# 1911 Encyclopædia Britannica/Number

**NUMBER**^{[1]} (through Fr. *nombre*, from Lat. *numerus*; from a root seen in Gr. νἑμειν to distribute), a word generally expressive of quantity, the fundamental meaning of which leads on analysis to some of the most difficult problems of higher mathematics.

1. The most elementary process of thought involves a distinction within an identity—the A and the not-A within the sphere throughout which these terms are intelligible. Again A may be a generic quality found in different modes Aa, Ab, Ac, &c.; for instance, colour in the modes, red, green, blue and so on. Thus the notions of “one,” “two” and the vague “many” are fundamental, and must have impressed themselves on the human mind at a very early period: evidence of this is found in the grammatical distinction of singular, dual and plural which occurs in ancient languages of widely different races. A more definite idea of number seems to have been gradually acquired by realizing the equivalence, as regards plurality, of different concrete groups, such as the fingers of the right hand and those of the left. This led to the invention of a set of names which in the first instance did not suggest a numerical system, but denoted certain recognized forms of plurality, just as blue, red, green, &c., denote recognized forms of colour. Eventually the conception of the series of natural numbers became sufficiently clear to lead to a systematic terminology, and the science of arithmetic was thus rendered possible. But it is only in quite recent times that the notion of number has been submitted to a searching critical analysis: it is, in fact, one of the most characteristic results of modern mathematical research that the term *number* has been made at once more precise and more extensive.

**Introduction (§ 1)**- Aggregates (§ 2)
- Order (§ 3)
- The Natural Scale (§ 4)
- Arithmetical Operations (§§ 5–6)
- Negative Numbers (§§ 7–9)
- Fractional Numbers (§§ 10–15)
- Irrational Numbers (§§ 16–18)
- Complex Numbers (§ 19)
- Transfinite Numbers (§§ 20–23)
- Theory of Numbers (§§ 24–25)
- Totients (§ 26)
- Residues and congruences (§ 27)
- The Theorems of Fermat and Wilson (§ 28)
- Exponents, Primitive Roots, Indices (§ 29)
- Linear Congruences (§ 30)
- Quadratic Residues (§ 31)
- Quadratic forms (§ 32)
- Method of Reduction (§ 33)
- Problem of Representation (§ 34)
- Automorphs (§ 35)
- Characters of a form or class (§ 36)
- Composition (§§ 37–38)
- Number of classes (§ 39)
- Bilinear Forms (§ 40)
- Higher Quadratic Forms (§ 41)
- Complex Numbers (Gaussian) (§§ 42–43)
- Algebraic Numbers (§§ 44–45)
- Ideals (§§ 46–48)
- Ideal Classes (§§ 49–50)
- Normal Fields (§ 51)
- Quadratic Fields (§ 52)
- Normal Residues (§ 53)
- Class-Number(§§ 54–55)
- Complex Quadratic Forms (§ 56)
- Kronecker's Method(§§ 57–58)
- Cyclotomy(§§ 59–61)
- Gauss's Sums(§§ 62–63)
- Higher Congruences(§§ 64–65)
- Forms of Higher Degree (§ 66)
- Results derived from Elliptic and Theta Functions(§§ 67–73)
- Frequency of Primes (§ 74)
- Arithmetical Functions (§ 75)
- Transcendental Numbers (§ 76)
- Miscellaneous Investigations(§§ 77–78)