Formula#In mathematics

In mathematics, a formula generally refers to an equation or inequality relating one mathematical expression to another, with the most important ones being mathematical theorems. For example, determining the volume of a sphere requires a significant amount of integral calculus or its geometrical analogue, the method of exhaustion.{{cite book

| last = Smith

| first = David E.

| author-link = David Eugene Smith

| year = 1958

| title = History of Mathematics

| publisher = Dover Publications

| location = New York

| isbn = 0-486-20430-8

}} However, having done this once in terms of some parameter (the radius for example), mathematicians have produced a formula to describe the volume of a sphere in terms of its radius:

: $V\; =\; \backslash frac\{4\}\{3\}\; \backslash pi\; r^3.$

Having obtained this result, the volume of any sphere can be computed as long as its radius is known. Here, notice 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 formulas which are larger and more complex.{{cite web

|url = https://math.stackexchange.com/q/24241

|title = Why do mathematicians use single letter variables?

|date = 28 February 2011

|website = math.stackexchange.com

|access-date = 31 December 2013

}} Mathematical formulas are often algebraic, analytical or in closed form.{{cite web

|url = https://www.andlearning.org/math-formula/

|title = List of Mathematical formulas

|date = 24 August 2018

|website = andlearning.org

}}

In a general context, formulas often represent mathematical models of real world phenomena, and as such can be used to provide solutions (or approximate solutions) to real world problems, with some being more general than others. For example, the formula

: $F\; =\; ma$

is an expression of Newton's second law, and is applicable to a wide range of physical situations. Other formulas, such as the use of the equation of a sine curve to model the movement of the tides in a bay, may be created to solve a particular problem. In all cases, however, formulas form the basis for calculations.

Expressions are distinct from formulas in the sense that they don't usually contain relations like equality (=) or inequality (<). Expressions denote a mathematical object, where as formulas denote a statement about mathematical objects.{{cite book|first=Robert R.|last=Stoll|year=1963|title=Set Theory and Logic|publisher=Dover Publications|location=San Francisco, CA|isbn=978-0-486-63829-4}}{{Citation

|last1=Hamilton

|first1=A. G.

|title=Logic for Mathematicians

|publisher=Cambridge University Press

|location=Cambridge

|edition=2nd

|isbn=978-0-521-36865-0

|year=1988}}{{dubious|Expressions containing equals signs|date=May 2023}} This is analogous to natural language, where a noun phrase refers to an object, and a whole sentence refers to a fact. For example, $8x-5$ is an expression, while $8x-5\; \backslash geq\; 3$ is a formula.

However, in some areas mathematics, and in particular in computer algebra, formulas are viewed as expressions that can be evaluated to true or false, depending on the values that are given to the variables occurring in the expressions. For example $8x-5\; \backslash geq\; 3$ takes the value false if {{mvar|x}} is given a value less than 1, and the value true otherwise. (See Boolean expression)

In mathematical logic, a formula (often referred to as a well-formed formula) is an entity constructed using the symbols and formation rules of a given logical language.{{Citation

|last=Rautenberg

|first=Wolfgang

|author-link=Wolfgang Rautenberg

|doi=10.1007/978-1-4419-1221-3

|title=A Concise Introduction to Mathematical Logic

|publisher=Springer Science+Business Media

|location=New York, NY

|edition=3rd

|isbn=978-1-4419-1220-6

|year=2010

}} For example, in first-order logic,

:$\backslash forall\; x\; \backslash forall\; y\; (P(f(x))\; \backslash rightarrow\backslash neg\; (P(x)\; \backslash rightarrow\; Q(f(y),x,z)))$

is a formula, provided that $f$ is a unary function symbol, $P$ a unary predicate symbol, and $Q$ a ternary predicate symbol.

{{main|Chemical formula}}

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.Atkins, P.W., Overton, T., Rourke, J., Weller, M. and Armstrong, F. Shriver and Atkins inorganic chemistry (4th edition) 2006 (Oxford University Press) {{isbn|0-19-926463-5}} 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{{sup sub|−|3}} denotes an ozone molecule consisting of three oxygen atoms{{Cite web|url=http://www.chm.bris.ac.uk/motm/ozone/CHEM.htm|title=Ozone Chemistry|website=www.chm.bris.ac.uk|access-date=2019-11-26}} and a net negative charge.

{{Image frame|width=300|content=H-\overset{\displaystyle H \atop |}{\underset

| the empirical formula C₂H₅

| the molecular formula C₄H₁₀ and

| the condensed formula (or semi-structural formula) CH₃CH₂CH₂CH₃.}}

|alt=A diagram showing one line of four connected Cs (carbon atoms) branching out to ten Hs (hydrogen atoms)}}

A chemical formula identifies each constituent element by its chemical symbol, and indicates the proportionate number of atoms of each element.

In empirical formulas, these proportions begin with a key element and then assign numbers of atoms of the other elements in the compound—as ratios to the key element. For molecular compounds, these ratio numbers can always be expressed as whole numbers. For example, the empirical formula of ethanol may be written C₂H₆O,{{Cite web|url=https://pubchem.ncbi.nlm.nih.gov/compound/702|title=Ethanol|last=PubChem|website=pubchem.ncbi.nlm.nih.gov|language=en|access-date=2019-11-26}} because the molecules of ethanol all contain two carbon atoms, six hydrogen atoms, and one oxygen atom. Some types of ionic compounds, however, cannot be written as empirical formulas which contains only the whole numbers. An example is boron carbide, whose formula of CB_n is a variable non-whole number ratio, with n ranging from over 4 to more than 6.5.

When the chemical compound of the formula consists of simple molecules, chemical formulas often employ ways to suggest the structure of the molecule. There are several types of these formulas, including molecular formulas and condensed formulas. A molecular formula enumerates the number of atoms to reflect those in the molecule, so that the molecular formula for glucose is C₆H₁₂O₆ rather than the glucose empirical formula, which is CH₂O. Except for the very simple substances, molecular chemical formulas generally lack needed structural information, and might even be ambiguous in occasions.

A structural formula is a drawing that shows the location of each atom, and which atoms it binds to.

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, could be written 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, making the stating of the name redundant.

Formulas used in science almost always require a choice of units.{{Cite book | isbn = 978-1466571143 | title = CRC Handbook of Chemistry and Physics, 94th Edition | editor1-last = Haynes | editor1-first= William M. | year = 2013 | orig-year = 1914 | publisher = CRC Press | location = Boca Raton }} Formulas are used to express relationships between various quantities, such as temperature, mass, or charge in physics; supply, profit, or demand in economics; or a wide range of other quantities in other disciplines.

An example of a formula used in science is Boltzmann's entropy formula. In statistical thermodynamics, it is a probability equation relating the entropy S of an ideal gas to the quantity W, which is the number of microstates corresponding to a given macrostate:

: $S\; =\; k\; \backslash cdot\; \backslash ln\; W$

where k is the Boltzmann constant, equal to {{physconst|k|ref=no}}, and W is the number of microstates consistent with the given macrostate.

• Formula editor
• Formula unit
• Law (mathematics)
• Mathematical notation
• Scientific law
• Chemical symbol
• Theorem
• Well-formed formula

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Category:Mathematical notation

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