Earliest Uses of Symbols of Number Theory

Last updated: May 24, 1999


Congruence of numbers. The congruent symbol used in number theory three
horizontal bars was introduced in print in 1801 by Carl Friedrich Gauss (1777-1855) in Disquisitions arithmeticae:
Numerorum congruentiam hoc signo, three horizontal bars, in posterum denotabimus, modulum ubi opus erit in clausulis adiungentes, -16 three horizontal bars 9 (mod. 5), -7 three
horizontal bars 15 (modo 11).
The citation above is from Disquisitiones 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 personal writings (Francis, page 82).

The number of primes less than x. Edmund Landau used pi(x) for the number of primes less than or equal to x in 1909 in Handbuch 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 particular domains of computation with letters.

Richard Dedekind (1831-1916) denoted the rationals by R and the reals by gothic R in Continuity and irrational numbers (1872). Dedekind also used K for the integers and J for complex numbers.

Helmut Hasse (1898-1979) used [capital gamma] for the integers and [capital rho] for the rationals in Höhere Algebra I and II, Berlin 1926. He kept to this notation in his later books on number theory. Hasse's choice of gamma and rho may have been determined by the initial letters of the German terms "ganze Zahl" (integer) and "rationale Zahl" (rational).

Otto Haupt used G0 for the integers and [capital rho]0 for the rationals in Einführung in die Algebra I and II, Leipzig 1929.

Bartel Leendert van der Waerden (1903-1996) used C for the integers and [capital gamma] for the rationals in Moderne Algebra I, Berlin 1930, but in editions during the sixties, he changed to Z and Q.

Edmund Landau (1877-1938) denoted the set of integers by a fraktur Z with a bar over it in Grundlagen 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 and Z for the set of integers are apparently due to Bourbaki. The letters stand for the German Quotient and Zahlen. These notations occur in Bourbaki's Algébre, Chapter 1.

Julio González Cabillón writes that he believes Bourbaki was responsible for both of the above symbols, quoting Weil, who wrote, "...it was high time to fix these notations once and for all, and indeed the ones we proposed, which introduced a number of modifications to the notations previously in use, met with general approval."

[Walter Felscher, Stacy Langton, Peter Flor, and A. J. Franco de Oliveira contributed to this entry.]

Euler's phi function. phi(m) was introduced by Carl Friedrich Gauss (1777-1855) in 1801 in his Disquisitiones Arithmeticae, articles 38, 39 (Cajori vol. 2, page 35, and Dickson, page 113-115).

The article "Number Theory" in the Encyclopaedia Britannica claims this symbol was introduced by Leonhard Euler (1707-1783). However Dickson (page 113) and Cajori (vol. 2, p. 35) say that Euler did not use a functional notation in Novi Comm. Ac. petrop., 8, 1760-1, 74, and Comm. Arith., 1, 274, and that Euler used piN in Acta Ac. Petrop., 4 II (or 8), 1780 (1755), 18, and Comm. Arith., 2, 127-133.

Sylvester, who used tau for this function, also believed that Euler used phi. He writes (in vol. IV p. 589 of his Collected Mathematical Papers) "I am in the habit of representing the totient of n by the symbol (tau) n, (tau) (taken from the initial of the word it denotes) being a less hackneyed letter than Euler's phi, which has no claim to preference over any other letter of the Greek alphabet, but rather the reverse." This information was taken from a post in sci.math by Robert Israel.

Quadratic reciprocity. Adrien-Marie Legendre introduced the notation that (D/p) = 1 if D is a quadratic residue of p, and (D/p) = -1 if D is a quadratic non-residue of p (Francis, page 85). The symbol was introduced in 1798, according to Peter Giblin, Primes and Programming, Cambridge University Press, 1993.

Mersenne numbers. Mersenne numbers are marked Mn by Allan Cunningham in 1911 in Mathematical Questions and Solutions from the Educational Times (Cajori vol. 2, page 41).

Fermat numbers. Fermat numbers are marked Fn in 1919 in L. E. Dickson's History of the Theory of Numbers (Cajori vol. 2, page 42).

The norm of a + bi. Dirichlet used N(a+bi) for the norm a2+b2 of the complex number a+bi in Crelle's Journal Vol. XXIV (1842) (Cajori vol. 2, page 33).

Galois field. Eliakim Hastings Moore used the symbol GF[qn] to represent the Galois field of order qn in 1893. The modern notation is "Galois-field of order qn" (Julio González Cabillón and Cajori vol. 2, page 41).

Sum of the divisors of n. Euler introduced the symbol the integral
symboln in a paper published in 1750 (DSB, article: "Euler").

Big-O notation was introduced by Paul Bachmann (1837-1920) in his Analytische Zahlentheorie in 1892. The actual O symbol is sometimes called a Landau symbol after Edmund Landau (1877-1938), who used this notation throughout his work.

Little-oh notation was first used by Edmund Landau (1877-1938) in 1909, according to the website of the University of Tennessee at Martin.


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