Definitions

# Formula

[fawr-myuh-luh]

In mathematics and in the sciences, a formula (plural: formulae, formulæ or formulas) is a concise way of expressing information symbolically (as in a mathematical or chemical formula), or a general relationship between quantities. One of many famous formulae is Albert Einstein's E = mc² (see special relativity).

## In mathematics

In mathematics, a formula is a key to solve an equation with variables. For example, the problem of determining the volume of a sphere is one that requires a significant amount of integral calculus to solve. However, having done this once, mathematicians can produce a formula to describe the volume in terms of some other parameter (the radius for example). This particular formula is:

$V =frac\left\{4\right\}\left\{3\right\} pi r^3.$

Having determined this result, and having a sphere of which we know the radius we can quickly and easily determine the 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.

In general mathematical use there is no essential difference in meaning with the term "expression", although the word "formula" tends to be reserved for an expression that "can stand on its own", that has a meaning outside of the immediate context in which it appears and a significance that can be grasped intuitively.

The majority of all mathematical study revolves around formulae in many different forms from quadratic equations to the equations of motion (mainly used in mechanical mathematics and physics). In a general context, formulae are applied to provide a mathematical solution for real world problems. Some may be general formulae designed to explain a phenomenon experienced everywhere - an example is force = mass × acceleration. It is a formula which applies anywhere in the universe. 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, a formula typically describes a calculation, such as addition, to be performed on two or more variables. A formula is often implicitly provided in the form of a computer instruction such as

Total fruit = number of Apples + number of Oranges.

In computer spreadsheet terminology, a formula is usually a text string containing cell references, e.g.

=A1+A2

where both A1 and A2 describe "cells" (column A, row 1 or 2) within the spreadsheet. The result appears within the cell containing the formula itself (possibly A3, at end of values in column A). The = sign precedes the right hand side of the formula indicating the cell contains a formula rather than data. The left hand side of the formula is, by convention, omitted because the result is always stored in the cell itself and would be redundant.

## Formula with prescribed units

A physical quantity can be expressed as the product of a number and a physical unit. A formula expresses a relationship between physical quantities. A necessary condition for a formula to be valid it 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

$V = 33.51~bold\left\{cm\right\}^\left\{3\right\}.$

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.

However, the vast majority of computations with measurements is 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 $V equiv mathrm\left\{VOL\right\}~bold\left\{tbsp\right\}$, where tbsp is the U.S. tablespoon (as seen in conversion of units) and VOL is the name for the number used by the computer. Similarly, the formula is to require $r equiv mathrm\left\{RAD\right\}~bold\left\{cm\right\}$. The derivation of the formula proceeds as:

$mathrm\left\{VOL\right\}~bold\left\{tbsp\right\} = frac\left\{4\right\}\left\{3\right\} pi mathrm\left\{RAD\right\}^3~ bold\left\{cm\right\}^3.$

Given that $1~bold\left\{tbsp\right\} = 14.787~bold\left\{cm\right\}^3$, the formula with prescribed units is

$mathrm\left\{VOL\right\} = 0.2833~mathrm\left\{RAD\right\}^3.$

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

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

$V = 0.2833~r^3.$
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.

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