## Binomial Sentence Examples

- With Descartes the use of exponents as now employed for denoting the powers of a quantity becomes systematic; and without some such step by which the homogeneity of successive powers is at once recognized, the
**binomial**theorem could scarcely have been detected. - Immediately on the completion of his Regne Animale in 1756, Brisson set about his Ornithologie, and it is only in the last two volumes of the latter that any reference is made to the tenth edition of the Systema Naturae, in which the
**binomial**method was introduced. - Gmelin availed himself of every publication he could, but he perhaps found his richest booty in the labours of Latham, neatly condensing his English descriptions into Latin diagnoses, and bestowing on them
**binomial**names. - (0B) = (e), &c. The
**binomial**coefficients appear, in fact, as symmetric functions, and this is frequently of importance. - Other forms are n-1 n-2 2 ax +nbx x +n(n-i)cx x +..., 1121 2 the
**binomial**coefficients C) being replaced by s!(e), and n 1, n-1 1 n-2 2 ax 1 +l i ox l 'x 2 + L ?cx 1 'x2+..., the special convenience of which will appear later. - Now the symbolic expression of the seminvariant can be expanded by the
**binomial**theorem so as to be exhibited as a sum of products of seminvariants, of lower degrees if alai 0-2a2 +...+crea0 can be broken up into any two portions (alai -1-0-2a2-1-ï¿½ï¿½ï¿½ +asas) +(as+1as +1 +o-8+2as+2+ï¿½ï¿½ï¿½ +ooae), such that Q1 +a2+... - Algebraic Forms;
**Binomial**; Combinatorial Analysis; Determin Ants; Equation; Continued Fraction; Function; Theory of groups; Logarithm; Number; Probability; Series. - (v.) Permutations and Combinations may be regarded as arithmetical recreations; they become important algebraically in reference to the
**binomial**theroem (ï¿½ï¿½ 41, 44)ï¿½ (vi.) Surds and Approximate Logarithms. - From the arithmetical point of view, surds present a greater difficulty than negative quantities and fractional numbers. - A multinomial consisting of two or of three terms is a
**binomial**or a trinomial. - This is important when we come to the
**binomial**theorem (ï¿½ 41, and cf. - The
**binomial**theorem gives a formula for writing down the coefficient of any stated term in the expansion of any stated power of a given**binomial**. - This is the
**binomial**theorem for a positive integral index. - 43ï¿½
**Binomial**Coefficients. - - The numbers denoted by n (r) in ï¿½ 41 are the
**binomial**coefficients shown in the table in ï¿½ 40; n (r) being the (r+ i) th number in the (n+ i) th row. - R!) = (n + r)(r) = (n+r)(n) (17)ï¿½ (iv.) By means of (17) the relations between the
**binomial**coefficients in the form p (4) may be replaced by others with the The most important relations are n[r] = n[r-i]+(n - I)(r) (r8); O[r] = 0 (19); n[r]-(n-s)[r] =n[r-i]+(n- I) [r-1]+...+ (n-s+I)[r-1] (20); n[r] =n[r-1]+(n-I)[r-1]+...+I[r-1] (21). - (v.) It should be mentioned that the notation of the
**binomial**'coefficients, and of the continued products such as n(n -1). - The
**binomial**theorem for positive integral index may then be written (x + y) n = -iyi +. - This property enables us to establish, by simple reasoning, certain relations between
**binomial**coefficients. - Application of
**Binomial**Theorem to Rational Integral Functions. - If we represent this expression by f (x), the expression obtained by changing x into x-+-h is f(x+h); and each term of this may be expanded by the
**binomial**theorem. - Relation of
**Binomial**Coefficients to Summation of Series. - The
**binomial**theorem may, for instance, be stated for (x+a)n alone; the formula for (x-a)" being obtained by writing it as {x+(-)a} n or Ix+(- a) } n, so that (x-a) n =x"- 1)xn-laF...+(-)rn(r)xn-rar+..., where + (-) r means - or + according as r is odd or even. - The use of negative coefficients leads to a difference between arithmetical division and algebraical division (by a multinomial), in that the latter may give rise to a quotient containing subtractive terms. The most important case is division by a
**binomial**, as illustrated by the following examples: - 2.10+1) 6.100+5.10+ 1(3.10+I 2.10+I) 6.100+I.10 - I (3.10 - I 6.100+3.10 6.100+3.10 2.10+ I - 2.10 - I 2.10 +I - 2.10 - I In (1) the division is both arithmetical and algebraical, while in (2) it is algebraical, the quotient for arithmetical division being 2.10+9. - It has been mentioned in ï¿½ 41 (ix.) that the
**binomial**theorem can be used for obtaining an approximate value for a power of a number; the most important terms only being taken into account. - There are extensions of the
**binomial**theorem, by means of which approximate calculations can be made of fractions, surds, and powers of fractions and of surds; the main difference being that the number of terms which can be taken into account is unlimited, so that, although we may approach nearer and nearer to the true value, we never attain it exactly. - Comparison with the table of
**binomial**coefficients in ï¿½ 43 suggests that, if m is any positive integer, (I +x)-m =Sr+Rr (25), where Sr=I -m[1]x+m[2]x2...+(-)rm[r]xr (26), Rr_(_)r+1xr+11m[r] (1Fx) - 1+(m - I[r](I+x) m) (27). - (iv.) To assimilate this to the
**binomial**theorem, we extend the definition of n (r) in (I) of ï¿½ 41 (i.) so as to cover negative integral values of n; and we then have (-m)(r)- iI m- = (-) rm [T] (28), so that, if n=--- -m, Sr1 +n(ox+n(2)x2+... - (ix.) The extension of n (r), and therefore of n [r ], to negative and fractional values of n, enables us to extend the applicability of the
**binomial**coefficients to the summation of series (ï¿½ 46 (ii.)). - This accounts for the fact that the same table of
**binomial**coefficients serves for the expansions of positive powers of i+x and of negative powers of i - x. - Consideration of the
**binomial**theorem for fractional index, or of the continued fraction representing a surd, or of theorems such as Wallis's theorem (ï¿½ 64), shows that a sequence, every term of which is rational, may have as its limit an irrational number, i.e. - He introduced the sign (=) for equality, and the terms
**binomial**and residual. - Continued fractions, one of the earliest examples of which is Lord Brouncker's expression for the ratio of the circumference to the diameter of a circle (see Circle), were elaborately discussed by John Wallis and Leonhard Euler; the convergency of series treated by Newton, Euler and the Bernoullis; the
**binomial**theorem, due originally to Newton and subsequently expanded by Euler and others, was used by Joseph Louis Lagrange as the basis of his Calcul des Fonctions. - Linnaeus by his
**binomial**system made it possible to write and speak with accuracy of any given species of plant or animal. **BINOMIAL**(from the Lat.- 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. - 2 3 4 The
**binomial**theorem was thus discovered as a development of John Wallis's investigations in the method of interpolation. - - For the history of the
**binomial**theorem, see John Collins, Commercium Epistolicum (1712); S. - It is to be noticed that each number is the sum of the numbers immediately 35 above and to the left of it; and 35 that the numbers along a line, termed a base, which cuts off an equal number of units along the top row and column are the co efficients in the
**binomial**expansion of (I+x) r - 1, where r represents the number of units cut off. - These ideas are further developed in various papers in the Bulletin and in his L'Anthropometrie, ou mesure des differentes facultes de l'homme (18'ji), in which he lays great stress on the universal applicability of the
**binomial**law, - according to which the number of cases in which, for instance, a certain height occurs among a large number of individuals is represented by an ordinate of a curve (the**binomial**) symmetrically situated with regard to the ordinate representing the mean result (average height). - - It is obvious that the Linnaean
**binomial**terminology and its subsequent trinomial refinement for species, sub-species, and varieties was adapted to express the differences between animals as they exist to-day, distributed contemporaneously over the surface of the earth, and that it is wholly inadapted to express either the minute gradations of successive generic series or the branchings of a genetically connected chain of life. - (1) Ile /3 c Tou irvpiov, On the Burning-Glass, where the focal properties of the parabola probably found a place; (2) Hepi On the Cylindrical Helix (mentioned by Proclus); (3) a comparison of the dodecahedron and the icosahedron inscribed in the same sphere; (4) `H Ka06Xov lrpa-yµareta, perhaps a work on the general principles of mathematics in which were included Apollonius' criticisms and suggestions for the improvement of Euclid's Elements; (5) ' (quick bringing-to-birth), in which, according to Eutocius, he showed how to find closer limits for the value of 7r than the 37 and 3,4-A of Archimedes; (6) an arithmetical work (as to which see Pappus) on a system of expressing large numbers in language closer to that of common life than that of Archimedes' Sand-reckoner, and showing how to multiply such large numbers; (7) a great extension of the theory of irrationals expounded in Euclid, Book x., from
**binomial**to multinomial and from ordered to unordered irrationals (see extracts from Pappus' comm. - Linnaeus' invention of
**binomial**nomenclature for designating species served systematic biology admirably, but at the same time, by attaching preponderating importance to a particular grade in classification, crystallized the doctrine of fixity. **Binomial**Theorem 11.2.3 116.- More generally, if we have obtained a as an approximate value for the pth root of N, the
**binomial**theorem gives as an approximate formula p,IN =a+6, where N = a P + pap - 19.