# Fractional Sentence Examples

When, by practice with logarithms, we become familiar with the correspondence between additions of length on the logarithmic scale (on a slide-rule) and multiplication of numbers in the natural scale (including

**fractional**numbers), A /5 acquires a definite meaning as the number corresponding to the extremity of a length x, on the logarithmic scale, such that 5 corresponds to the extremity of 2X.Of especial importance is Sidney Young,

**Fractional**Distillation (1903).A rigorous demonstration was wanting for many years, Leonhard Euler's proof for negative and

**fractional**values being faulty, and was finally given by Niels Heinrik Abel.The logarithms to base io of the first twelve numbers to 7 places of decimals are log 1 =0.0000000 log 5 log 2 = 0.3010300 log 6 log 3 =0.477 121 3 log 7 log 4 =0.6020600 log 8 The meaning of these results is that The integral part of a logarithm is called the index or characteristic, and the

**fractional**part the mantissa.**Fractional**silver coin is not legal tender above 20 pesos, and bronze and nickel coins not above 1 peso, but the government maintains conversion offices where such coins can be converted into silver pesos without loss.The following table gives the values of this constant and several expressions involving it Useful

**fractional**approximations are 22/7 and 355/113.The inactive mixture may be resolved into its active components by

**fractional**crystallization of the cinchonine salt, when the salt of the dextro modification separates first; or the ammonium salt may be fermented by Penicillium glaucum, when the laevo form is destroyed and the dextro form remains untouched; on the other' hand, Saccharomyces ellipsoideus destroys the dextro form, but does not touch the laevo form.Equations with

**Fractional**Coefficients.-As an example of a special form of equation we may take zx+ 3x = Io.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.By

**fractional**distillation is meant the separation of a mixture having components which boil at neighbouring temperatures.AdvertisementThe 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.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.**Fractional**portions of time were gauged by shadows, and time of day indicated by the position of the sun with reference to natural features.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.AdvertisementFor agreement with the terminology of

**fractional**numbers (§ 62) we shall describe such a quantity as a mixed quantity.To express the quantity in terms of £, it ought (20) (12) to be written £254 13 v 6; this would mean £254 1202 or £(254+20 20 + 6 12), and therefore would involve a

**fractional**number.Diagram of

**Fractional**Relation.-To find 17 of 14s.We can, however, denote the result of the process by a symbol, and deal with this symbol according to the laws of arithmetic. In this way we arrive at (i) negative numbers, (ii)

**fractional**numbers, (iii) surds, (iv) logarithms (in the ordinary sense of the word).The expressions 4 r are then

**fractional**numbers, their relation to n' n' n' ' ' ordinary or integral numbers being that /2 n times n times is equal to p times.AdvertisementHence we can treat the

**fractional**numbers which have any one denominator as 0 o constituting a number-series, as shown in the 2 adjoining diagram.A

**fractional**number is called a proper fraction or an improper fraction according as the numerator is or is not 3 less than the denominator; and an expression 4 such as 24 is called a mixed number.The series of halves 2 in the one case, and of thirds in the I O I other, are entirely different series of 2

**fractional**numbers, but we can corn ‚ 0 0 pare them by putting each in its proper position in relation to the series of sixths.We can thus find

**fractional**numbers equivalent to the sum or difference of any two**fractional**numbers.Generally, to find the sum or difference of two or more

**fractional**numbers, we must replace them by other**fractional**numbers having the same denominator; it is usually most convenient to take as this denominator the L.C.M.AdvertisementThe rule for multiplying a

**fractional**number by a**fractional**number is therefore the same as the rule for finding a fraction of a fraction.These debased currencies are usually at a premium over gold owing to the extreme scarcity of

**fractional**coinage.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.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.These constructed symbols may be called positive and negative coefficients; or a symbol such as (- p) may be called a negative number, in the same way that we call 3 a

**fractional**number.Also (d +1) mannonic acid can be split into the d and 1 acids by

**fractional**crystallization of the strychnine or brucine salts.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.If a complex mixture be operated upon, and a separation effected by collecting the distillates in several portions, the operation is termed "

**fractional**distillation."He extended the "law of continuity" as stated by Johannes Kepler; regarded the denominators of fractions as powers with negative exponents; and deduced from the quadrature of the parabola y=xm, where m is a positive integer, the area of the curves when m is negative or

**fractional**.Grienberger, using Snell's method, calculated the ratio correct to 39

**fractional**places. ?The separation of the mixed bases so obtained is effected by repeated

**fractional**crystallization, or by taking advantage of certain properties of the constituents.By Reason Also Of The

**Fractional**Excess Of The Length Of The Year Above 365 Days, It Likewise Happens That The Years Cannot All Contain The Same Number Of Days If The Epoch Of Their Commencement Remains Fixed; For The Day And The Civil Year Must Necessarily Be Considered As Beginning At The Same Instant; And Therefore The Extra Hours Cannot Be Included In The Year Till They Have Accumulated To A Whole Day.Lacombe in 1904 obtained the pure salts by

**fractional**crystallization of the nitric acid solution with magnesium nitrate in the presence of bismuth nitrate.From the tar distillate, the chrysene can be fractionally precipitated, and the fluoranthene can be separated from most of the pyrene by

**fractional**distillation in a partial vacuum.In either case the two hydrocarbons are finally separated by

**fractional**crystallization of their picrates, which are then decomposed by ammonia.The value of the silver peso in

**fractional**silver money is about nineteen pence; in a single coin about twenty pence.The silver coins are the peso of loo centavos and its

**fractional**parts of 20, 10 and 5 centavos.Antimony trichloride ("Butter of Antimony"), SbCl 31 is obtained by burning the metal in chlorine; by distilling antimony with excess of mercuric chloride; and by

**fractional**distillation of antimony tetroxide or trisulphide in hydrochloric acid solution.To check the exportation of silver coin, the

**fractional**denominations have been slightly debased.It is found among the products formed in the destructive distillation of wood, sugar, cellulose, &c., and for this reason it is always present in crude wood spirit, from which the greater portion of it may be recovered by

**fractional**distillation.The sulphates of the alkaloids thus obtained are not equally soluble in water, and the quinine sulphate can be separated by

**fractional**crystallization, being less soluble in water than the other sulphates.Phosphorus trichloride or phosphorous chloride, PC13, discovered by Gay-Lussac and Thenard in 1808, is obtained by passing a slow current of chlorine over heated red phosphorus or through a solution of ordinary phosphorus in carbon disulphide (purifying in the latter case by

**fractional**distillation).Thus to divide by a

**fractional**number we must multiply by the number obtained by interchanging the numerator and the denominator, i.e.This definition applies whether the original number is integral or

**fractional**.By means of the present and the preceding sections the rule given in § 63 can be extended to the statement that a

**fractional**number is equal to the number obtained by multiplying its numerator and its denominator by any**fractional**number.We can obtain negative

**fractional**numbers in the same way that we obtain negative integral numbers; thus - 4 or - 7A means that 4 or 7A is taken negatively.A

**fractional**number may be regarded as the result of a measuring division (§ 39) which cannot be performed exactly.Hence, if p and n are numbers, n is sometimes regarded as denoting the result of dividing p by n, whether p and n are integral or

**fractional**(mixed numbers being included in**fractional**).The idea and properties of a

**fractional**number having been explained, we may now call it, for brevity, a fraction.A mixed number, the

**fractional**part of which is a decimal fraction, is expressed by writing the integral part in front of the point, which is called the decimal point.Hence we may insert in the power-series numbers with

**fractional**indices, provided that the multiplication of these numbers follows the same law.To add or subtract

**fractional**numbers, we must reduce them to a common denominator; and similarly, to multiply or divide surds, we must express them as power-numbers with the same index.In treating a

**fractional**number, or the corresponding value of the quantity in the second column, as intermediate, we are in effect regarding the numbers 1, 2, 3,..The solubility of naphthalene by various oils has led some engineers to put in naphthalene washers, in which gas is brought into contact with a heavy tar oil or certain fractions distilled from it, the latter being previously mixed with some volatile hydrocarbon to replace in the gas those illuminating vapours which the oil dissolves out; and by

**fractional**distillation of the washing oil the naphthalene and volatile hydrocarbons are afterwards recovered.In June 1904, under the terms of an agreement with the American Secretary of War, Panama adopted the gold standard with the balboa, equivalent to an American gold dollar, as the unit; and promised to keep in a bank in the United States a deposit of American money equal to 15% of its issue of

**fractional**silver currency, which is limited to four and a half million balboas.At the same time it could only be but a

**fractional**part - as indeed the whole country under consideration could only be - of the Seistan of Persian history.In 1867 Rees Reece, taking advantage of the fact that two vapours of different boiling-points, when mixed, can be separated by means of

**fractional**condensation, brought out an absorption machine in which the distillate was very nearly anhydrous.Throughout most of a core the

**fractional**ionization depends on the**fractional**elemental abundances of sulfur and metals such as Na and Mg.If the keyword

**FRACTIONAL**is present, the translation is assumed to be in**fractional**coordinates, otherwise orthogonal angstroms.Designs include linear phase filters, a wideband phase shifter, and chirp filters with

**fractional**bandwidths exceeding 100% .The angular integral of the reduced matrix element is expressed in terms of recoupling coefficients and coefficients of

**fractional**parentage.Give examples of how the products of

**fractional**distillation of crude oil can be used.If val has more

**fractional**digits than can be represented in the DynFixed,**fractional**digits than can be represented in the DynFixed,**fractional**digits are truncated and the return value is false.Whether or not

**fractional**reserve banking was begun with the knowledge of the depositors, it soon became the way most banks did business.But if the difference is only

**fractional**, how will it help first-time buyers?A string looping through the hole can lead to massive particles with

**fractional**electric charge or to massive magnetic monopoles with multiple magnetic charges.The whole-number part of a logarithm is called the characteristic; the

**fractional**part is called the mantissa.From 1980, they were also produced in

**fractional**sizes of half ounce, quarter ounce, and tenth ounce.For example, consider the radix character, the symbol used to separate the integer portion of a number from the

**fractional**portion.For smaller and branch locations, it also supports 4 or 8 port analog and

**fractional**T1, E1 and PRI.The pulse echo waveform has a

**fractional**bandwidth of 50% .Under these conditions, it is no longer possible for the individual merchant, or for small groups of merchants, to acquaint themselves, by personal experience alone, with more than a

**fractional**part of the causes which affect the business in which they are engaged.Arithmetic, algebra, and the infinitesimal calculus, are sciences directly concerned with integral numbers, rational (or

**fractional**) numbers, and real numbers generally, which include incommensurable numbers.Many comparisons of the effectiveness of dephlegmating columns have been made (see Sidney Young,

**Fractional**Distillation, 1903).The hydrocarbon occurs in wood-tar and in petroleum, and is prepared commercially by

**fractional**distillation of the light oil fraction of the coal-tar distillate (see Coal Tar).This crude pyroligneous acid is then refined by

**fractional**distillation to yield a food grade version that is suitable for use in health products.Less than 2 years will be reimbursed on a

**fractional**basis.Stub matching is only desirable for relatively low

**fractional**bandwidths.The pulse echo waveform has a

**fractional**bandwidth of 50 %.In 1910, the invention and production of the

**fractional**horsepower electric motor spurred the development of small kitchen appliances.**Fractional**laser therapy treats older, white stretch marks.Another name for this procedure is

**fractional**laser resurfacing.Fraxel laser therapy is the latest technology in

**fractional**laser therapy.Of artificial productions the most fruitful and important is provided by the destructive or dry distillation of many organic substances; familiar examples are the distillation of coal, which yields ordinary lighting gas, composed of gaseous hydrocarbons, and also coal tar, which, on subsequent

**fractional**distillations, yields many liquid and solid hydrocarbons, all of high industrial value.Its fundamental principle is that, by a combination of glass scales with a micrometer screw, " the chief part of the distance to be measured is read off on the scale; the

**fractional**part of the scalespace is not estimated but measured by the screw."They may be separated by

**fractional**distillation.The next step is to suppose that

**fractional**numbers are represented in the same way.These figures are, with a

**fractional**increase in the Regular Army, applicable to-day.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.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 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.In the last twelve years of the 19th century the altilik currency was almost entirely withdrawn, and replaced by

**fractional**mejidie; a large proportion of the beshlik has also been withdrawn, but the metallik has not been touched.This layer is separated, and the phenol recovered by a process of

**fractional**distillation.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.Among the great variety of problems solved are problems leading to determinate equations of the first degree in one, two, three or four variables, to determinate quadratic equations, and to indeterminate equations of the first degree in one or more variables, which are, however, transformed into determinate equations by arbitrarily assuming a value for one of the required numbers, Diophantus being always satisfied with a rational, even if

**fractional**, result and not requiring a solution in integers.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.