Numerorum congruentiam hoc signo, , in posterum denotabimus, modulum ubi opus erit in clausulis adiungentes, -16 9 (mod. 5), -7 15 (modo 11).The citation above is from

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
(

**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 *G*^{0} 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.** (*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 *N* 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 . 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 , 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
*M _{n}* by Allan Cunningham in 1911 in

**Fermat numbers.** Fermat numbers are marked
*F _{n}* in 1919 in L. E. Dickson's

**The norm of a + bi.** Dirichlet used

**Galois field.** Eliakim Hastings Moore used the symbol
*GF*[*q ^{n}*] to represent the Galois field of
order

**Sum of the divisors of n.** Euler introduced the symbol

**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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