
Appendix D: Some Common Mathematical Symbols and Abbreviations

This Appendix contains a fairly long list of common mathematical symbols as well as a list of some common Latin abbreviations and phrases.  While you will not necessarily need all of the included symbols for your study of Linear Algebra, this list will hopefully nonetheless give you an idea of where much of our modern mathematical notation comes from.

Binary Relations

• $$=$$ (the equals sign) means "is the same as'' and was first introduced in the 1557 book The Whetstone of Witte by physician and mathematician Robert Recorde (c.~1510--1558).  He wrote, "I will sette as I doe often in woorke use, a paire of parralles, or Gemowe lines of one lengthe, thus: $$=\kern-1.75pt=\kern-1.75pt=\kern-1.75pt=\kern-1.75pt=$$, bicause noe 2 thynges can be moare equalle.''  (Recorde's equals sign was significantly longer than the one in modern usage and is based upon the idea of "Gemowe'' or "identical'' lines, where "Gemowe'' means "twin'' and comes from the same root as the name of the constellation "Gemini''.)
• Robert Recorde also introduced the plus sign, "$$+$$", and the minus sign, "$$-$$", in The Whetstone of Witte.
• $$<$$ (the less than sign) means "is strictly less than'', and $$>$$ (the greater than sign) means "is strictly greater than''. These first appeared in the book Artis Analyticae Praxis ad Aequationes Algebraicas Resolvendas ("The Analytical Arts Applied to Solving Algebraic Equations'') by mathematician and astronomer Thomas Harriot (1560--1621), which was published posthumously in 1631.
• Pierre Bouguer (1698--1758) later refined these to $$\leq$$ ("is less than or equals'') and $$\geq$$ ("is greater than or equals'') in 1734.  Bouger is sometimes called "the father of naval architecture'' due to his foundational work in the theory of naval navigation.
• $$:=$$ (the equal by definition sign) means "is equal by definition to''.  This is a common alternate form of the symbol "$$=_{\mathrm{\scriptscriptstyle Def}}$$", the latter having first appeared in the 1894 book Logica Matematica by logician Cesare Burali-Forti (1861--1931).  Other common alternate forms of the symbol "$$=_{\mathrm{\scriptscriptstyle Def}}$$" include "$$\stackrel{\mathrm{\scriptscriptstyle def}}{=}$$" and "$$\equiv$$", with "$$\equiv$$" being especially common in applied mathematics.
• $$\approx$$ (the approximately equals sign) means "is approximately equal to'' and was first introduced in the 1892 book Applications of Elliptic Functions by mathematician Alfred Greenhill (1847--1927).

Other modern symbols for "approximately equals'' include "$$\doteq$$" (read as "is nearly equal to''), "$$\cong$$" (read as "is congruent to''), "$$\simeq$$" (read as "is similar to''), "$$\asymp$$" (read as "is asymptotically equal to''), and "$$\propto$$" (read as "is proportional to'').  Usage varies, and these are sometimes used to denote varying degrees of "approximate equality'' within a given context.

Some Symbols from Mathematical Logic

• $$\therefore$$ (three dots) means "therefore'' and first appeared in print in the 1659 book Teusche Algebra ("Teach Yourself Algebra'') by mathematician Johann Rahn (1622--1676). Teusche Algebra also contains the first use of the obelus, "$$\div$$", to denote division.
• $$\because$$ (upside-down dots) means "because'' and seems to have first appeared in the 1805 book The Gentleman's Mathematical Companion. However, it is much more common (and less ambiguous) to just abbreviate "because'' as "b/c''.
• $$\ni$$ (the such that sign) means "under the condition that'' and first appeared in the 1906 edition of Formulaire de mathematiques by the logician Giuseppe Peano (1858--1932).  However, it is much more common (and less ambiguous) to just abbreviate "such that'' as "s.t.''.

There are two good reasons to avoid using "$$\ni$$" in place of ''such that''.  First of all, the abbreviation ''s.t.'' is significantly more suggestive of its meaning than is "$$\ni$$''.  Perhaps more importantly, though, the symbol "$$\ni$$" is now commonly used to mean ''contains as an element'', which is a logical extension of the usage of the unquestionably standard symbol "$$\in$$" to mean "is contained as an element in''.

• $$\Rightarrow$$ (the implies sign) means "logically implies that'', and $$\Leftarrow$$ (the is implied by sign) means "is logically implied by''.  Both have an unclear historical origin.  (E.g., "if it's raining, then it's pouring'' is equivalent to saying "it's raining $$\Rightarrow$$ it's pouring.'')
• $$\iff$$ (the iff symbol) means "if and only if'' (abbreviated "iff'') and is used to connect two logically equivalent mathematical statements.  (E.g., "it's raining iff it's pouring'' means simultaneously that "if it's raining, then it's pouring'' and that "if it's pouring, then it's raining''.  In other words, the statement "it's raining $$\iff$$ it's pouring'' means simultaneously that "it's raining $$\Rightarrow$$ it's pouring'' and "it's raining $$\Leftarrow$$ it's pouring''.) The abbreviation "iff'' is attributed to the mathematician Paul Halmos (1916--2006).
• $$\forall$$ (the universal quantifier) means "for all'' and was first used in the 1935 publication Untersuchungen ueber das logische Schliessen ("Investigations on Logical Reasoning'') by logician Gerhard Gentzen (1909--1945).  He called it the All-Zeichen ("all character'') by analogy to the symbol "$$\exists$$", which means "there exists''.
• $$\exists$$ (the {existential quantifier) means "there exists'' and was first used in the 1897 edition of Formulaire de mathematiques by the logician Giuseppe Peano (1858--1932).
• $$\Box$$ (the Halmos tombstone} or Halmos symbol) means "Q.E.D.'', which is an abbreviation of the Latin phrase quod erat demonstrandum ("which was to be proven''). "Q.E.D.'' has been the most common way to symbolize the end of a logical argument for many centuries, but the modern convention of the "tombstone'' is now generally preferred both because it is easier to write and because it is visually more compact. The symbol "$$\Box$$" was first made popular by mathematician Paul Halmos (1916--2006).

Some Notation from Set Theory

• $$\subset$$ (the is included in sign) means "is a subset of'' and $$\supset$$ (the includes sign) means "has as a subset''.  Both symbols were introduced in the 1890 book Vorlesungen uber die Algebra der Logik  ("Lectures on the Algebra of the Logic'') by logician Ernst Schroder (1841--1902).
• $$\in$$ (the is in sign) means "is an element of'' and first appeared in the 1895 edition of Formulaire de mathematiques by the logician Giuseppe Peano (1858--1932). Peano originally used the Greek letter "$$\epsilon$$" (viz.~the first letter of the Latin word est for "is''). The modern stylized version of this symbol was later introduced in the 1903 book Principles of Mathematics by logician and philosopher Betrand Russell (1872--1970).

It is also common to use the symbol "$$\ni$$" to mean "contains as an element'', which is not to be confused with the more archaic usage of "$$\ni$$" to mean "such that''.

• $$\cup$$ (the union sign) means "take the elements that are in either set'', and $$\cap$$ (the intersection sign) means "take the elements that the two sets have in common''. These were both introduced in the 1888 book Calcolo geometrico second l'Ausdehnungslehre di H. Grassmann preceduto dale operazioni della logica deduttiva ("Geometric Calculus based upon the teachings of H. Grassman, preceded by the operations of deductive logic'') by logician Giuseppe Peano (1858--1932).
• $$\emptyset$$ (the null set or empty set) means "the set without any elements in it'' and was first used in the 1939 book Elements de mathematiques by Nicolas Bourbaki. (Bourbaki is the collective pseudonym for a group of primarily European mathematicians who have written many mathematics books together.)  It was borrowed simultaneously from the Norwegian, Danish and Faroese alphabets by group member Andre Weil (1906--1998).
• $$\infty$$ (infinity) denotes "a quantity or number of arbitrarily large magnitude'' and first appeared in print in the 1655 publication De Sectionibus Conicus ("Tract on Conic Sections'') by mathematician John Wallis (1616--1703).  Possible explanations for Wallis' choice of "$$\infty$$" include its resemblance to the symbol "$$oo$$" (used by ancient Romans to denote the number 1000), to the final letter of the Greek alphabet $$\omega$$ (used symbolically to mean the "final'' number), and to a simple curve called a "lemniscate'', which can be endlessly traversed with little effort.

Some Important Numbers in Mathematics

• $$\pi$$ (the ratio of the circumference to the diameter of a circle) denotes the number $$3.141592653589\ldots$$, and was first used in the 1706 book Synopsis palmariorum mathesios ("A New Introduction to Mathematics'') by mathematician William Jones (1675--1749). The use of $$\pi$$ to denote this number was then popularized by the great mathematician Leonhard Euler (1707--1783) in his 1748 book Introductio in Analysin Infinitorum. (It is speculated that Jones chose the letter "$$\pi$$" because it is the first letter in the Greek word perimetron, $$\pi\epsilon\rho\iota\mu\epsilon\tau\rho o \nu$$, which roughly means "around''.)
• $$e$$ $$= \lim_{n \to \infty} (1 + \frac{1}{n})^{n}$$ (the natural logarithm base, also sometimes called Euler's number) denotes the number $$2.718281828459\ldots$$, and was first used in the 1728 manuscript Meditatio in Experimenta explosion tormentorum nuper instituta  ("Meditation on experiments made recently on the firing of cannon'') by Leonhard Euler.  (It is speculated that Euler chose "$$e$$" because it is the first letter in the Latin word for "exponential''.) The mathematician Edmund Landau (1877--1938) once wrote that, "The letter $$e$$ may now no longer be used to denote anything other than this positive universal constant.''
• $$i$$ $$= \sqrt{-1}$$ (the imaginary unit) was first used in the 1777 memoir Institutionum calculi integralis ("Foundations of Integral Calculus'') by Leonhard Euler. The five most important numbers in mathematics are widely considered to be (in order) $$0$$, $$1$$, $$i$$, $$\pi$$, and $$e$$. These numbers are even remarkably linked by the equation $$e^{i \pi} + 1 = 0$$, which the physicist Richard Feynman (1918--1988) once called "the most remarkable formula in mathematics".
• $$\gamma$$ $$= \lim_{n \to \infty} (\sum_{k=1}^{n} \frac{1}{k} - \ln{n})$$ (the Euler-Mascheroni constant, also known as just Euler's constant), "Annotations to Euler's Integral Calculus'') by geometer Lorenzo Mascheroni (1750--1800). The number $$\gamma$$ is widely considered to be the sixth most important important number in mathematics due to its frequent appearance in formulas from number theory and applied mathematics.  However, as of this writing, it is still not even known whether or not $$\gamma$$ is even an irrational number.
• $$\phi$$ $$= \frac{1}{2}(1+\sqrt{5})$$ (the golden ratio) denotes the number % $$1.618033988749 \ldots$$.  Its use was first attributed to % the American Mathematician Mr.  Mark Barr in The  Curves of Life: Being an Account of Spiral Formations and Their Application to Growth in Nature, to Science, and to Art: With Special Reference to the Manuscripts of Leonardo da Vinci (1914)} by Sir Theodore Andrea Cook (1867--1928): "The symbol $$\phi$$ was given to this proportion partly because it has a familiar sound to  those who wrestle constantly with $$\pi$$ and partly  because it is the $$1^{\textrm{st}}$$ letter of the name  of Pheidias, in whose sculpture this number is seen to prevail when the distance between salient points are measured."

The number $$\phi$$ is also often called the "divine proportion'' or the "golden proportion'', and it has been recognized since antiquity as an especially aesthetically pleasing ratio for the side lengths of a rectangle. Such  a rectangle is called a "golden rectangle''.

Some Common Latin Abbreviations and Phrases

• i.e. (id est) means "that is'' or "in other words''.  (It is used to paraphrase a statement that was just made, not to mean "for example'', and is always followed by a comma.)
•  e.g. (exempli gratia) means "for example''.  (It is usually used to give an example of a statement that was just made and is always followed by a comma.)
• viz. (videlicet) means "namely'' or "more specifically''.  (It is used to clarify a statement that was just made by providing more information and is never followed by a comma.)
• etc. (et cetera) means "and so forth'' or "and so on''.  (It is used to suggest that the reader should infer further examples from a list that has already been started and is usually not followed by a comma.)
• et al. (et alii) means "and others''. (It is used in place of listing multiple authors past the first and is never followed by a comma.)  The abbreviation "et al.''~can also be used in place of et alibi}, which means "and elsewhere''.
• cf. (conferre) means "compare to'' or "see also''.  (It is used either to draw a comparison or to refer the reader to somewhere else that they can find more information, and it is never followed by a comma.)
• q.v. (quod vide) means "which see'' or "go look it up if you're interested''.  (It is used to cross-reference a different written work or a different part of the same written work, and it is never followed by a comma.)  The plural form of "q.v.''~is "q.q.''
• v.s. (vide supra) means "see above''. (It is used to imply that more information can be found before the current point in a written work and is never followed by a comma.)
• N.B. (Nota Bene) means "note well'' or "pay attention to the following''.  (It is used to imply that the wise reader will pay especially careful attention to what follows and is never followed by a comma. Cf.~the abbreviation "verb.~sap.'')
• verb. sap. (verbum sapienti sat est) means "a word to the wise is enough'' or "enough has already been said''.  (It is used to imply that, while something may still be left unsaid, enough has been said for the reader to infer the entire meaning.)
• vs. (versus) means "against'' or "in contrast to''.  (It is used to contrast two things and is never followed by a comma.)  The abbreviation "vs.''~is also often written as "v.''
• c. (circa) means "around'' or "near''.  (It is used when giving an approximation, usually for a date, and is never followed by a comma.)  The abbreviation "c.''~is also commonly written as "ca.'', "cir.'', or "circ.''
• ex lib. (ex libris) means "from the  library of''.  (It is used to indicate ownership of a book and is never followed by a comma.).
• vice versa means "the other way around'' and is used to indicate that an implication can logically be reversed.  (This is sometimes abbreviated as "v.v.'')
• a fortiori means "from the stronger'' or "more importantly''.
• a priori means "from before the fact'' and refers to reasoning that is done while an event still has yet to happen.
• a posteriori means "from after the fact'' and refers to reasoning that is done after an event has already happened.
• ad hoc means "to this'' and refers to reasoning that is specific to an event as it is happening. (Such reasoning is regarded as not being generalizable to other situations.)
• ad infinitum means "to infinity'' or "without limit''.
• ad nauseam means "causing sea-sickness'' or "to excessive''.
• mutatis mutandis means "changing what needs changing'' or "with the necessary changes having been  made''.
• non sequitur means "it does not follow'' and refers to something that is out of place in a logical argument.  (This is sometimes abbreviated as "non seq.'')
• Me transmitte sursum, Caledoni! means, "Beam me up, Scotty!''
• Illud Latine dici non potest means "You  can't say that in Latin''.
• Quid quid latine dictum sit, altum videtur means something like, "Anything that is said in Latin will sound profound.''

Contributors

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