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Mathematical formula

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Mathematical formula

For other uses, see Formula (disambiguation).

In science, a formula is a concise way of expressing information symbolically as in a mathematical or chemical formula. The informal use of the term formula in science refers to the general construct of a relationship between given quantities. The plural of formula can be spelled either as formulas or formulae (from the original Latin).[1]

In mathematics, a formula is an entity constructed using the symbols and formation rules of a given logical language. For example, determining the volume of a sphere requires a significant amount of integral calculus or its geometrical analogue, the method of exhaustion; but, 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{4}{3} \pi r^3.

Having obtained this result, and knowing the radius of the sphere in question, we can quickly and easily determine its volume. Note that the volume V and the radius r are expressed as single letters instead of words or phrases. This convention, while less important in a relatively simple formula, means that mathematicians can more quickly manipulate larger and more complex formulae. Mathematical formulae are often algebraic, closed form, and/or analytical.

In modern chemistry, a chemical formula is a way of expressing information about the proportions of atoms that constitute a particular chemical compound, using a single line of chemical element symbols, numbers, and sometimes other symbols, such as parentheses, brackets, and plus (+) and minus (−) signs. For example, H2O is the chemical formula for water, specifying that each molecule consists of two hydrogen (H) atoms and one oxygen (O) atom. Similarly, O
3
denotes an ozone molecule consisting of three oxygen atoms and having a net negative charge.

In a general context, formulas 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, is applicable to a wide range of physical situations. Other formulas 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, formulas form the basis for all calculations.

Expressions are distinct from formulas in that they cannot contain an equals sign (=). Whereas formulas are comparable to sentences, expressions are more like phrases.

In computing

In computing, a formula typically describes a calculation, such as addition, to be performed on one or more variables. A formula is often implicitly provided in the form of a computer instruction such as.

Degrees Celsius = (5/9)*(Degrees Fahrenheit -32)

In computer spreadsheet 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 stating its name would be redundant.

Formulas 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 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

V = \frac{4}{3}\pi(2.0 \mbox{ cm})^3 = 33.47 \mbox{ cm}^{3}.

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 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 V \equiv \mathrm{VOL}~\bold{tbsp}, 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{RAD}~\bold{cm}. The derivation of the formula proceeds as:

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

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

\mathrm{VOL} = 0.2933~\mathrm{RAD}^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.

See also

References

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