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BINOMIAL (from the Lat. bi-, bis, twice, and nomen, a name or term), in mathematics, a word first introduced by Robert Recorde (1557) to denote a quantity composed of the sum or difference to two terms; as a + b, ab. The terms trinomial, quadrinomial, multinomial, &c., are applied to expressions composed similarly of three, four or many quantities.

The binomial theorem is a celebrated theorem, originally due to Sir Isaac Newton, by which any power of a binomial can be expressed as a series. In its modern form the theorem, which is true for all values of n, is written as

(x + a)n = xn + naxn−1 +   n·(n − 1)   a2xn−2   n·(n − 1)·(n − 2)   a3xn−3 ... + an.
1·2 1·2·3

The reader is referred to the article Algebra for the proof and applications of this theorem; here we shall only treat of the history of its discovery.

The original form of the theorem was first given in a letter, dated the 13th of June 1676, from Sir Isaac Newton to Henry Oldenburg for communication to Wilhelm G. Leibnitz, although Newton had discovered it some years previously. Newton there states that

(p + pq)m/n = pm/n +   m   aq +   mn   bq +   m − 2n   cq ... &c.,
n 2n 3n

where p + pq is the quantity whose mth/n power or root is required, p the first term of that quantity, and q the quotient of the rest divided by p, m/n the power, which may be a positive or negative integer or a fraction, and a, b, c, &c., the several terms in order, e.g.

a = pm/n, b =   m   aq, c =   mn   bq, and so on.
n 2n

In a second letter, dated the 24th of October 1676, to Oldenburg, Newton gave the train of reasoning by which he devised the theorem.

“In the beginning of my mathematical studies, when I was perusing the works of the celebrated Dr Wallis, and considering the series by the interpolation of which he exhibits the area of the circle and hyperbola (for instance, in this series of curves whose common base or axis is x, and the ordinates respectively (1 − xx)0/2, (1 − xx)1/2, (1 − xx)2/2, (1 − xx)3/2, &c.), I perceived that if the areas of the alternate curves, which are x, x1/3x3, x2/3x3 + 1/5x5, x3/3x3 + 3/5x51/7x7, &c., could be interpolated, we should obtain the areas of the intermediate ones, the first of which (1 − xx)1/2 is the area of the circle. Now in order to [do] this, it appeared that in all the series the first term was x; that the second terms 0/3x3, 1/3x3, 2/3x3, &c., were in arithmetical progression; and consequently that the first two terms of all the series to be interpolated would be x1/2x3/3, x3/2x3/3, x5/2x3/3, &c.

“Now for the interpolation of the rest, I considered that the denominators 1, 3, 5, &c., were in arithmetical progression; and that therefore only the numerical coefficients of the numerators were to be investigated. But these in the alternate areas, which are given, were the same with the figures of which the several powers of 11 consist, viz., of 110, 111, 112, 113, that is, the first 1; the second, 1, 1; the third, 1, 2, 1,; the fourth 1, 3, 3, 1; and so on. I enquired therefore how, in these series, the rest of the terms may be derived from the first two being given; and I found that by putting m for the second figure or term, the rest should be produced by the continued multiplication of the terms of this series m − 0/1 × m − 1/2 × m − 2/3 ..., &c. ... This rule I therefore applied to the series to be interpolated. And since, in the series for the circle, the second term was (1/2x3)/3, I put m = 1/2.... And hence I found the required area of the circular segment to be x1/2x3/31/8x5/51/16x7/7, &c. ... And in the same manner might be produced the interpolated areas of other curves; as also the area of the hyperbola and the other alternates in this series (1 + xx)0/2, (1 + xx)1/2, (1 + xx)2/2, &c. ... Having proceeded so far, I considered that the terms (1 − xx)0/2, (1 − xx)2/2, (1 − xx)4/2, (1 − xx)6/2, &c., that is 1, 1 − x2, 1 − 2x2 + x4, 1 − 3x2 + 3x4x6, &c., might be interpolated in the same manner as the areas generated by them, and for this, nothing more was required than to omit the denominators 1, 3, 5, 7, &c., in the terms expressing the areas; that is, the coefficients of the terms of the quantity to be interpolated (1 − xx)1/2 or (1 − xx)32, or generally (1 − xx)m will

be produced by the continued multiplication of this series m × (m−1)/2 × (m−2)/3 × (m−3)/4... &c.”

The binomial theorem was thus discovered as a development of John Wallis’s investigations in the method of interpolation. Newton gave no proof, and it was in the Ars Conjectandi (1713) that James Bernoulli’s proof for positive integral values of the exponent was first published, although Bernoulli must have discovered it many years previously. A rigorous demonstration was wanting for many years, Leonhard Euler’s proof for negative and fractional values being faulty, and was finally given by Niels Heinrik Abel.

The multi- (or poly-) nomial theorem has for its object the expansion of any power of a multinomial and was discussed in 1697 by Abraham Demoivre (see Combinatorial Analysis).

References.—For the history of the binomial theorem, see John Collins, Commercium Epistolicum (1712); S. P. Rigaud, The Correspondence of Scientific Men of the 17th Century (1841); M. Cantor, Geschichte der Mathematik (1894–1901).