Fractional sentence examples

fractional
  • is correct to the sixth fractional place.

    19
    14
  • They may be separated by fractional distillation.

    14
    15
  • Negative Quantities and Fractional Numbers.

    10
    8
  • PHENANTHRENE, C14H10, a hydrocarbon isomeric with anthracene, with which it occurs in the fraction of the coal tar distillate boiling between 270°-400° C. It may be separated from the anthracene oil by repeated fractional distillation, followed by fractional crystallization from alcohol (anthracene being the less soluble), and finally purified by oxidizing any residual anthracene with potassium bichromate and sulphuric acid (R.

    7
    6
  • 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.

    7
    7
  • 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.

    6
    5
  • These figures are, with a fractional increase in the Regular Army, applicable to-day.

    6
    10
  • In the last twelve years of the 19th century the altilik currency was almost entirely withdrawn, and replaced by fractional mejidie; a large proportion of the beshlik has also been withdrawn, but the metallik has not been touched.

    6
    13
  • This layer is separated, and the phenol recovered by a process of fractional distillation.

    6
    14
  • Petroleum spirit is tested for specific gravity, range of boilingpoints, and results of fractional distillation.

    6
    18
  • 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.

    5
    1
  • The extension of the methods to fractional numbers is part of the establishment of the laws governing these numbers (� 27 (ii.)).

    5
    2
  • Equations with Fractional Coefficients.-As an example of a special form of equation we may take zx+ 3x = Io.

    5
    4
  • such that 7 X 1 s = is., and can thence pass to the idea of fractional numbers.

    5
    4
  • Consideration of the binomial theorem for fractional index, or of the continued fraction representing a surd, or of theorems such as Wallis's theorem (� 64), shows that a sequence, every term of which is rational, may have as its limit an irrational number, i.e.

    5
    4
  • 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.

    5
    4
  • Sci., 1909 (iv.), 28, p. 347) claim to have separated two substances (of atomic weights 126.49 and 128.85 respectively) from tellurium, by fractional precipitation of tellurium chloride with water, but in the opinion of H.

    5
    4
  • 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.

    5
    5
  • The next step is to suppose that fractional numbers are represented in the same way.

    5
    8
  • (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.

    5
    11
  • (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.

    4
    3
  • The subject is here treated under the following subdivisions: (I) ordinary distillation, (2) distillation under reduced pressure, (3) fractional distillation, (4) distillation with steam, (5) theory of distillation, (6) dry distillation, (7) distillation in chemical technology and (8) commercial distillation of water.

    4
    3
  • Of especial importance is Sidney Young, Fractional Distillation (1903).

    4
    3
  • 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.

    4
    3
  • 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.

    4
    3
  • The following table gives the values of this constant and several expressions involving it Useful fractional approximations are 22/7 and 355/113.

    4
    3
  • rend., 1904 seq.) by fractional crystallization of the nickel double nitrates, the ethyl sulphates, and the bismuth double nitrates of the terbium earths.

    4
    4
  • 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.

    4
    4
  • 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."

    4
    6
  • the idea of (-5) as a number with which we can perform such operations as multiplication comes later (� 49)� (ii.) On the other hand, the conception of a fractional number follows directly from the use of fractions, involving the subdivision of a unit.

    4
    7
  • The month may be divided in two ways: a fractional part may be taken (decad or pentad), as in East Africa or Ancient Egypt (moon-week), or the week may be settled without regard to the length of the month (market-week, &c.).

    4
    8
  • 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.

    4
    11
  • 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.

    4
    11
  • 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.

    4
    12
  • 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.

    4
    13
  • 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.

    4
    13
  • 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.

    4
    14
  • 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.

    3
    2
  • (vii.) The use of fractional indices follows directly from this parallelism.

    3
    2
  • 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.

    3
    2
  • Fractional Expressions.

    3
    2
  • (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 (� 46 (ii.)).

    3
    2
  • Also (d +1) mannonic acid can be split into the d and 1 acids by fractional crystallization of the strychnine or brucine salts.

    3
    2
  • 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.

    3
    2
  • If a complex mixture be operated upon, and a separation effected by collecting the distillates in several portions, the operation is termed "fractional distillation."

    3
    2
  • Anschiitz, independently of one another, in troduced "distillation under reduced pressure"; and "fractional distillation" was greatly aided by the columns of Wurtz (1855), E.

    3
    2
  • By fractional distillation is meant the separation of a mixture having components which boil at neighbouring temperatures.

    3
    2
  • The theory of fractional distillation, or the behaviour of liquid mixtures when heated to their boiling-points, is more complex.

    3
    2
  • 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.

    3
    2
  • 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.

    3
    2
  • 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.

    3
    2
  • These debased currencies are usually at a premium over gold owing to the extreme scarcity of fractional coinage.

    3
    3
  • 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.

    3
    3
  • (v.) Permutations and Combinations may be regarded as arithmetical recreations; they become important algebraically in reference to the binomial theroem (�� 41, 44)� (vi.) Surds and Approximate Logarithms. - From the arithmetical point of view, surds present a greater difficulty than negative quantities and fractional numbers.

    3
    3
  • (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.

    3
    6
  • (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 (� 64).

    3
    6
  • 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.

    3
    10
  • 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.

    3
    11
  • 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.

    3
    11
  • Fractional portions of time were gauged by shadows, and time of day indicated by the position of the sun with reference to natural features.

    2
    1
  • 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.

    2
    1
  • Division of Fractional Numbers 6.4 67.

    0
    0
  • Negative Fractional Numbers 6.5 68.

    0
    0
  • Genesis of Fractional Numbers 6.6 76.

    0
    0
  • 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.

    0
    0
  • For agreement with the terminology of fractional numbers (§ 62) we shall describe such a quantity as a mixed quantity.

    0
    0
  • Diagram of Fractional Relation.-To find 17 of 14s.

    0
    0
  • 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).

    0
    0
  • 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.

    0
    0
  • Hence 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.

    0
    0
  • 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.

    0
    0
  • Fractional Numbers with different Denominators.

    0
    0
  • 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.

    0
    0
  • We can thus find fractional numbers equivalent to the sum or difference of any two fractional numbers.

    0
    0
  • 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.

    0
    0
  • of the original fractional numbers (cf.

    0
    0
  • - A fraction (or fractional number), the numerator or denominator of which is a fractional number, is called a complex fraction (or fractional number), to distinguish it from a simple fraction, which is a fraction having integers for numerator and denominator.

    0
    0
  • The rule for multiplying a fractional number by a fractional number is therefore the same as the rule for finding a fraction of a fraction.

    0
    0
  • (v.) The use of the solidus / separating two numbers is for convenience of printing fractions or fractional numbers.

    0
    1
  • 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.

    0
    1
  • 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.

    0
    1
  • 6 X 2 16) sides, he proved that the ratio was > 3.1415926535 and < 3' 14159 26 537, so that its value became known (in 1579) correctly to io fractional places.

    0
    1
  • Grienberger, using Snell's method, calculated the ratio correct to 39 fractional places. ?

    0
    1
  • In 1699 Abraham Sharp, on the suggestion of Edmund Halley, took Gregory's series, and, putting tan 0=11/3, found the ratio equal to 1 112 (I { 5 32 - 7 33 +...), from which he calculated it correct to 71 fractional places.'

    0
    1
  • The separation of the mixed bases so obtained is effected by repeated fractional crystallization, or by taking advantage of certain properties of the constituents.

    0
    1
  • 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.

    0
    1
  • Lacombe in 1904 obtained the pure salts by fractional crystallization of the nitric acid solution with magnesium nitrate in the presence of bismuth nitrate.

    0
    1
  • not even avoid fractional exponents ("Racine cubique de serait en circle"), and is ignorant only of negative exponents.

    0
    1
  • 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.

    0
    1
  • In either case the two hydrocarbons are finally separated by fractional crystallization of their picrates, which are then decomposed by ammonia.

    0
    1
  • civ., 1881, p. 450) in which barium monoxide is heated in a current of air, forming the dioxide, which when the retorts are exhausted yields up oxygen and leaves a residue of monoxide; but this method is now being superseded, its place being taken by the fractional distillation of liquid air (The Times, Engin.

    0
    1
  • The table gives their partial dispersions for six different regions of the spectrum also expressed (in brackets below) as fractional parts of the dispersion from C to F.

    0
    1
  • The value of the silver peso in fractional silver money is about nineteen pence; in a single coin about twenty pence.

    0
    1
  • The silver coins are the peso of loo centavos and its fractional parts of 20, 10 and 5 centavos.

    0
    1
  • 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.

    0
    1
  • To check the exportation of silver coin, the fractional denominations have been slightly debased.

    0
    1
  • It is obtained from the higher boiling fractions, after separation of naphthalene and anthracene, by fractional distillation, the portion boiling between 290-340° C. being taken.

    0
    1
  • 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.

    0
    1
  • 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.

    0
    1
  • 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).

    0
    1
  • Three isomeric hydrocarbons of this formula exist; they occur in the light oil fraction of the coal: tar distillate, but they cannot be separated by fractional distillation owing to the closeness of their boiling points.

    0
    1
  • 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.

    0
    1
  • Fractional Numbers 6.2 65.

    0
    1
  • Multiplication of Fractional Numbers 6.3 66.

    0
    1
  • a fractional number) x such that 4 times x is equal tot But jtimes 7 times x is, by the last section, equal to x.

    0
    1
  • Thus to divide by a fractional number we must multiply by the number obtained by interchanging the numerator and the denominator, i.e.

    0
    1
  • This definition applies whether the original number is integral or fractional.

    0
    1
  • 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.

    0
    1
  • 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.

    0
    1
  • A fractional number may be regarded as the result of a measuring division (§ 39) which cannot be performed exactly.

    0
    1
  • 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).

    0
    1
  • The idea and properties of a fractional number having been explained, we may now call it, for brevity, a fraction.

    0
    1
  • 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.

    0
    1
  • Hence we may insert in the power-series numbers with fractional indices, provided that the multiplication of these numbers follows the same law.

    0
    1
  • By analogy with the notation of fractional numbers, N will be = N X N; and, generally, NI will mean the product of p numbers, the product of q of which is equal to N.

    0
    1
  • 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.

    0
    1
  • £I same method can be applied in other cases by regarding a quantity expressed in several denominations as a fractional number of units of the largest denomination mentioned; thus 7s.

    0
    1
  • fractional or decimal) numbers not shown in the table.

    0
    1
  • 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,..

    0
    1
  • 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.

    0
    1
  • With the exception of camphene, all the terpenes are liquids, boiling approximately between 160° and 190° C., so that it is almost impossible to separate them from the various essential oils by fractional distillation.

    0
    1
  • 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.

    0
    1
  • 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.

    0
    1
  • 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.

    0
    1
  • The pure acid may be obtained by fractional distillation as a colourless liquid of very unpleasant smell, boiling at 30° C., and extremely explosive.

    0
    1
  • Throughout most of a core the fractional ionization depends on the fractional elemental abundances of sulfur and metals such as Na and Mg.

    0
    1
  • If the keyword FRACTIONAL is present, the translation is assumed to be in fractional coordinates, otherwise orthogonal angstroms.

    0
    1
  • Designs include linear phase filters, a wideband phase shifter, and chirp filters with fractional bandwidths exceeding 100% .

    0
    1
  • fractional reserve banking has, since the outset of western industrialization, initiated production for the purpose of trade and exchange.

    0
    1
  • The angular integral of the reduced matrix element is expressed in terms of recoupling coefficients and coefficients of fractional parentage.

    0
    1
  • coefficients of fractional parentage.

    0
    1
  • Give examples of how the products of fractional distillation of crude oil can be used.

    0
    1
  • fractional entitlements to Offer Shares will not be allotted.

    0
    1
  • fractional factorials with factors at two and/or four levels.

    0
    1
  • fractional distillation of crude oil can be used.

    0
    1
  • fractional parentage (cfp) cards can be taken with their original format from Comp.

    0
    1
  • fractional bandwidth only gives us 5 TV channels of 10 MHz each.

    0
    1
  • fractional coordinates are assumed in which case the nearest image to the defect center is taken.

    0
    1
  • 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.

    0
    1
  • Whether or not fractional reserve banking was begun with the knowledge of the depositors, it soon became the way most banks did business.

    0
    1
  • But if the difference is only fractional, how will it help first-time buyers?

    0
    1
  • lieu of fractional entitlements.

    0
    1
  • A string looping through the hole can lead to massive particles with fractional electric charge or to massive magnetic monopoles with multiple magnetic charges.

    0
    1
  • The whole-number part of a logarithm is called the characteristic; the fractional part is called the mantissa.

    0
    1
  • From 1980, they were also produced in fractional sizes of half ounce, quarter ounce, and tenth ounce.

    0
    1
  • Unusual features: The input coefficients of fractional parentage (cfp) cards can be taken with their original format from Comp.

    0
    1
  • For example, consider the radix character, the symbol used to separate the integer portion of a number from the fractional portion.

    0
    1
  • pyroligneous acid is then refined by fractional distillation to yield a food grade version that is suitable for use in health products.

    0
    1
  • Abstract R.N. Palmer and D. Jaksch, High field fractional quantum Hall effect in optical lattices, Phys.

    0
    1
  • radix character, the symbol used to separate the integer portion of a number from the fractional portion.

    0
    1
  • reimbursed on a fractional basis.

    0
    1
  • stub matching is only desirable for relatively low fractional bandwidths.

    0
    1
  • For smaller and branch locations, it also supports 4 or 8 port analog and fractional T1, E1 and PRI.

    0
    1
  • The pulse echo waveform has a fractional bandwidth of 50% .

    0
    1
  • 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.

    0
    1
  • 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.

    0
    1
  • 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.

    0
    1
  • PHENANTHRENE, C14H10, a hydrocarbon isomeric with anthracene, with which it occurs in the fraction of the coal tar distillate boiling between 270°-400° C. It may be separated from the anthracene oil by repeated fractional distillation, followed by fractional crystallization from alcohol (anthracene being the less soluble), and finally purified by oxidizing any residual anthracene with potassium bichromate and sulphuric acid (R.

    0
    1
  • The extension of the methods to fractional numbers is part of the establishment of the laws governing these numbers (� 27 (ii.)).

    0
    1
  • The latter is the more advanced process, implying some knowledge of the laws of fractional numbers, as well as an application of the associative law (� 26 (i.)).

    0
    1
  • the idea of (-5) as a number with which we can perform such operations as multiplication comes later (� 49)� (ii.) On the other hand, the conception of a fractional number follows directly from the use of fractions, involving the subdivision of a unit.

    0
    1
  • (v.) Permutations and Combinations may be regarded as arithmetical recreations; they become important algebraically in reference to the binomial theroem (�� 41, 44)� (vi.) Surds and Approximate Logarithms. - From the arithmetical point of view, surds present a greater difficulty than negative quantities and fractional numbers.

    0
    1
  • (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 (� 64).

    0
    1
  • (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 (� 46 (ii.)).

    0
    1
  • Consideration of the binomial theorem for fractional index, or of the continued fraction representing a surd, or of theorems such as Wallis's theorem (� 64), shows that a sequence, every term of which is rational, may have as its limit an irrational number, i.e.

    0
    1
  • Many comparisons of the effectiveness of dephlegmating columns have been made (see Sidney Young, Fractional Distillation, 1903).

    0
    1
  • In the case of fractional numbers (i.e.

    0
    1
  • Carlyle refers to the opinion of genealogists that Cromwell "was indubitably either the ninth or the tenth or some other fractional part of half a cousin of Charles Stuart," but this has been completely exploded by Walter Rye in the Genealogist (" The Steward Genealogy and Cromwell's Royal Descent," new series, vol.

    0
    1
  • It is obtained from the higher boiling fractions, after separation of naphthalene and anthracene, by fractional distillation, the portion boiling between 290-340° C. being taken.

    0
    1
  • Phosphorus combines with nitrogen and chlorine to form several polymeric substances of the general formula (PNC1 2) x, where x may be 1, 3, 4, 5, 6, 7, or 11; they may be obtained by heating the pentachloride with ammonium chloride in a sealed tube and separating the mixture by fractional distillation (H.

    0
    1
  • Thus 2 is equal to -, and a is equal to -16°, and conversely; in other words, any fractional number is equivalent to the fractional number obtained by multiplying or dividing the numerator and denominator by any integer.

    0
    1
  • 100.00 These elements in tar are built up into an enormous number of compounds (see Coal Tar), and its value as a by-product may be gathered from the fact that on fractional distillation it yields - (I) benzene and its homologues, from which aniline, the source of most of the coal-tar colours, can be derived; (2) carbolic acid, from which picric acid, used as a dye, a powerful explosive, and to give the bitter flavour to some kinds of beer, is made, also many most valuable disinfectants; (3) naphthalene, used for disinfecting, and also as the "Albo-carbon" employed in an enriching burner for gas; (4) pitch, extensively used in path-making, from which such bodies as anthracene and saccharin can be extracted.

    0
    1
  • With the exception of camphene, all the terpenes are liquids, boiling approximately between 160° and 190° C., so that it is almost impossible to separate them from the various essential oils by fractional distillation.

    0
    1
  • 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).

    0
    1
  • The pure acid may be obtained by fractional distillation as a colourless liquid of very unpleasant smell, boiling at 30° C., and extremely explosive.

    0
    1
  • This crude pyroligneous acid is then refined by fractional distillation to yield a food grade version that is suitable for use in health products.

    0
    1
  • Less than 2 years will be reimbursed on a fractional basis.

    0
    1
  • Stub matching is only desirable for relatively low fractional bandwidths.

    0
    1
  • The pulse echo waveform has a fractional bandwidth of 50 %.

    0
    1
  • In 1910, the invention and production of the fractional horsepower electric motor spurred the development of small kitchen appliances.

    0
    1
  • Fractional laser therapy treats older, white stretch marks.

    0
    1
  • Another name for this procedure is fractional laser resurfacing.

    0
    1
  • Fraxel laser therapy is the latest technology in fractional laser therapy.

    0
    1
Browse other sentences examples →