Formula

# Formula

Discussion

Encyclopedia
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a formula (plural: formulae or formulas) is an entity constructed using the symbols and formation rules of a given logical language.

In science
Science
Science is a systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe...

, a formula is a concise way of expressing information symbolically (as in a mathematical or chemical formula
Chemical formula
A chemical formula or molecular formula is a way of expressing information about the atoms that constitute a particular chemical compound....

), or a general relationship between quantities. Colloquial use of the term in mathematics often refers to a similar construct.

Such formulae are the key to solving an equation
Equation
An equation is a mathematical statement that asserts the equality of two expressions. In modern notation, this is written by placing the expressions on either side of an equals sign , for examplex + 3 = 5\,asserts that x+3 is equal to 5...

with variables. For example, determining the volume
Volume
Volume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains....

of a sphere
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...

requires a significant amount of integral calculus; but, having done this once, mathematicians can produce a formula to describe the volume in terms of some other parameter (the radius
In classical geometry, a radius of a circle or sphere is any line segment from its center to its perimeter. By extension, the radius of a circle or sphere is the length of any such segment, which is half the diameter. If the object does not have an obvious center, the term may refer to its...

for example). This particular formula is:
Having obtained this result, and knowing the radius of the sphere in question, we can quickly and easily determine its volume. Note that the quantities V, the volume, and r the radius are expressed as single letters. This convention, while less important in a relatively simple formula, means that mathematicians can more quickly manipulate larger and more complex formulae.

Expression
Expression (mathematics)
In mathematics, an expression is a finite combination of symbols that is well-formed according to rules that depend on the context. Symbols can designate numbers , variables, operations, functions, and other mathematical symbols, as well as punctuation, symbols of grouping, and other syntactic...

s are distinct from formulae in that they cannot contain an equals sign; whereas formulae are comparable to sentences, expressions are more like phrases.

In a general context, formulae are applied to provide a mathematical solution for real world problems. Some may be general: F = ma, which is one expression of Newton's second law
Newton's laws of motion
Newton's laws of motion are three physical laws that form the basis for classical mechanics. They describe the relationship between the forces acting on a body and its motion due to those forces...

, is applicable to a wide range of physical situations. Other formulae may be specially created to solve a particular problem; for example, using the equation of a sine curve to model the movement of the tides in a bay. In all cases however, formulae form the basis for all calculations.

## In computing

In computing
Computing
Computing is usually defined as the activity of using and improving computer hardware and software. It is the computer-specific part of information technology...

, a formula typically describes a calculation
Calculation
A calculation is a deliberate process for transforming one or more inputs into one or more results, with variable change.The term is used in a variety of senses, from the very definite arithmetical calculation of using an algorithm to the vague heuristics of calculating a strategy in a competition...

, such as addition, to be performed on one or more variables. A formula is often implicitly provided in the form of a computer
Computer
A computer is a programmable machine designed to sequentially and automatically carry out a sequence of arithmetic or logical operations. The particular sequence of operations can be changed readily, allowing the computer to solve more than one kind of problem...

instruction such as.
Degrees Celsius = (5/9)*(Degrees Fahrenheit -32)

A spreadsheet is a computer application that simulates a paper accounting worksheet. It displays multiple cells usually in a two-dimensional matrix or grid consisting of rows and columns. Each cell contains alphanumeric text, numeric values or formulas...

software, a formula indicating how to compute the value of a cell, say A3, is written such as
=A1+A2

where A1 and A2 refer to other cells (column A, row 1 or 2) within the spreadsheet. This is a shortcut for the "paper" form A3 = A1+A2 where A3 is, by convention, omitted because the result is always stored in the cell itself and would be redundant.

## Formulae with prescribed units

A physical quantity
Physical quantity
A physical quantity is a physical property of a phenomenon, body, or substance, that can be quantified by measurement.-Definition of a physical quantity:Formally, the International Vocabulary of Metrology, 3rd edition defines quantity as:...

can be expressed as the product of a number and a physical unit
Units of measurement
A unit of measurement is a definite magnitude of a physical quantity, defined and adopted by convention and/or by law, that is used as a standard for measurement of the same physical quantity. Any other value of the physical quantity can be expressed as a simple multiple of the unit of...

. A formula expresses a relationship between physical quantities. A necessary condition for a formula to be valid is that all terms have the same dimension, meaning every term in the formula could be potentially converted to contain the identical unit (or product of identical units).

In the example above, for the volume of a sphere, we may wish to compute with r = 2.0 cm, which yields

There is vast educational training about retaining units in computations, and converting units to a desirable form, such as in units conversion by factor-label
Units conversion by factor-label
Many, if not most, parameters and measurements in the physical sciences and engineering are expressed as a numerical quantity and a corresponding dimensional unit; for example: 1000 kg/m³, 100 kPa/bar, 50 miles per hour, 1000 Btu/lb. Converting from one dimensional unit to another is often...

.

However, the vast majority of computations with measurements are done in computer programs with no facility for retaining a symbolic computation of the units. Only the numerical quantity is used in the computation. This requires that the universal formula be converted to a formula that is intended to be used only with prescribed units, meaning the numerical quantity is implicitly assumed to be multiplying a particular unit. The requirements about the prescribed units must be given to users of the input and the output of the formula.

For example suppose the formula is to require that , where tbsp is the U.S. tablespoon (as seen in conversion of units
Conversion of units
Conversion of units is the conversion between different units of measurement for the same quantity, typically through multiplicative conversion factors.- Process :...

) and VOL is the name for the number used by the computer. Similarly, the formula is to require
. The derivation of the formula proceeds as:

Given that ,
the formula with prescribed units is

The formula is not complete without words such as:
Other possible words are "VOL is the ratio of to tbsp and RAD is the ratio of to cm."

The formula with prescribed units could also appear with simple symbols,
perhaps even the identical symbols as in the original dimensional formula:

and the accompanying words could be: "where V is volume (tbsp) and r is radius (cm)".

If the physical formula is not dimensionally homogeneous, and therefore erroneous,
the falsehood becomes apparent in the impossibility
to derive a formula with prescribed units. It would not be possible to
derive a formula consisting only of numbers and dimensionless ratios.