Binomial Formula

BINOMIAL FORMULA. Binomial theorem is the name at tached to an algebraic formula widely used for transforming and simplifying algebraic expressions and also in processes of approxi mation.

Positive integral exponents. If a and b stand for any real numbers, and n is a positive integer, then the binomial (a-l-b), raised to the nth power, can be "expanded" by the formula, I) 2) (a+b). =an-Fnan-lb+ an-2b2-1- a"---31)3+ 2' 31 (I) By actual multiplication the expressions are easily obtained :— (a+b)2 = a2-1-2ab-Fb2, (a+br = a3-1-3a2H-3ab2d-b3, (a+b)4 = a4 4a3b 6a2b2 4ab3 b4 The same results are obtained by substituting for n in Formula I successively the values 2, 3, and 4. In other words, Formula is a generalization of the three special cases obtained by multi plication. By the aid of Formula it is possible to dispense with the process of multiplication, which becomes tedious when n is large; the expansion can be written down provided the changes in the exponents and coefficients of the successive terms are ob served. These changes are seen to be in accordance with the following laws: (I) the exponents of a are for the successive terms n, n-1, n-2, ...I, and finally o (not written, because a° =1); that is, the exponent of a is n for the first term, and diminishes by unity in each successive term; (2) the exponent of b is o in the first term (not written, because b°-= I), and is I, 2, 3, ...n-r, n for the terms following; (3) the coefficients are formed according to a slightly more complicated rule. The coefficient of the first term is 1, and that of the second term is n. The coefficient of the third term is obtained by multiplying the coefficient of the second term (n) by the exponent of a in that term (n-i), and dividing the product by 2, the number of that term in the series; thus one obtainsn(n— I). The coefficient of the fourth 2 term is similarly obtained from the third term. Or, generally, the coefficient of the rth term is obtained by multiplying the co efficient of the (r-i)th term by the exponent of a in the (r-i)th term, and dividing the product thus obtained by r-I. In this manner there follows the general expression for the rth term n(n--I)... (n—r+2) of the expansion, namely an-H-'br-1 by the (r 1)1 aid of which any term may be written. Thus, if we desire the 4th term, we substitute 4 for r and obtain that term as given in Formula 1. Observing these three rules, it is possible to write down the expansion of a binomial for any power.

Fractional and negative exponents. When the exponent n is a positive fraction, a negative fraction, or a negative integer, the binomial formula retains the same form as above, except that there is no last term bn. In other words, the series now has an un limited number of terms and is an "infinite series." That there is no end to the series becomes evident from the fact that the coefficient for the rth term, n(n-1)-• (n—r+2), does not (r— become zero when the exponent n is not a positive integer. For example, if n=3, none of the factors, (n-1 ), (n-2)..., (n-r+2), vanishes, and hence the rth term does not become zero, no matter how large a positive integral value r may represent. For the simpler case which we considered first, when n is a positive integer, say 5, it is easy to see that a term beyond the 5th, say the 7th term in the expansion, has in the numerator a factor (5-7+2) which is zero; hence there really is no 7th term. The same factor zero occurs when r is taken to represent the 8th, gth, or a term still higher. Thus, for n=5, there is a last term in the expansion, namely b5. Thus far we have considered only rational values of the numbers a, b, and n, but the binomial formula is applicable to irrational values.

Sometimes it is convenient to change the form of the binomial formula by letting a=-- 1, and b=x, a variable. Since all the powers of I are 1, the expression becomes, if we also insert the rth term, n(n— I) (1-Fx)n = 1-Fnx+ 2! X2+...± n(n — 1)(n — z) . ..(n — r 2) (2) (r--1)! When the expansion is an infinite series, the theory of the binomial formula becomes much more difficult. In fact, the ex panded form on the right side of the equality sign can be said to be equal in arithmetical value to (a+b)n or to (i+x)n, on the left side, only when the series is "convergent"; that is, when the sum obtained by adding more and more successive terms of the series approaches a definite finite value as a limit. When that sum does not approach a limit, the infinite series is not con vergent, but "divergent," and the series is no longer equal in arithmetical value to (a+b)n or (i+x)n. In other words, the sign of equality in the formula holds true only when the infinite series II. is convergent. The series II. is convergent when x is numerically less than 1; it is divergent when x is numerically greater than 1. The intermediate case when x is numerically equal to is less important.

Exponents complex numbers. A still further generalization of the binomial formula is obtained by letting one or all of the letters a, b, x, n represent imaginary or complex numbers, of the type c+id, where c and d are real, and where The expansion is an infinite series when n is a complex number. Tests of convergence have been found, similar to the ones given above.

History.

The history of the binomial theorem possesses points of interest. For values of a, b, and n, which may be in tegral or fractional, the formula was first given by Sir Isaac Newton in a letter of June 13, 1676, and explained further in a letter of Oct. 24, 1676, both addressed to H. Oldenburg, then secretary of the Royal Society of London. Newton's notation was somewhat different from that of to-day. The statement is frequently made that the binomial theorem is engraved on New ton's tomb in Westminster Abbey, but there is no such inscrip tion. Newton gave no proof of the theorem, but verified it by actual multiplication and root extraction. The beginnings of the binomial theorem are found long before the time of Newton. The Indians and Arabs used the expansion of (a+b)2 and (a-4)3 for root extractions. Vieta knew the expansion of (a+b)4. B. Pascal derived the binomial coefficients from the method of what is called the "arithmetical triangle," published posthumously in 1665, but the arithmetical triangle is found be fore Pascal as early as 13o3 in a Chinese tract by Chu Shih-Chieh.

The first proof of the binomial formula for the case of positive integral exponents was given by Jakob (James) Bernoulli 1705) in his posthumous work, the Ars conjectandi, 1713, p. 89. When the formula is an infinite series, rigorous proofs are much more difficult, because they necessarily involve considerations of convergence. The general proof given by Leonhard Euler (Novi comment. Petrop., vol. xix. for P. 103) and other writers of the 18th century dealt mainly with the form of the expansion and did not adequately consider the question of the equality of the arithmetical values on the two sides of the equa tion. The earliest one to give a rigorous proof of the general case was the Norwegian mathematician Niels Henrick Abel, in Crelle's Journal, vol. i., 1826, p. 311. It is a remarkable article of 28 large quarto pages, and it establishes the theorem not only when the exponent n is integral, or fractional, or negative, but also when the exponent is a complex number, and when a and b may likewise be complex numbers. (F. CA.)