would be:

where $s$ is the numeric value of $t$ when expressed in seconds and $d$ is the numeric value of $D = 5 T$ when expressed in metres.

Generally, the use of numerical-value equations is discouraged.

Constants
The dimensionless constants that arise in the results obtained, such as the $T$ in the Poiseuille's Law problem and the $t$ in the spring problems discussed above, come from a more detailed analysis of the underlying physics and often arise from integrating some differential equation. Dimensional analysis itself has little to say about these constants, but it is useful to know that they very often have a magnitude of order unity. This observation can allow one to sometimes make "back of the envelope" calculations about the phenomenon of interest, and therefore be able to more efficiently design experiments to measure it, or to judge whether it is important, etc.

Formalisms
Paradoxically, dimensional analysis can be a useful tool even if all the parameters in the underlying theory are dimensionless, e.g., lattice models such as the Ising model can be used to study phase transitions and critical phenomena. Such models can be formulated in a purely dimensionless way. As we approach the critical point closer and closer, the distance over which the variables in the lattice model are correlated (the so-called correlation length, $D$) becomes larger and larger. Now, the correlation length is the relevant length scale related to critical phenomena, so one can, e.g., surmise on "dimensional grounds" that the non-analytical part of the free energy per lattice site should be $d$, where $C$ is the dimension of the lattice.

It has been argued by some physicists, e.g., Michael J. Duff, that the laws of physics are inherently dimensionless. The fact that we have assigned incompatible dimensions to Length, Time and Mass is, according to this point of view, just a matter of convention, borne out of the fact that before the advent of modern physics, there was no way to relate mass, length, and time to each other. The three independent dimensionful constants: $κ$, $χ$, and $~ 1/χd$, in the fundamental equations of physics must then be seen as mere conversion factors to convert Mass, Time and Length into each other.

Just as in the case of critical properties of lattice models, one can recover the results of dimensional analysis in the appropriate scaling limit; e.g., dimensional analysis in mechanics can be derived by reinserting the constants $d$, $c$, and $ħ$ (but we can now consider them to be dimensionless) and demanding that a nonsingular relation between quantities exists in the limit $G$, $ħ$ and $c$. In problems involving a gravitational field the latter limit should be taken such that the field stays finite.

Dimensional equivalences
Following are tables of commonly occurring expressions in physics, related to the dimensions of energy, momentum, and force.

Programming languages
Dimensional correctness as part of type checking has been studied since 1977. Implementations for Ada and C++ were described in 1985 and 1988. Kennedy's 1996 thesis describes an implementation in Standard ML, and later in F#. There are implementations for Haskell, OCaml, and Rust, Python, and a code checker for Fortran.

Griffioen's 2019 thesis extended Kennedy's Hindley–Milner type system to support Hart's matrices. McBride and Nordvall-Forsberg show how to use dependent types to extend type systems for units of measure.

Mathematica 13.2 has a function for transformations with quantities named NondimensionalizationTransform that applies a nondimensionalization transform to an equation. Mathematica also has a function to find the dimensions of a unit such as 1 J named UnitDimensions. Mathematica also has a function that will find dimensionally equivalent combinations of a subset of physical quantities named DimensionalCombations. Mathematica can also factor out certain dimension with UnitDimensions by specifying an argument to the function UnityDimensions. For example, you can use UnityDimensions to factor out angles. In addition to UnitDimensions, Mathematica can find the dimensions of a QuantityVariable with the function QuantityVariableDimensions.

Affine quantities
Some discussions of dimensional analysis implicitly describe all quantities as mathematical vectors. In mathematics scalars are considered a special case of vectors; vectors can be added to or subtracted from other vectors, and, inter alia, multiplied or divided by scalars. If a vector is used to define a position, this assumes an implicit point of reference: an origin. While this is useful and often perfectly adequate, allowing many important errors to be caught, it can fail to model certain aspects of physics. A more rigorous approach requires distinguishing between position and displacement (or moment in time versus duration, or absolute temperature versus temperature change).

Consider points on a line, each with a position with respect to a given origin, and distances among them. Positions and displacements all have units of length, but their meaning is not interchangeable: This illustrates the subtle distinction between affine quantities (ones modeled by an affine space, such as position) and vector quantities (ones modeled by a vector space, such as displacement).
 * adding two displacements should yield a new displacement (walking ten paces then twenty paces gets you thirty paces forward),
 * adding a displacement to a position should yield a new position (walking one block down the street from an intersection gets you to the next intersection),
 * subtracting two positions should yield a displacement,
 * but one may not add two positions.
 * Vector quantities may be added to each other, yielding a new vector quantity, and a vector quantity may be added to a suitable affine quantity (a vector space acts on an affine space), yielding a new affine quantity.
 * Affine quantities cannot be added, but may be subtracted, yielding relative quantities which are vectors, and these relative differences may then be added to each other or to an affine quantity.

Properly then, positions have dimension of affine length, while displacements have dimension of vector length. To assign a number to an affine unit, one must not only choose a unit of measurement, but also a point of reference, while to assign a number to a vector unit only requires a unit of measurement.

Thus some physical quantities are better modeled by vectorial quantities while others tend to require affine representation, and the distinction is reflected in their dimensional analysis.

This distinction is particularly important in the case of temperature, for which the numeric value of absolute zero is not the origin 0 in some scales. For absolute zero,
 * −273.15 °C ≘ 0 K = 0 °R ≘ −459.67 °F,

where the symbol ≘ means corresponds to, since although these values on the respective temperature scales correspond, they represent distinct quantities in the same way that the distances from distinct starting points to the same end point are distinct quantities, and cannot in general be equated.

For temperature differences,
 * 1 K = 1 °C ≠ 1 °F = 1 °R.

(Here °R refers to the Rankine scale, not the Réaumur scale). Unit conversion for temperature differences is simply a matter of multiplying by, e.g., 1 °F / 1 K (although the ratio is not a constant value). But because some of these scales have origins that do not correspond to absolute zero, conversion from one temperature scale to another requires accounting for that. As a result, simple dimensional analysis can lead to errors if it is ambiguous whether 1 K means the absolute temperature equal to −272.15 °C, or the temperature difference equal to 1 °C.

Orientation and frame of reference
Similar to the issue of a point of reference is the issue of orientation: a displacement in 2 or 3 dimensions is not just a length, but is a length together with a direction. (In 1 dimension, this issue is equivalent to the distinction between positive and negative.) Thus, to compare or combine two dimensional quantities in multi-dimensional Euclidean space, one also needs a bearing: they need to be compared to a frame of reference.

This leads to the extensions discussed below, namely Huntley's directed dimensions and Siano's orientational analysis.

Huntley's extensions
Huntley has pointed out that a dimensional analysis can become more powerful by discovering new independent dimensions in the quantities under consideration, thus increasing the rank $$m$$ of the dimensional matrix.

He introduced two approaches:
 * The magnitudes of the components of a vector are to be considered dimensionally independent. For example, rather than an undifferentiated length dimension L, we may have Lx represent dimension in the x-direction, and so forth. This requirement stems ultimately from the requirement that each component of a physically meaningful equation (scalar, vector, or tensor) must be dimensionally consistent.
 * Mass as a measure of the quantity of matter is to be considered dimensionally independent from mass as a measure of inertia.

Directed dimensions
As an example of the usefulness of the first approach, suppose we wish to calculate the distance a cannonball travels when fired with a vertical velocity component $$v_\text{y}$$ and a horizontal velocity component $v_\text{x}$, assuming it is fired on a flat surface. Assuming no use of directed lengths, the quantities of interest are then $R$, the distance travelled, with dimension L, $v_\text{x}$, $v_\text{y}$, both dimensioned as T−1L, and $g$ the downward acceleration of gravity, with dimension T−2L.

With these four quantities, we may conclude that the equation for the range $R$ may be written:
 * $$R \propto v_\text{x}^a\,v_\text{y}^b\,g^c .$$

Or dimensionally
 * $$\mathsf{L} = \left(\mathsf{T}^{-1}\mathsf{L}\right)^{a+b} \left(\mathsf{T}^{-2}\mathsf{L}\right)^c$$

from which we may deduce that $$a + b + c = 1$$ and $a + b + 2c = 0$, which leaves one exponent undetermined. This is to be expected since we have two fundamental dimensions T and L, and four parameters, with one equation.

However, if we use directed length dimensions, then $$v_\mathrm{x}$$ will be dimensioned as T−1L$G$, $$v_\mathrm{y}$$ as T−1L$c → ∞$, $R$ as L$ħ → 0$ and $g$ as T−2L$G → 0$. The dimensional equation becomes:

\mathsf{L}_\mathrm{x} = \left({\mathsf{T}^{-1}}{\mathsf{L}_\mathrm{x}}\right)^a \left({\mathsf{T}^{-1}}{\mathsf{L}_\mathrm{y}}\right)^b \left({\mathsf{T}^{-2}}{\mathsf{L}_\mathrm{y}}\right)^c $$ and we may solve completely as $E$, $F$ and $d$. The increase in deductive power gained by the use of directed length dimensions is apparent.

Huntley's concept of directed length dimensions however has some serious limitations:
 * It does not deal well with vector equations involving the cross product,
 * nor does it handle well the use of angles as physical variables.

It also is often quite difficult to assign the L, L$S$, L$t$, L$P$, symbols to the physical variables involved in the problem of interest. He invokes a procedure that involves the "symmetry" of the physical problem. This is often very difficult to apply reliably: It is unclear as to what parts of the problem that the notion of "symmetry" is being invoked. Is it the symmetry of the physical body that forces are acting upon, or to the points, lines or areas at which forces are being applied? What if more than one body is involved with different symmetries?

Consider the spherical bubble attached to a cylindrical tube, where one wants the flow rate of air as a function of the pressure difference in the two parts. What are the Huntley extended dimensions of the viscosity of the air contained in the connected parts? What are the extended dimensions of the pressure of the two parts? Are they the same or different? These difficulties are responsible for the limited application of Huntley's directed length dimensions to real problems.

Quantity of matter
In Huntley's second approach, he holds that it is sometimes useful (e.g., in fluid mechanics and thermodynamics) to distinguish between mass as a measure of inertia (inertial mass), and mass as a measure of the quantity of matter. Quantity of matter is defined by Huntley as a quantity only to inertial mass, while not implicating inertial properties. No further restrictions are added to its definition.

For example, consider the derivation of Poiseuille's Law. We wish to find the rate of mass flow of a viscous fluid through a circular pipe. Without drawing distinctions between inertial and substantial mass, we may choose as the relevant variables:

There are three fundamental variables, so the above five equations will yield two independent dimensionless variables:
 * $$\pi_1 = \frac{\dot{m}}{\eta r}$$
 * $$\pi_2 = \frac{p_\mathrm{x}\rho r^5}{\dot{m}^2}$$

If we distinguish between inertial mass with dimension $$M_\text{i}$$ and quantity of matter with dimension $$M_\text{m}$$, then mass flow rate and density will use quantity of matter as the mass parameter, while the pressure gradient and coefficient of viscosity will use inertial mass. We now have four fundamental parameters, and one dimensionless constant, so that the dimensional equation may be written:
 * $$C = \frac{p_\mathrm{x}\rho r^4}{\eta \dot{m}}$$

where now only $ρ$ is an undetermined constant (found to be equal to $$\pi/8$$ by methods outside of dimensional analysis). This equation may be solved for the mass flow rate to yield Poiseuille's law.

Huntley's recognition of quantity of matter as an independent quantity dimension is evidently successful in the problems where it is applicable, but his definition of quantity of matter is open to interpretation, as it lacks specificity beyond the two requirements he postulated for it. For a given substance, the SI dimension amount of substance, with unit mole, does satisfy Huntley's two requirements as a measure of quantity of matter, and could be used as a quantity of matter in any problem of dimensional analysis where Huntley's concept is applicable.

Siano's extension: orientational analysis
Angles are, by convention, considered to be dimensionless quantities (although the wisdom of this is contested ). As an example, consider again the projectile problem in which a point mass is launched from the origin $m$ at a speed $v$ and angle $p$ above the x-axis, with the force of gravity directed along the negative y-axis. It is desired to find the range $L$, at which point the mass returns to the x-axis. Conventional analysis will yield the dimensionless variable $I$, but offers no insight into the relationship between $ω$ and $p$.

Siano has suggested that the directed dimensions of Huntley be replaced by using orientational symbols $V$ to denote vector directions, and an orientationless symbol 10. Thus, Huntley's L$T$ becomes L1$N$ with L specifying the dimension of length, and $A$ specifying the orientation. Siano further shows that the orientational symbols have an algebra of their own. Along with the requirement that $I$, the following multiplication table for the orientation symbols results:

The orientational symbols form a group (the Klein four-group or "Viergruppe"). In this system, scalars always have the same orientation as the identity element, independent of the "symmetry of the problem". Physical quantities that are vectors have the orientation expected: a force or a velocity in the z-direction has the orientation of $t$. For angles, consider an angle $η$ that lies in the z-plane. Form a right triangle in the z-plane with $r$ being one of the acute angles. The side of the right triangle adjacent to the angle then has an orientation $S$ and the side opposite has an orientation $q$. Since (using $ϕ$ to indicate orientational equivalence) $E$ we conclude that an angle in the xy-plane must have an orientation $B$, which is not unreasonable. Analogous reasoning forces the conclusion that $ε$ has orientation $μ$ while $V$ has orientation 10. These are different, so one concludes (correctly), for example, that there are no solutions of physical equations that are of the form $p$, where $C$ and $θ$ are real scalars. An expression such as $$\sin(\theta+\pi/2)=\cos(\theta)$$ is not dimensionally inconsistent since it is a special case of the sum of angles formula and should properly be written:

\sin\left(a\,1_\text{z} + b\,1_\text{z}\right) = \sin\left(a\,1_\text{z}) \cos(b\,1_\text{z}\right) + \sin\left(b\,1_\text{z}) \cos(a\,1_\text{z}\right), $$ which for $$a = \theta$$ and $$b = \pi/2$$ yields $θ$. Siano distinguishes between geometric angles, which have an orientation in 3-dimensional space, and phase angles associated with time-based oscillations, which have no spatial orientation, i.e. the orientation of a phase angle is $a$.

The assignment of orientational symbols to physical quantities and the requirement that physical equations be orientationally homogeneous can actually be used in a way that is similar to dimensional analysis to derive more information about acceptable solutions of physical problems. In this approach, one solves the dimensional equation as far as one can. If the lowest power of a physical variable is fractional, both sides of the solution is raised to a power such that all powers are integral, putting it into normal form. The orientational equation is then solved to give a more restrictive condition on the unknown powers of the orientational symbols. The solution is then more complete than the one that dimensional analysis alone gives. Often, the added information is that one of the powers of a certain variable is even or odd.

As an example, for the projectile problem, using orientational symbols, $m$, being in the xy-plane will thus have dimension $A$ and the range of the projectile $b$ will be of the form:
 * $$R = g^a\,v^b\,\theta^c\text{ which means }\mathsf{L}\,1_\mathrm{x} \sim

\left(\frac{\mathsf{L}\,1_\text{y}}{\mathsf{T}^2}\right)^a \left(\frac{\mathsf{L}}{\mathsf{T}}\right)^b\,1_\mathsf{z}^c.\,$$

Dimensional homogeneity will now correctly yield $p$ and $m$, and orientational homogeneity requires that $\sin(\theta\,1_\text{z} + [\pi/2]\,1_\text{z}) = 1_\text{z}\cos(\theta\,1_\text{z})$. In other words, that $1_0$ must be an odd integer. In fact, the required function of theta will be $v$ which is a series consisting of odd powers of $R$.

It is seen that the Taylor series of $F$ and $t$ are orientationally homogeneous using the above multiplication table, while expressions like $S$ and $L$ are not, and are (correctly) deemed unphysical.

Siano's orientational analysis is compatible with the conventional conception of angular quantities as being dimensionless, and within orientational analysis, the radian may still be considered a dimensionless unit. The orientational analysis of a quantity equation is carried out separately from the ordinary dimensional analysis, yielding information that supplements the dimensional analysis.

Related areas of mathematics

 * Covariance and contravariance of vectors
 * Exterior algebra
 * Geometric algebra
 * Quantity calculus

title: Dimensional analysis


text:
December 25

Pre-1600

 * 36 – Forces of Emperor Guangwu of the Eastern Han, under the command of Wu Han, conquer the separatist Chengjia empire, reuniting China.
 * 274 – A temple to Sol Invictus is dedicated in Rome by Emperor Aurelian.
 * 333 – Roman Emperor Constantine the Great elevates his youngest son Constans to the rank of Caesar.
 * 336 – First documentary sign of Christmas celebration in Rome.
 * 350 – Vetranio meets Constantius II at Naissus (Serbia) and is forced to abdicate his imperial title. Constantius allows him to live as a private citizen on a state pension.
 * 508 – Clovis I, king of the Franks, is baptized into the Catholic faith at Reims, by Saint Remigius.
 * 597 – Augustine of Canterbury and his fellow-labourers baptise in Kent more than 10,000 Anglo-Saxons.
 * 800 – The coronation of Charlemagne as Holy Roman Emperor, in Rome.
 * 820 – Eastern Emperor Leo V is murdered in a church of the Great Palace of Constantinople by followers of Michael II.
 * 1000 – The foundation of the Kingdom of Hungary: Hungary is established as a Christian kingdom by Stephen I of Hungary.
 * 1013 – Sweyn Forkbeard takes control of the Danelaw and is proclaimed king of England.
 * 1025 – Coronation of Mieszko II Lambert as king of Poland.
 * 1046 – Henry III is crowned Holy Roman Emperor by Pope Clement II.
 * 1066 – William the Conqueror, Duke of Normandy is crowned king of England, at Westminster Abbey, London.
 * 1076 – Coronation of Bolesław II the Generous as king of Poland.
 * 1100 – Baldwin of Boulogne is crowned the first King of Jerusalem in the Church of the Nativity in Bethlehem.
 * 1130 – Count Roger II of Sicily is crowned the first king of Sicily.
 * 1261 – Eleven-year-old John IV Laskaris of the restored Eastern Roman Empire is deposed and blinded by orders of his co-ruler Michael VIII Palaiologos.
 * 1492 – The carrack Santa María, commanded by Christopher Columbus, runs onto a reef off Haiti due to an improper watch.
 * 1553 – Battle of Tucapel: Mapuche rebels under Lautaro defeat the Spanish conquistadors and executes the governor of Chile, Pedro de Valdivia.
 * 1559 – Pope Pius IV is elected, four months after his predecessor's death.

1601–1900

 * 1758 – Halley's Comet is sighted by Johann Georg Palitzsch, confirming Edmund Halley's prediction of its passage. This was the first passage of a comet predicted ahead of time.
 * 1766 – Mapuches in Chile launch a series of surprise attacks against the Spanish starting the Mapuche uprising of 1766.
 * 1776 – American Revolutionary War: George Washington and the Continental Army cross the Delaware River at night to attack Hessian forces serving Great Britain at Trenton, New Jersey, the next day.
 * 1793 – General "Mad Anthony" Wayne and a 300 man detachment identify the site of St. Clair's 1791 defeat by the large number of unburied human remains at modern Fort Recovery, Ohio.
 * 1809 – Dr. Ephraim McDowell performs the first ovariotomy, removing a 22-pound tumor.
 * 1814 – Rev. Samuel Marsden holds the first Christian service on land in New Zealand at Rangihoua Bay.
 * 1815 – The Handel and Haydn Society, oldest continually performing arts organization in the United States, gives its first performance.
 * 1826 – The Eggnog Riot at the United States Military Academy concludes after beginning the previous evening.
 * 1831 – The Great Jamaican Slave Revolt begins; up to 20% of Jamaica's slaves mobilize in an ultimately unsuccessful fight for freedom.
 * 1837 – Second Seminole War: American general Zachary Taylor leads 1,100 troops against the Seminoles at the Battle of Lake Okeechobee.
 * 1868 – Pardons for ex-Confederates: United States President Andrew Johnson grants an unconditional pardon to all Confederate veterans.

1901–present

 * 1914 – World War I: A series of unofficial truces occur across the Western Front to celebrate Christmas.
 * 1915 – The National Protection War breaks out against the Empire of China, as military leaders Cai E and Tang Jiyao proclaim the independence of Yunnan and begin a campaign to restore the Republic.
 * 1927 – B. R. Ambedkar and his followers burn copies of the Manusmriti in Mahad, Maharashtra, to protest its treatment of Dalit people.
 * 1932 – A magnitude 7.6 earthquake in Gansu, China kills 275 people.
 * 1941 – Admiral Chester W. Nimitz, appointed commander of the U.S. Pacific Fleet on December 17, arrives at Pearl Harbor.
 * 1941  – World War II: Battle of Hong Kong ends, beginning the Japanese occupation of Hong Kong.
 * 1941  – Admiral Émile Muselier seizes the archipelago of Saint Pierre and Miquelon, which become the first part of France to be liberated by the Free French Forces.
 * 1946 – The first European self-sustaining nuclear chain reaction is initiated within the Soviet Union's F-1 nuclear reactor.
 * 1950 – The Stone of Scone, traditional coronation stone of British monarchs, is taken from Westminster Abbey by Scottish nationalist students. It later turns up in Scotland on April 11, 1951.
 * 1951 – A bomb explodes at the home of Harry T. Moore and Harriette V. S. Moore, early leaders of the Civil Rights Movement, killing Harry instantly and fatally wounding Harriette.
 * 1962 – The Soviet Union conducts its final above-ground nuclear weapon test, in anticipation of the 1963 Partial Nuclear Test Ban Treaty.
 * 1963 – Turkish Cypriot Bayrak Radio begins transmitting in Cyprus after Turkish Cypriots are forcibly excluded from Cyprus Broadcasting Corporation.
 * 1968 – Apollo program: Apollo 8 performs the first successful Trans-Earth injection (TEI) maneuver, sending the crew and spacecraft on a trajectory back to Earth from Lunar orbit.
 * 1968  – Kilvenmani massacre: Forty-four Dalits (untouchables) are burnt to death in Kizhavenmani village, Tamil Nadu, a retaliation for a campaign for higher wages by Dalit laborers.
 * 1976 – EgyptAir Flight 664, a Boeing 707-366C, crashes on approach to Don Mueang International Airport, killing 71 people.
 * 1977 – Israeli Prime Minister Menachem Begin meets in Egypt with its president Anwar Sadat.
 * 1986 – Iraqi Airways Flight 163, a Boeing 737-270C, is hijacked and crashes in Arar, Saudi Arabia, killing 63 people.
 * 1989 – Romanian Revolution: Deposed President of Romania Nicolae Ceaușescu and his wife, Elena, are condemned to death and executed after a summary trial.
 * 1991 – Mikhail Gorbachev resigns as President of the Soviet Union (the union itself is dissolved the next day). Ukraine's referendum is finalized and Ukraine officially leaves the Soviet Union.
 * 1999 – Cubana de Aviación Flight 310, a Yakovlev Yak-42, crashes near Bejuma, Carabobo State, Venezuela, killing 22 people.
 * 2003 – UTA Flight 141, a Boeing 727-223, crashes at the Cotonou Airport in Benin, killing 141 people.
 * 2003  – The ill-fated Beagle 2 probe, released from the Mars Express spacecraft on December 19, stops transmitting shortly before its scheduled landing.
 * 2004 – The Cassini orbiter releases Huygens probe which successfully landed on Saturn's moon Titan on January 14, 2005.
 * 2012 – An Antonov An-72 plane crashes close to the city of Shymkent, killing 27 people.
 * 2012  – Air Bagan Flight 011, a Fokker 100, crashes on approach to Heho Airport in Heho, Myanmar, killing two people.
 * 2016 – A Russian Defence Ministry Tupolev Tu-154 carrying members of the Alexandrov Ensemble crashes into the Black Sea shortly after takeoff, killing all 92 people on board.
 * 2019 – Twenty people are killed and thousands are left homeless by Typhoon Phanfone in the Philippines.
 * 2020 – An explosion in Nashville, Tennessee, occurs, leaving three civilians in the hospital.
 * 2021 – The James Webb Space Telescope is launched.

Pre-1600

 * 1250 – John IV Laskaris, Byzantine emperor (d. 1305)
 * 1281 – Alice de Lacy, 4th Countess of Lincoln (d. 1348)
 * 1400 – John Sutton, 1st Baron Dudley, Lord Lieutenant of Ireland (d. 1487)
 * 1424 – Margaret Stewart, Dauphine of France (d. 1445)
 * 1461 – Christina of Saxony, Queen consort of Denmark (d. 1521)
 * 1490 – Francesco Marinoni, Italian Roman Catholic priest (d. 1562)
 * 1493 – Antoinette de Bourbon, French noblewoman (d. 1583)
 * 1505 – Christine of Saxony, German noblewoman (d. 1549)
 * 1564 – Johannes Buxtorf, German Calvinist theologian (d. 1629)
 * 1583 – Orlando Gibbons, English organist and composer (d. 1625)
 * 1584 – Margaret of Austria, Queen of Spain (d. 1611)

1601–1900

 * 1601 – Ernest I, Duke of Saxe-Gotha (d. 1675)
 * 1628 – Noël Coypel, French painter and educator (d. 1707)
 * 1642 (OS) – Isaac Newton, English physicist and mathematician (d. 1726/1727)
 * 1652 – Archibald Pitcairne, Scottish physician, anatomist, and scholar (d. 1713)
 * 1665 – Lady Grizel Baillie, Scottish-English poet and songwriter (d. 1746)
 * 1674 – Thomas Halyburton, Scottish minister and theologian (d. 1712)
 * 1686 – Giovanni Battista Somis, Italian violinist and composer (d. 1763)
 * 1700 – Leopold II, Prince of Anhalt-Dessau (d. 1758)
 * 1711 – Jean-Joseph de Mondonville, French violinist and composer (d. 1772)
 * 1716 – Johann Jakob Reiske, German physician and scholar (d. 1774)
 * 1717 – Pope Pius VI (d. 1799)
 * 1728 – Johann Adam Hiller, German composer and conductor (d. 1804)
 * 1730 – Philip Mazzei, Italian-American physician and philosopher (d. 1816)
 * 1745 – Chevalier de Saint-Georges, Caribbean-French violinist, composer, and conductor (d. 1799)
 * 1757 – Benjamin Pierce, American general and politician, 17th Governor of New Hampshire (d. 1839)
 * 1766 – Christmas Evans, Welsh Nonconformist preacher (d. 1838)
 * 1771 – Dorothy Wordsworth, English diarist and poet (d. 1855)
 * 1776 – Sydney, Lady Morgan, Irish author and poet (d. 1859)
 * 1810 – L. L. Langstroth, American apiarist, clergyman and teacher (d. 1895)
 * 1821 – Clara Barton, American nurse and humanitarian, founder of the American Red Cross (d. 1912)
 * 1825 – Stephen F. Chadwick, American lawyer and politician, 5th Governor of Oregon (d. 1895)
 * 1829 – Patrick Gilmore, Irish-American composer and bandleader (d. 1892)
 * 1856 – Pud Galvin, American baseball player and manager (d. 1902)
 * 1861 – Francis Henry Buzzacott, American hunter, explorer and army scout famous for writing Buzzacott's Masterpiece (d. 1947)
 * 1861 – Madan Mohan Malaviya, Indian educator, lawyer, and politician, President of the Indian National Congress (d. 1946)
 * 1865 – Evangeline Booth, English 4th General of The Salvation Army (d. 1950)
 * 1869 – Charles Finger, English-American journalist and author (d. 1941)
 * 1872 – Helena Rubinstein, Polish-American businesswoman and philanthropist (d. 1965)
 * 1873 – Otto Frederick Hunziker, Swiss-American agriculturalist and educator (d. 1959)
 * 1874 – Lina Cavalieri, Italian soprano and actress (d. 1944)
 * 1875 – Francis Aveling, Canadian psychologist and priest (d. 1941)
 * 1875  – Theodor Innitzer, Austrian cardinal (d. 1955)
 * 1876 – Muhammad Ali Jinnah, Indian-Pakistani lawyer and politician, 1st Governor-General of Pakistan (d. 1948)
 * 1876  – Adolf Otto Reinhold Windaus, German chemist and academic, Nobel Prize laureate (d. 1959)
 * 1878 – Louis Chevrolet, American race car driver and businessman, co-founded Chevrolet (d. 1941)
 * 1878  – Noël, Countess of Rothes, philanthropist, social leader and heroine of Titanic disaster (d. 1956)
 * 1878  – Joseph M. Schenck, Russian-American film producer (d. 1961)
 * 1883 – Hugo Bergmann, Czech-Israeli philosopher and academic (d. 1975)
 * 1883  – Hana Meisel, Belarusian-Israeli agronomist and politician (d. 1972)
 * 1884 – Samuel Berger, American boxer (d. 1925)
 * 1884  – Evelyn Nesbit, American model and actress (d. 1967)
 * 1886 – Malak Hifni Nasif, Egyptian poet and activist (d. 1918)
 * 1886  – Kid Ory, American trombonist and bandleader (d. 1973)
 * 1887 – Conrad Hilton, American entrepreneur (d. 1979)
 * 1889 – Lila Bell Wallace, American publisher and philanthropist, co-founded Reader's Digest (d. 1984)
 * 1890 – Noel Odell, English geologist and mountaineer (d. 1987)
 * 1891 – Kenneth Anderson, Indian-English general and politician, Governor of Gibraltar (d. 1959)
 * 1891  – Clarrie Grimmett, New Zealand-Australian cricketer (d. 1980)
 * 1899 – Humphrey Bogart, American actor (d. 1957)

1901–present

 * 1901 – Princess Alice, Duchess of Gloucester (d. 2004)
 * 1902 – William Bell, American tuba player and educator (d. 1971)
 * 1902  – Barton MacLane, American actor, playwright, and screenwriter (d. 1969)
 * 1903 – Antiochos Evangelatos, Greek composer and conductor (d. 1981)
 * 1904 – Gerhard Herzberg, German-Canadian physicist and chemist, Nobel Prize laureate (d. 1999)
 * 1906 – Lew Grade, Baron Grade, Ukrainian-English film producer (d. 1998)
 * 1906  – Ernst Ruska, German physicist and academic, Nobel Prize laureate (d. 1988)
 * 1907 – Cab Calloway, American singer-songwriter and bandleader (d. 1994)
 * 1907  – Mike Mazurki, Ukrainian-American wrestler and actor (d. 1990)
 * 1907  – Glenn McCarthy, American businessman, founded the Shamrock Hotel (d. 1988)
 * 1908 – Quentin Crisp, English author and illustrator (d. 1999)
 * 1908  – Ernest L. Massad, American general (d. 1993)
 * 1908  – Jo-Jo Moore, American baseball player (d. 2001)
 * 1909 – Zora Arkus-Duntov, Belgian-American engineer (d. 1996)
 * 1911 – Louise Bourgeois, French-American sculptor and painter (d. 2010)
 * 1913 – Candy Candido, American singer, bass player, and voice actor (d. 1999)
 * 1913  – Tony Martin, American singer (d. 2012)
 * 1914 – James Fletcher Jnr, New Zealand businessman (d. 2007)
 * 1914  – Oscar Lewis, American anthropologist of Latin America (d. 1970)
 * 1915 – Pete Rugolo, Italian-American composer and producer (d. 2011)
 * 1916 – Ahmed Ben Bella, Algerian soldier and politician, 1st President of Algeria (d. 2012)
 * 1917 – Arseny Mironov, Russian scientist, engineer, pilot, oldest active researcher in aircraft aerodynamics and flight testing (d. 2019)
 * 1917  – Lincoln Verduga Loor, Ecuadorian journalist and politician (d. 2009)
 * 1918 – Anwar Sadat, Egyptian lieutenant and politician, 3rd President of Egypt, Nobel Prize laureate (d. 1981)
 * 1919 – Naushad Ali, Indian composer and director (d. 2006)
 * 1919  – Paul David, Canadian cardiologist and politician, founded the Montreal Heart Institute (d. 1999)
 * 1919  – Noele Gordon, English actress (d. 1985)
 * 1921 – Zaib-un-Nissa Hamidullah, Indian-Pakistani journalist and author (d. 2000)
 * 1921  – Steve Otto, Polish-Canadian lawyer and politician (d. 1989)
 * 1922 – William Demby, American author (d. 2013)
 * 1923 – René Girard, French-American historian, philosopher, and critic (d. 2015)
 * 1923  – Louis Lane, American conductor and educator (d. 2016)
 * 1924 – Rod Serling, American screenwriter and producer, created The Twilight Zone (d. 1975)
 * 1924  – Atal Bihari Vajpayee, Indian poet and politician, 10th Prime Minister of India (d. 2018)
 * 1925 – Carlos Castaneda, Peruvian-American anthropologist and author (d. 1998)
 * 1925  – Ned Garver, American baseball player (d. 2017)
 * 1925  – Sam Pollock, Canadian businessman (d. 2007)
 * 1926 – Enrique Jorrín, Cuban violinist and composer (d. 1987)
 * 1927 – Nellie Fox, American baseball player and coach (d. 1975)
 * 1927  – Leo Kubiak, American basketball and baseball player
 * 1927  – Ram Narayan, Indian sarangi player
 * 1928 – Irish McCalla, American actress and model (d. 2002)
 * 1928  – Dick Miller, American actor, director, and screenwriter (d. 2019)
 * 1929 – Christine M. Jones, American educator and politician (d. 2013)
 * 1929  – China Machado, Chinese-born Portuguese-American fashion model, editor and television producer (d. 2016)
 * 1929  – Chris Kenner, American singer and songwriter (d. 1976)
 * 1930 – Emmanuel Agassi, Iranian-American boxer and coach (d. 2021)
 * 1930  – Armenak Alachachian, Armenian basketball player and coach (d. 2017)
 * 1930  – Mary Rose Tuitt, Montserrat politician (d. 2005)
 * 1932 – Mabel King, American actress and singer (d. 1999)
 * 1933 – Basil Heatley, English runner (d. 2019)
 * 1935 – Sadiq al-Mahdi, Sudanese politician, Prime Minister of Sudan (d. 2020)
 * 1935  – Stephen Barnett, American scholar and academic (d. 2009)
 * 1935  – Jeanne Hopkins Lucas, American educator and politician (d. 2007)
 * 1936 – Princess Alexandra, The Honourable Lady Ogilvy
 * 1936  – Ismail Merchant, Indian-English director and producer (d. 2005)
 * 1937 – Maung Aye, Burmese military officer
 * 1937  – O'Kelly Isley Jr., American R&B/soul singer-songwriter (d. 1986)
 * 1938 – Duane Armstrong, American painter
 * 1938  – Noel Picard, Canadian ice hockey player (d. 2017)
 * 1939 – Ghulam Ahmad Bilour, Pakistani businessman and politician
 * 1939  – Bob James, American keyboard player, songwriter, and producer
 * 1939  – Akong Rinpoche, Tibetan-Chinese spiritual leader (d. 2013)
 * 1940 – Hilary Spurling, English journalist and author
 * 1941 – Kenneth Calman, Scottish physician and academic
 * 1942 – Françoise Dürr, French tennis player and coach
 * 1942  – Barbara Follett, English politician
 * 1942  – Barry Goldberg, American keyboard player, songwriter, and producer
 * 1942  – Enrique Morente, Spanish singer-songwriter (d. 2010)
 * 1943 – Wilson Fittipaldi Júnior, Brazilian race car driver and businessman (d. 2024)
 * 1943  – Ravish Malhotra, Indian pilot and military officer
 * 1943  – Eve Pollard, English journalist and author
 * 1943  – Hanna Schygulla, German actress
 * 1943  – Jacqui McShee, English singer
 * 1944 – Kenny Everett, British comedian and broadcaster (d. 1995)
 * 1944  – Jairzinho, Brazilian footballer
 * 1944  – Sam Strahan, New Zealand rugby player (d. 2019)
 * 1945 – Rick Berman, American screenwriter and producer
 * 1945  – Mike Pringle, Zambian-Scottish lawyer and politician
 * 1945  – Noel Redding, English singer-songwriter and bass player (d. 2003)
 * 1945  – Ken Stabler, American football player and sportscaster (d. 2015)
 * 1946 – Jimmy Buffett, American singer-songwriter, guitarist, producer, and actor (d. 2023)
 * 1948 – Merry Clayton, American singer and actress
 * 1948  – Kay Hymowitz, American sociologist and writer
 * 1948  – Barbara Mandrell, American singer-songwriter and actress
 * 1948  – Joel Santana, Brazilian footballer and manager
 * 1949 – Simone Bittencourt de Oliveira, Brazilian singer
 * 1949  – Nawaz Sharif, Pakistani politician, 12th Prime Minister of Pakistan
 * 1949  – Sissy Spacek, American actress
 * 1950 – Peter Boardman, English mountaineer and author (d. 1982)
 * 1950  – Karl Rove, American political strategist and activist
 * 1950  – Manny Trillo, Venezuelan baseball player and manager
 * 1952 – Tolossa Kotu, Ethiopian runner and coach
 * 1952  – CCH Pounder, Guyanese-American actress
 * 1952  – Desireless, French singer and songwriter
 * 1953 – Kaarlo Maaninka, Finnish runner
 * 1954 – Annie Lennox, Scottish singer-songwriter and pianist
 * 1957 – Mansoor Akhtar, Pakistani cricketer
 * 1957  – Chris Kamara, English footballer and sportscaster
 * 1957  – Shane MacGowan, English-Irish singer-songwriter (d. 2023)
 * 1958 – Cheryl Chase, American voice actress and singer
 * 1958  – Rickey Henderson, American baseball player and coach
 * 1958  – Konstantin Kinchev, Russian singer-songwriter and guitarist
 * 1958  – Alannah Myles, Canadian singer-songwriter and actress
 * 1959 – Michael P. Anderson, American colonel, pilot, and astronaut (d. 2003)
 * 1959  – Ramdas Athawale, Indian poet and politician
 * 1961 – Íngrid Betancourt, Colombian political scientist and politician
 * 1962 – Francis Dunnery, English musician
 * 1964 – Ian Bostridge, English tenor
 * 1964  – Gary McAllister, Scottish footballer and manager
 * 1965 – Ed Davey, English politician, Leader of the Liberal Democrats
 * 1965  – Dmitri Mironov, Russian ice hockey player
 * 1965  – David Rath, Czech physician and politician
 * 1966 – Toshi Arai, Japanese race car driver
 * 1967 – Jason Thirsk, American bass player (d. 1996)
 * 1968 – Helena Christensen, Danish model and actress
 * 1968  – Jim Dowd, American ice hockey player
 * 1969 – Nicolas Godin, French musician
 * 1969  – Noel Goldthorpe, Australian rugby league player
 * 1969  – Frederick Onyancha, Kenyan runner
 * 1970 – Emmanuel Amunike, Nigerian footballer and manager
 * 1970  – Rodney Dent, American basketball player
 * 1971 – Dido, English singer-songwriter
 * 1971  – Noel Hogan, Irish musician and songwriter
 * 1971  – Justin Trudeau, Canadian educator and politician, 23rd Prime Minister of Canada
 * 1972 – Mac Powell, American singer-songwriter and producer
 * 1972  – Qu Yunxia, Chinese runner
 * 1973 – Robbie Elliott, English footballer and coach
 * 1973  – Chris Harris, American wrestler
 * 1973  – Daisuke Miura, Japanese baseball player and coach
 * 1973  – Alexandre Trudeau, Canadian journalist and director
 * 1975 – Hideki Okajima, Japanese baseball player
 * 1975  – Choi Sung-yong, South Korean footballer and manager
 * 1975  – Marcus Trescothick, English cricketer
 * 1976 – Tuomas Holopainen, Finnish keyboard player, songwriter, and producer
 * 1976  – Atko Väikmeri, Estonian footballer
 * 1976  – Armin van Buuren, Dutch DJ and record producer
 * 1977 – Ali Tandoğan, Turkish footballer
 * 1977  – Israel Vázquez, Mexican boxer
 * 1978 – Simon Jones, Welsh cricketer
 * 1978  – Joel Porter, Australian footballer and manager
 * 1978  – Jeremy Strong, American actor
 * 1979 – Ferman Akgül, Turkish singer-songwriter
 * 1979  – Laurent Bonnart, French footballer
 * 1979  – Robert Huff, English race car driver
 * 1979  – Hyun Young-min, South Korean footballer
 * 1980 – Laura Sadler, English actress (d. 2003)
 * 1980  – Marcus Trufant, American football player
 * 1981 – Trenesha Biggers, American wrestler and model
 * 1981 – Camille Herron, American ultramarathon runner
 * 1981  – Christian Holst, Danish-Faroese footballer
 * 1981  – Willy Taveras, Dominican baseball player
 * 1982 – Shawn Andrews, American football player
 * 1982  – Rob Edwards, Welsh footballer
 * 1982  – Ethan Kath, Canadian keyboard player, songwriter and producer
 * 1982  – Chris Rene, American singer-songwriter and producer
 * 1984 – Chris Cahill, Samoan footballer
 * 1984  – Alastair Cook, English cricketer
 * 1984  – Jessica Origliasso, Australian singer, actress, and fashion designer
 * 1984  – Lisa Origliasso, Australian singer, actress, and fashion designer
 * 1984  – Chris Richard, American basketball player
 * 1985 – Martin Mathathi, Kenyan runner
 * 1985  – Rusev, Bulgarian-American professional wrestler
 * 1985  – Perdita Weeks, Welsh actress
 * 1987 – Ceyhun Gülselam, Turkish footballer
 * 1987  – Demaryius Thomas, American football player (d. 2021)
 * 1988 – Eric Gordon, American basketball player
 * 1988  – Lukas Hinds-Johnson, German rugby player
 * 1988  – Joãozinho, Brazilian footballer
 * 1991 – Avu-chan, Japanese musician, songwriter, actor, model and producer
 * 1992 – Mitakeumi Hisashi, Japanese sumo wrestler
 * 1993 – Emi Takei, Japanese actress, fashion model and singer
 * 1999 – Adut Akech, South Sudanese-Australian fashion model

Pre-1600

 * 304 – Saint Anastasia
 * 795 – Pope Adrian I
 * 820 – Emperor Leo V
 * 936 – Zhang Jingda, general of Later Tang
 * 940 – Makan ibn Kaki, Iranian general
 * 1147 – Guy II, Count of Ponthieu (b. c. 1120)
 * 1156 – Peter the Venerable, French abbot and saint (b. 1092)
 * 1156  – Sverker the Elder, king of Sweden
 * 1294 – Mestwin II, Duke of Pomerania
 * 1395 – Elisabeth, Countess of Neuchâtel, Swiss ruler
 * 1406 – Henry III of Castile (b. 1379)
 * 1505 – George Grey, 2nd Earl of Kent, English politician (b. 1454)
 * 1553 – Pedro de Valdivia, Spanish explorer and politician, 1st Royal Governor of Chile (b. 1500)

1601–1900

 * 1634 – Lettice Knollys, English noblewoman (b. 1543)
 * 1635 – Samuel de Champlain, French soldier, geographer, and explorer (b. 1567)
 * 1676 – William Cavendish, 1st Duke of Newcastle, English soldier and politician, Lord Lieutenant of Nottinghamshire (b. 1592)
 * 1676  – Matthew Hale, English lawyer and jurist, Lord Chief Justice of England and Wales (b. 1609)
 * 1683 – Kara Mustafa Pasha, Ottoman general and politician, 111th Grand Vizier of the Ottoman Empire (b. 1634)
 * 1708 – Jørgen Thormøhlen, German-Norwegian merchant (b. c.1640)
 * 1730 – Henry Scott, 1st Earl of Deloraine, Scottish peer and general (b. 1676)
 * 1758 – James Hervey, English priest and author (b. 1714)
 * 1784 – Yosa Buson, Japanese poet and painter (b. 1716)
 * 1796 – Velu Nachiyar, Queen of Sivagangai (b. 1730)
 * 1824 – Barbara von Krüdener, German mystic and author (b. 1764)
 * 1824  – William Lawless, Irish revolutionary, later French Army general (b. 1772)
 * 1868 – Linus Yale, Jr., American engineer and businessman (b. 1821)
 * 1875 – Young Tom Morris, Scottish golfer (b. 1851)
 * 1880 – Fridolin Anderwert, Swiss lawyer and politician, President of the Swiss National Council (b. 1828)

1901–present

 * 1916 – Albert Chmielowski, Polish saint, founded the Albertine Brothers (b. 1845)
 * 1921 – Vladimir Korolenko, Russian journalist, author, and activist (b. 1853)
 * 1925 – Karl Abraham, German psychoanalyst and author (b. 1877)
 * 1926 – Emperor Taishō of Japan (b. 1879)
 * 1928 – Miles Burke, American boxer (b. 1885)
 * 1930 – Jakob Mändmets, Estonian journalist and author (b. 1871)
 * 1933 – Francesc Macià, Catalan colonel and politician, 122nd President of Catalonia (b. 1859)
 * 1935 – Paul Bourget, French author and critic (b. 1852)
 * 1938 – Karel Čapek, Czech author and playwright (b. 1890)
 * 1940 – Agnes Ayres, American actress (b. 1898)
 * 1941 – Richard S. Aldrich, American lawyer and politician (b. 1884)
 * 1944 – George Steer, South African-English journalist and author (b. 1909)
 * 1946 – W. C. Fields, American actor, comedian, juggler, and screenwriter (b. 1880)
 * 1947 – Gaspar G. Bacon, American lawyer and politician, 51st Lieutenant Governor of Massachusetts (b. 1886)
 * 1949 – Leon Schlesinger, American animator and producer, founded Warner Bros. Cartoons (b. 1884)
 * 1950 – Neil Francis Hawkins, English politician (b. 1903)
 * 1952 – Margrethe Mather, American photographer (b. 1886)
 * 1953 – Patsy Donovan, Irish-American baseball player and manager (b. 1865)
 * 1953  – William Haselden, British cartoonist (b. 1872)
 * 1956 – Robert Walser, Swiss author and playwright (b. 1878)
 * 1957 – Charles Pathé, French record producer, founded Pathé Records (b. 1863)
 * 1961 – Owen Brewster, American captain, lawyer, and politician, 54th Governor of Maine (b. 1888)
 * 1961  – Otto Loewi, German-American pharmacologist and academic, Nobel Prize laureate (b. 1873)
 * 1963 – Tristan Tzara, Romanian-French poet, playwright, painter, and critic (b. 1896)
 * 1970 – Michael Peto, Hungarian-English photographer and journalist (b. 1908)
 * 1973 – İsmet İnönü, Turkish general and politician, 2nd President of Turkey (b. 1884)
 * 1973  – Gabriel Voisin, French pilot and engineer (b. 1880)
 * 1975 – Gaston Gallimard, French publisher, founded Éditions Gallimard (b. 1881)
 * 1975  – Gunnar Kangro, Estonian mathematician and author (b. 1913)
 * 1977 – Charlie Chaplin, English actor and director (b. 1889)
 * 1979 – Joan Blondell, American actress and singer (b. 1906)
 * 1979  – Jordi Bonet, Canadian painter and sculptor (b. 1932)
 * 1980 – Fred Emney, English actor and comedian (b. 1900)
 * 1983 – Joan Miró, Spanish painter and sculptor (b. 1893)
 * 1988 – Shōhei Ōoka, Japanese author and critic (b. 1909)
 * 1988  – Edward Pelham-Clinton, 10th Duke of Newcastle, English entomologist and lepidopterist (b. 1920)
 * 1989 – Benny Binion, American poker player and businessman (b. 1904)
 * 1989  – Elena Ceaușescu, Romanian politician, First Lady of Romania (b. 1916)
 * 1989  – Nicolae Ceaușescu, Romanian general and politician, 1st President of Romania (b. 1918)
 * 1989  – Betty Garde, American actress (b. 1905)
 * 1989  – Frederick F. Houser, American judge and politician, 34th Lieutenant Governor of California (b. 1905)
 * 1989  – Billy Martin, American baseball player and manager (b. 1928)
 * 1989  – Robert Pirosh, American director and screenwriter (b. 1910)
 * 1991 – Wilbur Snyder, American football player and wrestler (b. 1929)
 * 1992 – Monica Dickens, British-American nurse and author (b. 1915)
 * 1993 – Pierre Victor Auger, French physicist and academic (b. 1899)
 * 1994 – Zail Singh, Indian politician, 7th President of India (b. 1916)
 * 1995 – Emmanuel Levinas, Lithuanian-French philosopher and academic (b. 1906)
 * 1995  – Dean Martin, American singer and actor (b. 1917)
 * 1995  – Chang Kee-ryo, Korean surgeon (b. 1914)
 * 1996 – Bill Hewitt, Canadian sportscaster (b. 1928)
 * 1997 – Anatoli Boukreev, Kazakh mountaineer and explorer (b. 1958)
 * 1997  – Denver Pyle, American actor (b. 1920)
 * 1998 – John Pulman, English snooker player (b. 1923)
 * 2000 – Neil Hawke, Australian cricketer and footballer (b. 1939)
 * 2000  – Willard Van Orman Quine, American philosopher and academic (b. 1908)
 * 2001 – Alfred A. Tomatis, French otolaryngologist and academic (b. 1920)
 * 2003 – Nicholas Mavroules, American politician (b. 1929)
 * 2004 – Gennadi Strekalov, Russian engineer and astronaut (b. 1940)
 * 2005 – Derek Bailey, English guitarist (b. 1930)
 * 2005  – Robert Barbers, Filipino police officer, lawyer, and politician, 15th Secretary of the Interior and Local Government (b. 1944)
 * 2005  – Birgit Nilsson, Swedish operatic soprano (b. 1918)
 * 2005  – Joseph Pararajasingham, Sri Lankan journalist, businessman, and politician (b. 1934)
 * 2006 – James Brown, American singer-songwriter (b. 1933)
 * 2007 – Des Barrick, English cricketer (b. 1927)
 * 2007  – Jim Beauchamp, American baseball player and coach (b. 1939)
 * 2008 – Eartha Kitt, American singer and actress (b. 1927)
 * 2009 – Vic Chesnutt, American singer-songwriter and guitarist (b. 1964)
 * 2010 – Carlos Andrés Pérez, Venezuelan politician, 66th President of Venezuela (b. 1922)
 * 2011 – Giorgio Bocca, Italian journalist (b. 1920)
 * 2011  – Jim Sherwood, American saxophonist (b. 1942)
 * 2011  – Simms Taback, American author and illustrator (b. 1932)
 * 2012 – Erico Aumentado, Filipino journalist, lawyer, and politician (b. 1940)
 * 2012  – Halfdan Hegtun, Norwegian radio host and politician (b. 1918)
 * 2012  – Joe Krivak, American football player and coach (b. 1935)
 * 2012  – Turki bin Sultan, Saudi Arabian politician (b. 1959)
 * 2012  – Şerafettin Elçi, Turkish lawyer, politician, government minister (b. 1938)
 * 2013 – Anthony J. Bryant, American historian and author (b. 1961)
 * 2013  – David R. Harris, English geographer, anthropologist, archaeologist and academic (b. 1930)
 * 2013  – Wayne Harrison, English footballer (b. 1967)
 * 2013  – Mike Hegan, American baseball player and sportscaster (b. 1942)
 * 2013  – Lola Lange, Canadian rural feminist and appointee to the Royal Commission on the Status of Women (b. 1922)
 * 2013  – Mel Mathay, Filipino politician, 8th Mayor of Quezon City (b. 1932)
 * 2014 – Ricardo Porro, Cuban-French architect (b. 1925)
 * 2014  – Geoff Pullar, English cricketer (b. 1935)
 * 2014  – David Ryall, English actor (b. 1935)
 * 2015 – George Clayton Johnson, American author and screenwriter (b. 1929)
 * 2015  – Dorothy M. Murdock, American author and historian (b. 1961)
 * 2016 – Valery Khalilov, Russian military musician and composer (b. 1952)
 * 2016  – George Michael, British singer and songwriter (b. 1963)
 * 2016  – Vera Rubin, American astronomer (b. 1928)
 * 2017 – D. Herbert Lipson, American magazine publisher (Philadelphia, Boston) (b. 1929)
 * 2018 – Sulagitti Narasamma, Indian midwife (b. 1920)
 * 2019 – Ari Behn, Norwegian writer (b. 1972)
 * 2020 – K. C. Jones, American basketball player and coach (b. 1932)
 * 2021 – Wayne Thiebaud, American artist (b. 1920)
 * 2022 – Fabián O'Neill, Uruguayan footballer (b. 1973)
 * 2023 – Jim Breaks, British wrestler (b. 1940)

Holidays and observances

 * Children's Day (Cameroon, Central African Republic, Chad, Equatorial Guinea, Democratic Republic of the Congo, Gabon, Republic of Congo)
 * Christian feast day:
 * Anastasia of Sirmium (Catholic Church)
 * Stephen (Armenian Apostolic Church)
 * December 25 (Eastern Orthodox liturgics)
 * Christmas Day, Christian festival commemorating the birth of Jesus. (Internationally observed)
 * Tulsi Pujan Diwas (India)
 * Constitution Day (Taiwan)
 * Good Governance Day (India)
 * Quaid-e-Azam's Day (Pakistan)
 * Takanakuy (Chumbivilcas Province, Peru)

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: December 25