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.

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.

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.

For 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.

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.

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.

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.

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.