**Divisibility.** In the unpublished notes of his 1925 number theory seminar, Hardy used
*a*|*b* several times in reference to work by Polya. [Jim Tattersall]

In 1927 Edmund Landau used *a*|*b* in
*Elementary
Number Theory*. [Jim Tattersall]

**Congruence of numbers.** The congruent symbol used in
number theory ≡ was introduced in print in 1801 by Carl Friedrich
Gauss (1777-1855) in *Disquisitiones arithmeticae*:

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

However, Gauss had used the symbol much earlier in his personal writings (Francis, page 82).

See MODULUS on Words page.

**The number of primes less than x.** Edmund Landau used
π(

**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.

In 1872 Richard Dedekind denoted the rationals by R and
the reals by blackletter R in *Stetigkeit und irrationale Zahlen* (1872) (*Continuity and irrational numbers*
*Works,* **3**, 315-334.
Dedekind also used K for the integers and J for complex numbers.

In 1888 Richard Dedekind denoted the natural numbers by *N* in *Was ist und was sollen die Zahlen,* §6.

In 1889 Giuseppe Peano cited Dedekind’s book in his *Arithmetices prinicipia nova methodo exposita,* and used the same symbol for the positive integers as Dedekind. Peano used *N, R,* and *Q, * 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 his *Formulaire de mathématiques,*
Peano used *N* for the positive integers, *n*
for integers, *N _{0}* for the positive integers and
zero,

In 1897 Peano used *N*_{1} instead of *N.* [Wilfried Neumaier]

In 1926 Helmut Hasse (1898-1979) used Γ for the integers and
Ρ (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).

In 1929 Otto Haupt used *G*^{0} for the integers and
Ρ^{0} (capital rho) for the rationals in *Einführung in die
Algebra I and II,* Leipzig 1929.

In 1930 Bartel Leendert van der Waerden used C for the integers
and Γ for the rationals in *Moderne Algebra I,*
Berlin 1930, but in editions during the sixties, he changed to Z and
Q.

In 1930 Edmund Landau 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 N. Bourbaki. (N. Bourbaki was a group
of mostly French mathematicians which began meeting in the 1930s,
aiming to write a thorough unified account of all mathematics.) 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, Wilfried Neumaier, and A. J. Franco de Oliveira contributed to this entry.]

** C for the set of complex numbers.** William C. Waterhouse
wrote to a history of mathematics mailing list in 2001:

Checking things I have available, I found C used for the complex numbers in an early paper by Nathan Jacobson:Structure and Automorphisms of Semi-Simple Lie Groups in the Large,The second edition of Birkhoff and MacLane,Annals of Math.40 (1939), 755-763.Survey of Modern Algebra(1953), also uses C (but is not using the Bourbaki system: it has J for integers, R for rationals, R^# for reals). I have't seen the first edition (1941), but I would expect to find C used there too. I'm sure I remember C used in this sense in a number of other American books published around 1950.I think the first Bourbaki volume published was the results summary on set theory, in 1939, and it does not contain any symbol for the complex numbers. Of course Bourbaki had probably chosen the symbols by that time, but I think in fact the first appearance of (bold-face) C in Bourbaki was in the formal introduction of complex numbers in Chapter 8 of the topology book (first published in 1947).

**Euler's phi function (totient function).** The symbol φ(*m*) for the number of integers less than *m*
that are relatively prime to *m* was introduced by Carl Friedrich Gauss (1777-1855) in 1801 in his
*Disquisitiones arithmeticae*
articles 38, 39 (p. 30) (Cajori vol. 2, page 35, and Dickson, page 113-115).

The function was first studied by Leonhard Euler
(1707-1783), although 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 π*N* in *Acta Ac. Petrop.,* 4 II (or 8), 1780 (1755), 18, and *Comm. Arith.,*
2, 127-133. Shapiro agrees, writing: "He did not employ any symbol for
the function until 1780, when he used the notation π*n.*"

Sylvester, who introduced the name *totient* for the function, seems to have believed
that Euler had used φ. He writes in 1888
(
vol. IV p. 589 of his *Collected Mathematical
Papers*) "I am in the habit of representing the totient of *n*
by the symbol τ *n*, τ (taken from the initial of the word it
denotes) being a less hackneyed letter than Euler's φ, 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.

See TOTIENT on Words page.

**Legendre symbol (quadratic reciprocity).**
Adrien-Marie Legendre introduced the notation
= 1 if *D* is a quadratic residue of *p,* and
= -1 if *D* is a quadratic non-residue of *p.*

According to Hardy & Wright's *An Introduction to the
Theory of Numbers,* "Legendre introduced 'Legendre's symbol' in his
*Essai sur la theorie des nombres,* first published in 1798. See, for example,
§135 of the second edition (1808)." In the third edition on
Gallica this is on p.197.

However, according to William J. Leveque in *Fundamentals of Number Theory,*
"Legendre introduced his symbol in an article in 1785, and at the same time stated the
reciprocity law without using the symbol."

[Both of these citations were provided by Paul Pollack.]

**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

In 1888, James Joseph Sylvester continued the use of Euler's notation
*n* (Shapiro).

Allan Cunningham used σ(*N*) to represent the
sum of the proper divisors of *N* in *Proceedings of the
London Mathematical Society* 35 (1902-03):

The Repetition of the Sum-Factor Operation. Abstract of an informal communication 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, but excludingN). It was found that, with most numbers, σ^{n}N= 1, when the operation (σ) is repeated often enough. There is a small class for which σ^{n}N=P(aperfectnumber), and then repeats; another small class for which σ^{n}N=A, σ^{n + 1}N=B, whereA, Bareamicablenumbers, and then repeats (A, Balternately); another small class for which (even whenNissmall, < 1000) σ^{n}Nincreases beyond the practical power of calculation.

In 1927 Landau chose the notation *S*(*n*) (Shapiro).

L. E. Dickson used *s*(*n*) for the sum of the divisors of
*n* (Cajori vol. 2, page 29).

**The Möbius function.** Möbius' work appeared in 1832
but the µ symbol was not used.

The notation µ(*n*) was introduced by Franz Mertens
(1840-1927) in 1874 in "Über einige asymptotische Gesetze der
Zahlentheorie," *Crelle's Journal* (Shapiro).

**Big-O** and **little-o notation.** According
to Wladyslaw Narkiewicz in *The 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 appeared first 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 the first time in Bachmann's book. The symbol o(·) appears first in Landau (1909a). Earlier this relation has been usually denoted by {·}.The references are to Paul Bachmann (1837-1920) and his

[Paul Pollack contributed to this entry.]