# Science / Origins of mathematical notation

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## Can you name the first mathematicians to use the given notation?

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NotationOriginatorEarliest known usage
Vertical bars for the absolute value of x (|x|)Zur Theorie der Potenzreihen, 1841
Bra-ket notation (⟨| , |⟩) in physicsA New Notation for Quantum Mechanics, 1939
Elongated 'S' (∫) for integralsAnalyseos tetragonisticae pars secunda, 1675
Vertical bars around the entries of a matrix for its determinant Cambridge Mathematical Journal, Vol. II, p. 267-271, 1841
Exclamation mark (!) for factorialÉléments d'arithmétique universelle, 1808
Curly d (∂) for partial derivativesMemoire sur la manière de distinguer les maxima des minima dans le Calcul des Variations, 1786
Square 'tombstone' symbol (∎/□) for the end of a proof20th century
Greek letter pi (π) for the constant 3.1415...Synopsis palmariorum mathesios, 1706
Less than and greater than symbolsArtis Analyticae Praxis ad Aequationes Algebraicas Resolvenda, published 1631
f(x) for a function f of a variable xCommentarii Academiae Scientiarum Petropolitanae, 1734
Prime (') notation for total derivativesThéorie des fonctions analytiques, 1797
Addition (+) and subtraction (-) signsBehende und hüpsche Rechenung auff allen Kauffmanschafft, 1489
Equals sign (=)The Whetstone of Witte, 1557
NotationOriginatorEarliest known usage
Perpendicularity sign (⊥)Cursus mathematicus, 1634
Obelus (÷) for division, therefore symbol (∴)Teutsche Algebra, 1659
Multiplication symbol (×); plus-minus signClavis Mathematicae, published 1631
Curly braces ({, }) for enclosing elements of setsEin Beitrag zur Mannifaltigkeitslehre, 1878
Three horizontal lines (≡) for congruence of integersDisquisitiones arithmeticae, 1801
Nabla symbol (∇) for vectorial derivatives, etc.19th century
Stylized Greek letter epsilon (∈) for 'is an element of'Arithmetices prinicipia nova methodo exposita, 1899
Infinity symbol (∞)De sectionibus conicis, 1655
Empty set symbol (∅)Éléments de mathématique, 1939
Square root/radical symbol (√)Die Coss, 1525
Universal quantifier symbol (∀)Untersuchungen ueber das logische Schliessen, 1935
Big-O notation for asymptotic boundsAnalytische Zahlentheorie, 1894

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