# Binomial Sentence Examples

- Determinants composed of
**binomial**coefficients have been studied by V. - Relation of
**Binomial**Coefficients to Summation of Series. - 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. - - For the history of the
**binomial**theorem, see John Collins, Commercium Epistolicum (1712); S. - - 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. - 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). - 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. **Binomial**name Equus caballus Linnaeus, 1758 The horse (Equus caballus, sometimes seen as a subspecies of the Wild Horse, Equus ferus caballus) is a large odd-toed ungulate mammal, one of ten modern species of the genus Equus.- (0B) = (e), &c. The
**binomial**coefficients appear, in fact, as symmetric functions, and this is frequently of importance. - (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. - 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. - (Avenarius, 1863.) (General type.) (Becquerel, 1863.) (Tait, 1870.) (Barus, 1889.) (Holborn and Wien, 1892.) (Paschen, 3893.) (Steele, 1894.) (Holman, 3896.) (Stanfield, 1898.) (Holborn and Day, 1899.) See sec. 15.) For moderate ranges of temperature the
**binomial**formula of M. - 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. - This property enables us to establish, by simple reasoning, certain relations between
**binomial**coefficients. - 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. - Linnaeus by his
**binomial**system made it possible to write and speak with accuracy of any given species of plant or animal. - 2 3 4 The
**binomial**theorem was thus discovered as a development of John Wallis's investigations in the method of interpolation. **Binomial**Theorem 11.2.3 116.- 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. - 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. - Algebraic Forms;
**Binomial**; Combinatorial Analysis; Determin Ants; Equation; Continued Fraction; Function; Theory of groups; Logarithm; Number; Probability; Series. - 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**. - - 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. - (v.) It should be mentioned that the notation of the
**binomial**'coefficients, and of the continued products such as n(n -1). - 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. - 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. - 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). - (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.)). - 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. - 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. - 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+... - A multinomial consisting of two or of three terms is a
**binomial**or a trinomial. - This is the
**binomial**theorem for a positive integral index. - 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). - The
**binomial**theorem for positive integral index may then be written (x + y) n = -iyi +. - Application of
**Binomial**Theorem to Rational Integral Functions. - He introduced the sign (=) for equality, and the terms
**binomial**and residual. - 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. - 43ï¿½
**Binomial**Coefficients.