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.
(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.