## Fractional Sentence Examples

- We find that fractions follow certain laws corresponding exactly with those of integral multipliers, and we are therefore able to deal with the
**fractional**numbers as if they were integers. - The caesium and rubidium are separated from this by repeated
**fractional**crystallization of their double platinum chlorides, which are much less soluble in water than those of the other alkali metals (R. - Obtained by
**fractional**distillation of the aqueous distillate, special precautions being necessary owing to the excessively poisonous nature of the free acid: K 4 Fe(NC),±3H 2 SO 4 = 2K2S04+FeS04+6HCN. - Is correct to the sixth
**fractional**place. - Rend., 1869, 69, p. 169) obtained the sodium salt by the action of zinc on a concentrated solution of sodium bisulphite: Zn + 4NaHSO 3 = Na 2 S 2 O 4 + ZnSO 3 + Na 2 SO 3 + 2H 2 O, the salt being separated from the sulphites formed by
**fractional**precipitation. - The membership in each house, however, is slightly above these figures, owing to a system of
**fractional**representation and to the constitutional amendment of 1903 which allows each county at least one representative in the House of Representatives. - It is generally understood that the products of
**fractional**distillation, even in the laboratory, are not identical with the hydrocarbons present in the crude oil, but are in part produced by the action of heat upon them. - In an experiment on 3500 grams of paraffin produced from shale (melting point 44'5° C.) they obtained nearly 4 litres of liquid hydrocarbons, which they subjected to
**fractional**distillation, and on examining the fraction distilling below loo° C., they found it to consist mainly of olefines. - In the routine examination of crude petroleum it is customary to determine the specific gravity, and the amount of water and earthy matter in suspension; the oil is also frequently subjected to a process of
**fractional**distillation in order to ascertain whether there has been any addition of distilled products or residue. - Petroleum spirit is tested for specific gravity, range of boilingpoints, and results of
**fractional**distillation. - In the separation of the constituents of the complex mixture of oxides obtained from the " rare earth " minerals, the methods generally forced upon chemists are those of
**fractional**precipitation or crystallization; the striking resemblances of the compounds of these elements rarely admitting of a complete separation by simple precipitation and filtration. - The
**fractional**mejidie coins (5, 2 and 1 piastres) are quoted at a separate rate in the market, usually at a premium over the 20-piastre piece. - The oxides are then converted into double sulphates which are separated from each other by repeated
**fractional**crystallization or by**fractional**precipitation with ammonia or some other base. - If µ be the molecular conductivity, and its value at infinite dilution, the
**fractional**number of molecules dissociated is k /µop, which we may write as a. - They may be separated by
**fractional**distillation. - The extension of the methods to
**fractional**numbers is part of the establishment of the laws governing these numbers (ï¿½ 27 (ii.)). - Equations with
**Fractional**Coefficients.-As an example of a special form of equation we may take zx+ 3x = Io. - (iii.) The general statement of the laws of operation of fractions is perhaps best deferred until we come to
**fractional**numbers, when letters can be used to express the laws of multiplication and division of such numbers. - Negative Quantities and
**Fractional**Numbers. - The idea of (-5) as a number with which we can perform such operations as multiplication comes later (ï¿½ 49)ï¿½ (ii.) On the other hand, the conception of a
**fractional**number follows directly from the use of fractions, involving the subdivision of a unit. - (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. - Thus the concrete fact required to enable us to pass arithmetically from the conception of a
**fractional**number to the conception of a surd is the fact of performing calculations by means of logarithms. - (vii.) The use of
**fractional**indices follows directly from this parallelism. - This suggests that we should write; la as a l l3; and we find that the use of
**fractional**indices in this way satisfies the laws of integral indices. **Fractional**Expressions.- (v.) The further extension to
**fractional**values (positive or negative) of n depends in the first instance on the establishment of a method of algebraical evolution which bears the same relation to arithmetical evolution (calculation of a surd) that algebraical division bears to arithmetical division. - (vi.) The definition of no.), which has already been extended in (iv.) above, has to be further extended so as to cover
**fractional**values of n, positive or negative. - (vii.) The comparison of the numerical value of I-Fn(1)x +n(242+...+n(r)xr, when n is
**fractional**, with that of (i+x)n, involves advanced methods (ï¿½ 64). - (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 next step is to suppose that
**fractional**numbers are represented in the same way. - The following table gives the values of this constant and several expressions involving it Useful
**fractional**approximations are 22/7 and 355/113. - A large number of their statements are rough (2, 18, 33), being based on the working equivalence of the bath or epha with the Attic metretes, from which are sometimes drawn
**fractional**statements which seem more accurate than they are. - Chemie, 1885, 6, 477) into two components (known respectively as neodymium and praseodymium) by repeated
**fractional**crystallization of the double nitrate of ammonium and didymium in nitric acid. **Fractional**portions of time were gauged by shadows, and time of day indicated by the position of the sun with reference to natural features.- This water cannot be entirely removed by
**fractional**distillation, and to prepare anhydrous or "absolute" alcohol the commercial product must be allowed to stand over some dehydrating agent, such as caustic lime, baryta, anhydrous copper sulphate, &c., and then distilled. - If a complex mixture be operated upon, and a separation effected by collecting the distillates in several portions, the operation is termed "
**fractional**distillation." - Anschiitz, independently of one another, in troduced "distillation under reduced pressure"; and "
**fractional**distillation" was greatly aided by the columns of Wurtz (1855), E. - By
**fractional**distillation is meant the separation of a mixture having components which boil at neighbouring temperatures. - The theory of
**fractional**distillation, or the behaviour of liquid mixtures when heated to their boiling-points, is more complex. - At one time it was thought that these mixtures of constant boiling-point (an extended list is given in Young's
**Fractional**Distillation) were definite compounds. - The crude cresol obtained from tar cannot be separated into its different constituents by
**fractional**distillation, since the boiling points of the three isomers are very close together.