Divisibility. In the unpublished notes of his 1925 number theory seminar, Hardy useda|b several times in reference to work by Polya. [Jim Tattersall]
In 1927 Edmund Landau useda|b inElementaryNumber Theory. [Jim Tattersall]
Congruence of numbers. The congruent symbol used innumber theory ≡ was introduced in print in 1801 by Carl FriedrichGauss (1777-1855) inDisquisitiones arithmeticae:
Numerorum congruentiam hoc signo, ≡, in posterumdenotabimus, modulum ubi opus erit in clausulis adiungentes, -16 ≡ 9(mod. 5), -7 ≡ 15 (mod. 11).The citation above is fromDisquisitiones arithmeticae(Leipzig, 1801), art. 2;Werke, Vol. I (Gottingen, 1863), p.10 (Cajori vol. 2, page 35).
However, Gauss had used the symbol much earlier in his personalwritings (Francis, page 82).
SeeMODULUS onWords page.
The number of primes less thanx. Edmund Landau usedπ(x) for the number of primes less than or equal tox in 1909 inHandbuch der Lehre von der Verteilung der Primzahlen(Cajori vol. 2, page 36).
Letters for the sets of rational and real numbers.The authors of classical textbooks such as Weber and Fricke did not denote particulardomains of computation with letters.
In 1872 Richard Dedekind denoted the rationals by R andthe reals by blackletter R inStetigkeit und irrationale Zahlen (1872) (Continuity and irrational numbersWorks,3, 315-334.Dedekind also used K for the integers and J for complex numbers.
In 1888 Richard Dedekind denoted the natural numbers byN inWas ist und was sollen die Zahlen, §6.
In 1889 Giuseppe Peano cited Dedekind’s book in hisArithmetices prinicipia nova methodo exposita, and used the same symbol for the positive integers as Dedekind. Peano usedN, R, andQ, and showed their meaning in a table on page 23:
| N | numerus integer positivus |
| R | num. rationalis positivus |
| Q | quantitas, sive numerus realis positivus |
In 1895 in hisFormulaire de mathématiques,Peano usedN for the positive integers,nfor integers,N0 for the positive integers andzero,R for positive rational numbers,r for rationalnumbers,Q for positive real numbers,q for realnumbers, andQ0 for positive real numbers and zero[Cajori vol. 2, page 299].
In 1897 Peano usedN1 instead ofN. [Wilfried Neumaier]
In 1926 Helmut Hasse (1898-1979) used Γ for the integers andΡ (capital rho) for the rationals inHöhere Algebra I andII, Berlin 1926. He kept to this notation in his later books onnumber theory. Hasse's choice of gamma and rho may have beendetermined by the initial letters of the German terms "ganze Zahl"(integer) and "rationale Zahl" (rational).
In 1929 Otto Haupt usedG0 for the integers andΡ0 (capital rho) for the rationals inEinführung in dieAlgebra I and II, Leipzig 1929.
In 1930 Bartel Leendert van der Waerden used C for the integersand Γ for the rationals inModerne Algebra I,Berlin 1930, but in editions during the sixties, he changed to Z andQ.
In 1930 Edmund Landau denoted the set of integers by a fraktur Zwith a bar over it inGrundlagen der Analysis (1930, p. 64).He does not seem to introduce symbols for the sets of rationals,reals, or complex numbers.
Q for the set of rational numbers andZ for the set ofintegers are apparently due to N. Bourbaki. (N. Bourbaki was a groupof mostly French mathematicians which began meeting in the 1930s,aiming to write a thorough unified account of all mathematics.) Theletters stand for the GermanQuotient andZahlen. Thesenotations occur in Bourbaki'sAlgébre, Chapter 1.
Julio González Cabillón writes that he believesBourbaki was responsible for both of the above symbols, quoting Weil,who wrote, "...it was high time to fix these notations once and forall, and indeed the ones we proposed, which introduced a number ofmodifications to the notations previously in use, met with generalapproval."
[Walter Felscher, Stacy Langton, Peter Flor,Wilfried Neumaier, and A. J. Franco deOliveira contributed to this entry.]
C for the set of complex numbers. William C. Waterhousewrote to a history of mathematics mailing list in 2001:
Checking things I have available, I found C used for thecomplex numbers in an early paper by Nathan Jacobson:Structure and Automorphisms of Semi-Simple Lie Groups inthe Large,Annals of Math. 40 (1939), 755-763.The second edition of Birkhoff and MacLane,Survey of ModernAlgebra (1953), also uses C (but is not using the Bourbakisystem: it has J for integers, R for rationals, R^# for reals). Ihave't seen the first edition (1941), but I would expect to find Cused there too. I'm sure I remember C used in this sense in a numberof other American books published around 1950.I think the first Bourbaki volume published was the results summaryon set theory, in 1939, and it does not contain any symbol for thecomplex numbers. Of course Bourbaki had probably chosen the symbolsby that time, but I think in fact the first appearance of (bold-face)C in Bourbaki was in the formal introduction of complex numbers inChapter 8 of the topology book (first published in1947).
Euler's phi function (totient function). The symbol φ(m) for the number of integers less thanmthat are relatively prime tom was introduced by Carl Friedrich Gauss (1777-1855) in 1801 in hisDisquisitiones arithmeticaearticles 38, 39 (p. 30) (Cajori vol. 2, page 35, and Dickson, page 113-115).
The function was first studied byLeonhard Euler(1707-1783), although Dickson (page 113) and Cajori (vol. 2, p. 35) say that Euler did not use a functionalnotation inNovi Comm. Ac. petrop., 8, 1760-1, 74, andComm. Arith.,1, 274, and that Euler used πN inActa Ac. Petrop., 4 II (or 8), 1780 (1755), 18, andComm. Arith.,2, 127-133. Shapiro agrees, writing: "He did not employ any symbol forthe function until 1780, when he used the notation πn."
Sylvester, who introduced the nametotient for the function, seems to have believedthat Euler had used φ. He writes in 1888(vol. IV p. 589 of hisCollected MathematicalPapers) "I am in the habit of representing the totient ofnby the symbol τn, τ (taken from the initial of the word itdenotes) being a less hackneyed letter than Euler's φ, which hasno claim to preference over any other letter of the Greek alphabet, but ratherthe reverse." This information was taken from a post in sci.math by RobertIsrael.
SeeTOTIENT onWords page.
Legendre symbol (quadratic reciprocity).Adrien-Marie Legendre introduced the notation
= 1 ifD is a quadratic residue ofp, and = -1 ifD is a quadratic non-residue ofp.
According to Hardy & Wright'sAn Introduction to theTheory of Numbers, "Legendre introduced 'Legendre's symbol' in hisEssai sur la theorie des nombres, first published in 1798. See, for example,§135 of the second edition (1808)." In the third edition onGallica this is on p.197.
However, according to William J. Leveque inFundamentals of Number Theory,"Legendre introduced his symbol in an article in 1785, and at the same time stated thereciprocity law without using the symbol."
[Both of these citations were provided by Paul Pollack.]
Mersenne numbers. Mersenne numbers are markedMn by Allan Cunningham in 1911 inMathematicalQuestions and Solutions from the Educational Times (Cajori vol.2, page 41).
Fermat numbers. Fermat numbers are markedFn in 1919 in L. E. Dickson'sHistory of theTheory of Numbers (Cajori vol. 2, page 42).
The norm ofa +bi. Dirichlet usedN(a+bi)for the norma2+b2of the complex numbera+bi inCrelle's Journal Vol. XXIV(1842) (Cajori vol. 2, page 33). SeeNORM onWords page
Galois field. Eliakim Hastings Moore used the symbolGF[qn] to represent the Galois field oforderqn in 1893. The modern notation is"Galois-field of orderqn" (Julio GonzálezCabillón and Cajori vol. 2, page 41).
Sum of the divisors ofn. Euler introduced the symbol
n in a paper published in 1750 (DSB, article:"Euler").
In 1888, James Joseph Sylvester continued the use of Euler's notationn (Shapiro).
Allan Cunningham used σ(N) to represent thesum of the proper divisors ofN inProceedings of theLondon Mathematical Society 35 (1902-03):
The Repetition of the Sum-Factor Operation. Abstract of an informalcommunication made by Lieut.-Col. A. Cuningham, June 12th, 1902.[Cajori, vol. 2, page 29, and Paul Pollack]Let σ(N) denote the sum of the sub-factors ofN (including 1, butexcludingN). It was found that, with most numbers, σnN = 1, whenthe operation (σ) is repeated often enough. There is a small class forwhich σnN =P (aperfect number), and then repeats; anothersmall class for which σnN =A, σn + 1N =B,whereA, B areamicable numbers, and then repeats (A, B alternately); another smallclass for which (even whenN issmall, < 1000) σnN increasesbeyond the practical power of calculation.
In 1927 Landau chose the notationS(n) (Shapiro).
L. E. Dickson useds(n) for the sum of the divisors ofn (Cajori vol. 2, page 29).
The Möbius function. Möbius' work appeared in 1832but the µ symbol was not used.
The notation µ(n) was introduced by Franz Mertens(1840-1927) in 1874 in "Über einige asymptotische Gesetze derZahlentheorie,"Crelle's Journal (Shapiro).
Big-O andlittle-o notation. Accordingto Wladyslaw Narkiewicz inThe Development of Prime Number Theory:
The symbols O(·) and o(·) are usually called the Landau symbols.This name is only partially correct, since it seems that the first of them appearedfirst in the second volume of P. Bachmann's treatise on number theory (Bachmann,1894). In any case Landau (1909a, p. 883) states that he had seen it for thefirst time in Bachmann's book. The symbol o(·) appears first in Landau (1909a).Earlier this relation has been usually denoted by {·}.The references are toPaul Bachmann (1837-1920) and hisAnalytische Zahlentheorie and toEdmund Landau (1877-1938) and hisHandbuch der Lehre von der Verteilung der Primzahlen.
[Paul Pollack contributed to this entry.]
