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 "excerpt": "A formula is an expression in mathematical symbols stating a fact, rule, or relationship among quantities; in logic it is a well-formed expression denoting a truth value.",
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 "markdown": "# Formula\n\nA formula is an expression written in mathematical or logical symbols that states a fact, rule, or principle, most commonly a relationship among quantities such as constants, variables, and operators.<sup>[1](https://mathworld.wolfram.com/Formula.html)</sup> In mathematical logic the word has a stricter meaning: a syntactically correct expression of a formalized language, built over a signature (the formal list of sorts, function and predicate symbols defining a logic language), a set of variables, and a logic, which represents a truth value rather than an ordinary value.<sup>[2](https://encyclopediaofmath.org/wiki/Formula)</sup> In everyday mathematical practice the term is looser. A formula usually takes the form of an algebraic statement with an equality or inequality, containing constants, mathematical operators, and variables or parameters written as letters.<sup>[3](https://www.treccani.it/enciclopedia/formula_(Enciclopedia-della-Matematica)/)</sup>\n\n| Key fact | Detail |\n|---|---|\n| General meaning | A fact, rule, or principle expressed in mathematical symbols; examples include equations, equalities, identities, inequalities, and asymptotic expressions<sup>[1](https://mathworld.wolfram.com/Formula.html)</sup> |\n| Logical meaning | A well-formed expression of a formalized language over a signature, variables, and a logic, representing truth values<sup>[2](https://encyclopediaofmath.org/wiki/Formula)</sup> |\n| Typical parts | Constants, operators, variables, and parameters; the variable isolated on the left of an equality is called the subject<sup>[3](https://www.treccani.it/enciclopedia/formula_(Enciclopedia-della-Matematica)/)</sup><sup> • </sup><sup>[4](https://www.amsi.org.au/teacher_modules/Formulas.html)</sup> |\n| First systematic algebraic notation | François Viète, *In artem analyticam isagoge* (Tours, 1591), vowels for unknowns, consonants for knowns<sup>[5](https://mathshistory.st-andrews.ac.uk//Biographies/Viete/)</sup> |\n| Modern letter convention | Descartes, *La Géométrie* (1637): late-alphabet letters for unknowns, early-alphabet letters for knowns; the equals sign dates to Recorde, 1557<sup>[6](https://mathshistory.st-andrews.ac.uk/HistTopics/History_overview/)</sup> |\n| Physical constraint | Every equation expressing a physical law must be dimensionally consistent: all terms share dimensions and function arguments are dimensionless<sup>[7](https://pressbooks.online.ucf.edu/osuniversityphysics/chapter/1-4-dimensional-analysis/)</sup> |\n| Machine verification | NIST's LaCASt translator converts DLMF formulae to Maple and Mathematica for automated checking; one test subset covered 4,078 semantic LaTeX formulae<sup>[8](https://www.nist.gov/publications/comparative-verification-validation-digital-mathematical-libraries-and-computer-algebra)</sup> |\n\n## Formulas in logic\n\nIn formal logic a formula is defined by its syntax alone. Given a signature \\( \\Sigma = (S, F) \\) and a set \\( P \\) of predicate symbols whose range is the set \\( \\mathbb{B} \\) of truth values of the underlying logic, formulas are the expressions generated inductively from these ingredients. They resemble terms, which denote sort values, but because predicate and logic symbols enter their definition, formulas denote truth values instead.<sup>[2](https://encyclopediaofmath.org/wiki/Formula)</sup>\n\nThe inductive definition is explicit in predicate calculus. Any atomic formula is a formula, and if \\( A \\) and \\( B \\) are formulas and \\( y \\) is an object variable, then \\( (\\neg A) \\), \\( (A \\mathbin\\& B) \\), \\( (A \\lor B) \\), \\( (A \\supset B) \\), \\( (\\forall y\\ A) \\), and \\( (\\exists y\\ A) \\) are formulas.<sup>[9](https://encyclopediaofmath.org/wiki/Logical_formula)</sup> Exact definitions differ from one formalized language to another; the Encyclopedia of Mathematics points to axiomatic set theory, formal arithmetic, predicate calculus, and type theory as settings where the concept is pinned down precisely.<sup>[2](https://encyclopediaofmath.org/wiki/Formula)</sup>\n\n## Formulas in mathematical practice\n\nOutside logic, the word covers several related kinds of symbolic statement. MathWorld defines a formula as a fact, rule, or principle expressed in mathematical symbols, with equations, equalities, identities, inequalities, and asymptotic expressions as examples.<sup>[1](https://mathworld.wolfram.com/Formula.html)</sup> The distinction among these rests on how the statement behaves under substitution. An equation true for all values of its variables is an identity, such as \\( (a+b)^{2} = a^{2} + 2ab + b^{2} \\); an equation true only for particular values is one to be solved, such as \\( 3x + 1 = 4 \\), whose solution is \\( x = 1 \\).<sup>[10](https://www.math.utah.edu/online/1010/algebra/)</sup>\n\n**The subject of a formula.** School and applied mathematics usually present a formula with a single variable on the left of the equal sign, called the subject. In \\( E = mc^{2} \\) the subject is \\( E \\); in \\( A = \\pi r^{2} \\) it is \\( A \\). Given values for every other variable, the subject's value follows by substitution. [Euler's formula](https://www.edgepedia.org/eulers-formula) \\( V - E + F = 2 \\), which connects the vertices, edges, and faces of a convex polyhedron, has no subject as written, but any of its three variables can be made the subject by rearrangement; for a cube, \\( E = 12 \\), \\( V = 8 \\), and \\( F = 6 \\).<sup>[4](https://www.amsi.org.au/teacher_modules/Formulas.html)</sup>\n\nFormulas serve two practical purposes: computing one quantity from the others, and showing how one quantity changes as the others change. In \\( A = \\pi r^{2} \\) the area grows as the square of the radius, while in Newton's law \\( F = G\\, m_{1} m_{2} / r^{2} \\) the force decreases as the distance increases, an inverse-square law.<sup>[4](https://www.amsi.org.au/teacher_modules/Formulas.html)</sup>\n\n## How formula notation developed\n\nAlgebraic writing passed through three stages: rhetorical, in which problems were stated entirely in prose; syncopated, using shorthand abbreviations alongside words; and symbolic. Algebra itself began in [Babylonia](https://www.edgepedia.org/babylonia) and Egypt around 1700 BC, the Greek mathematician [Diophantus](https://www.edgepedia.org/diophantus) was a pioneer of the syncopated stage, and Indian mathematicians developed a syncopated notation independently.<sup>[11](https://onlinelibrary.wiley.com/doi/10.1111/j.1949-8594.2000.tb17262.x)</sup> Indian workers also introduced symbols for several unknowns, for the square, the square root, and the subtrahend; Brahmagupta's seventh-century notation wrote \\( 3x^{2} + 10x - 8 = x^{2} + 1 \\) in an entirely different form from today's.<sup>[12](https://encyclopediaofmath.org/wiki/Mathematical_symbols)</sup>\n\n**Viète and Descartes.** The first systematic algebraic notation appeared in François Viète's *In artem analyticam isagoge*, published at Tours in 1591. Viète used letters as symbols for quantities, both known and unknown, taking vowels for unknowns and consonants for knowns. He was still bound by a condition of homogeneity of dimension: an equation such as \\( x^{3} + x = 1 \\) made no geometric sense to him, because a cube cannot be added to a line, so he wrote equations like \\( A^{3} + B^{2}A = B^{2}Z \\), in which every term has dimension three.<sup>[5](https://mathshistory.st-andrews.ac.uk//Biographies/Viete/)</sup> The modern convention, in which letters near the end of the alphabet denote unknowns and letters near the beginning denote knowns, was introduced by [René Descartes](https://www.edgepedia.org/rene-descartes) in *La Géométrie* in 1637; Viète's convention fell out of use.<sup>[6](https://mathshistory.st-andrews.ac.uk/HistTopics/History_overview/)</sup> Other familiar pieces of formula notation arrived piecemeal: the equals sign from Robert Recorde in 1557, juxtaposition \\( ax \\) for a product, and modern exponents from Descartes in 1637, with James Hume's 1636 edition of Viète placing raised Roman-numeral exponents in nearly the modern position.<sup>[6](https://mathshistory.st-andrews.ac.uk/HistTopics/History_overview/)</sup><sup> • </sup><sup>[13](https://mathshistory.st-andrews.ac.uk//Miller/mathsym/operation/)</sup> The summation symbol \\( \\Sigma \\) was first used by [Leonhard Euler](https://www.edgepedia.org/leonhard-euler) in 1755 in *Institutiones calculi differentialis*.<sup>[13](https://mathshistory.st-andrews.ac.uk//Miller/mathsym/operation/)</sup>\n\nBy the end of the sixteenth century, with Viète, and especially during the first half of the seventeenth century with Harriot, Fermat, and Descartes, mathematicians began treating algebra symbolically, and through the seventeenth and eighteenth centuries they increasingly used formulas to represent quantitative dynamic relationships across the sciences.<sup>[14](https://old.maa.org/press/maa-reviews/taming-the-unknown-a-history-of-algebra-from-antiquity-to-the-early-twentieth-century)</sup>\n\n## Formulas in science and dimensional consistency\n\nA formula that expresses a physical relationship must satisfy a structural constraint beyond its algebra. Each term in the equation must have the same dimensions as the other terms, and the arguments of any standard mathematical functions appearing in it must be dimensionless. This property, dimensional consistency, holds for every equation expressing a physical law or principle, and it gives a practical tool: it checks whether a remembered or derived formula can be correct, and it can even suggest plausible forms when the quantities involved are known.<sup>[7](https://pressbooks.online.ucf.edu/osuniversityphysics/chapter/1-4-dimensional-analysis/)</sup>\n\nThe check works term by term. For \\( s = vt + 0.5at^{2} \\), with \\( [v] = \\mathrm{LT}^{-1} \\) and \\( [a] = \\mathrm{LT}^{-2} \\), both right-hand terms reduce to length \\( \\mathrm{L} \\), matching \\( [s] \\), so the equation is consistent. For \\( s = vt^{2} + 0.5at \\) the terms come out as \\( \\mathrm{LT} \\) and \\( \\mathrm{LT}^{-1} \\), matching neither each other nor \\( [s] \\).<sup>[7](https://pressbooks.online.ucf.edu/osuniversityphysics/chapter/1-4-dimensional-analysis/)</sup> [Dimensional analysis](https://www.edgepedia.org/dimensional-analysis) has limits: an equation such as \\( x = at + bt^{2} \\) passes term by term, but the analysis cannot determine the coefficients or distinguish among terms carrying the same dimensions, so it cannot guess the full formula a priori.<sup>[15](https://physics.suparnpatra.com/classes/xi/ch02-units-and-measurements/limitations-of-dimensions)</sup> In modeling, dimensional homogeneity has a further consequence: a dimensionally homogeneous equation is independent of the units of measurement used.<sup>[16](https://ku.ac.bd/discipline/uploads/offered-course-material/%5BDym_C.L.%5D_Principles_of_mathematical_modeling.pdf)</sup>\n\n## Verification and digital formula collections\n\nLarge reference collections gather formulas in machine-readable form. The NIST Digital Library of Mathematical Functions (DLMF), the web companion to the NIST Handbook of Mathematical Functions, provides a reference tool for researchers in applied mathematics, the physical sciences, and engineering who encounter special functions.<sup>[17](https://dlmf.nist.gov/front/introduction)</sup> Because a single typographical error can invalidate a published formula, correctness is treated as a measurable property. NIST's LaCASt translator converts DLMF formulae written in semantic LaTeX into the languages of the computer algebra systems [Maple](https://www.edgepedia.org/maple-software) and Mathematica, enabling automated verification and computation of the formulae; one test data subset comprised 4,078 semantic LaTeX DLMF formulae.<sup>[8](https://www.nist.gov/publications/comparative-verification-validation-digital-mathematical-libraries-and-computer-algebra)</sup><sup> • </sup><sup>[18](https://arxiv.org/abs/2109.08899)</sup> This work treats the correctness, accuracy, and reliability of both digital mathematical libraries and computer algebra systems as crucial attributes of modern research, which depends on both.<sup>[8](https://www.nist.gov/publications/comparative-verification-validation-digital-mathematical-libraries-and-computer-algebra)</sup>\n\nThe Latin plural of formula is *formulae*, though the anglicized *formulas* is now more common in English usage.<sup>[1](https://mathworld.wolfram.com/Formula.html)</sup>\n\n## References\n\n1. [Formula, Wolfram MathWorld](https://mathworld.wolfram.com/Formula.html)\n2. [Formula, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Formula)\n3. [Formula, Enciclopedia della Matematica, Treccani](https://www.treccani.it/enciclopedia/formula_(Enciclopedia-della-Matematica)/)\n4. [Formulas, The Improving Mathematics Education in Schools (TIMES) Project, AMSI](https://www.amsi.org.au/teacher_modules/Formulas.html)\n5. [François Viète (1540–1603), MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk//Biographies/Viete/)\n6. [History overview, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/HistTopics/History_overview/)\n7. [1.4 Dimensional Analysis, University Physics Volume 1, OpenStax](https://pressbooks.online.ucf.edu/osuniversityphysics/chapter/1-4-dimensional-analysis/)\n8. [Comparative Verification & Validation of Digital Mathematical Libraries and Computer Algebra Systems, NIST](https://www.nist.gov/publications/comparative-verification-validation-digital-mathematical-libraries-and-computer-algebra)\n9. [Logical formula, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Logical_formula)\n10. [What is Algebra, Math 1010 Online, University of Utah](https://www.math.utah.edu/online/1010/algebra/)\n11. [Lynn Stallings (2000), A Brief History of Algebraic Notation, School Science and Mathematics 100: 230–235](https://onlinelibrary.wiley.com/doi/10.1111/j.1949-8594.2000.tb17262.x)\n12. [Mathematical symbols, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Mathematical_symbols)\n13. [Earliest Uses of Symbols of Operation, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk//Miller/mathsym/operation/)\n14. [Taming the Unknown: A History of Algebra from Antiquity to the Early Twentieth Century, MAA Reviews](https://old.maa.org/press/maa-reviews/taming-the-unknown-a-history-of-algebra-from-antiquity-to-the-early-twentieth-century)\n15. [Limitations of Dimensional Analysis, Physics Lab](https://physics.suparnpatra.com/classes/xi/ch02-units-and-measurements/limitations-of-dimensions)\n16. [C. L. Dym, Principles of Mathematical Modeling](https://ku.ac.bd/discipline/uploads/offered-course-material/%5BDym_C.L.%5D_Principles_of_mathematical_modeling.pdf)\n17. [DLMF: Mathematical Introduction, NIST Digital Library of Mathematical Functions](https://dlmf.nist.gov/front/introduction)\n18. [Automated Symbolic and Numerical Testing of DLMF Formulae using Computer Algebra Systems, arXiv:2109.08899](https://arxiv.org/abs/2109.08899)\n\n---\n*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics*\n\n*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*\n\n*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*\n\nLicense: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license\n",
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