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fractions

fractions Sentence Examples

  • Napier thus had complete command over decimal fractions and the use of the decimal point.

  • 2), with a float plate on its upper side, carrying three indicating dials, recording respectively fractions, units and tens of miles (up to a hundred).

  • For forty years after the death of its founder it remained united under the authority of a series of grand khans chosen from among his descendants, and then it began to fall to pieces till the various fractions of it became independent khanates.

  • AS, the Roman unit of weight and measure, divided into 12 unciae (whence both "ounce" and "inch"); its fractions being deunx i 2, dextans, dodrans 4 i bes 3, septunx T7-2-, semis z, quincunx A, triens 3 i quadrans 4, sextans s, sescuncia $, uncia r i g.

  • Beneath are the official Liverpool quotations of " futures," as they appeared on the morning of the 19th of April 1906: A merican Deliveries, any port, basis of middling, good ordinary clause (the fractions are given in moths of a penny).

  • In order to separate the distillate into various fractions, and to remove as much of it as possible free from condensed steam, it is now usual to employ condensing appliances of special form with outlets for running off the different fractions.

  • If we express the pressure, volume and temperature as fractions of the critical constants, then, calling these fractions the " reduced " pressure, volume and temperature, and denoting them by 7r, 0 and 0 respectively, the characteristic equation becomes (7+3/0 2)(30-i) =80; which has the same form for all substances.

  • Obviously, therefore, liquids are comparable when the pressures, volumes and temperatures are equal fractions of the critical constants.

  • For the subjects of this heading see the articles DIFFERENTIAL EQUATIONS; FOURIER'S SERIES; CONTINUED FRACTIONS; FUNCTION; FUNCTION OF REAL VARIABLES; FUNCTION COMPLEX; GROUPS, THEORY OF; INFINITESIMAL CALCULUS; MAXIMA AND MINIMA; SERIES; SPHERICAL HARMONICS; TRIGONOMETRY; VARIATIONS, CALCULUS OF.

  • The amounts of Turkish gold, silver and debased coinage in circulation are approximately £T16,500,000, in gold, £T8,70o,000 (940,000,000 piastres at 108) in silver mejidies and fractions, and 200,000,000 piastres in beshlik and metallik.

  • The Liber abaci, which fills 459 printed pages, contains the most perfect methods of calculating with whole numbers and with fractions, practice, extraction of the square and cube roots, proportion, chain rule, finding of proportional parts, averages, progressions, even compound interest, just as in the completest mercantile arithmetics of our days.

  • Leonardo, making use of fractions of the sexagesimal scale, gives X = I° 221 7 42" i 33 iv 4v 40 vi, after having demonstrated, by a discussion founded on the 10th book of Euclid, that a solution by square roots is impossible.

  • We have to multiply a01; -alas+a2 by ao, -aif32+a2 and we obtain ao (3 - aoal(f31N2 +01133) +aoa2(SI+13) -i-a?31a2 - aIa2(31 + 02) + al, 131+02 = b, 131132 = b t'i +s2 = 2bob2, and clearing of fractions R 1,5 = (a o b 2 - a2 b o) 2 + (a i b o - aobi) (aib2 - a2b1).

  • In astronomical practice the masses of the planets are commonly expressed as fractions of the mass of the sun, the latter being taken as unity.

  • Unfortunately these eclipses are not sudden but slowly changing phenomena, so that they cannot be observed without an error of at least several seconds, and not infrequently important fractions of a minute.

  • In algebra he discovered the method of approximating to the real roots of an equation by means of continued fractions, and imagined a general process of solving algebraical equations of every degree.

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

  • The deductions follow directly from the definitions, and such mechanical processes as "clearing of fractions " find no place (� 21 (ii.)).

  • (ii.) Perhaps the worst thing we can do, from the point of view of intelligibility, is to " clear of fractions " by multiplying both sides by 6.

  • (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.

  • (iii.) There are important theorems as to the relative value of fractions; e.g.

  • 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.

  • 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.

  • Thus, if we have an equation P=Q, where P and Q are numbers involving fractions, we can clear of fractions, not by multiplying P and Q by a number m, but by applying the equal multiples P and Q to a number m as unit.

  • There are extensions of the binomial theorem, by means of which approximate calculations can be made of fractions, surds, and powers of fractions and of surds; the main difference being that the number of terms which can be taken into account is unlimited, so that, although we may approach nearer and nearer to the true value, we never attain it exactly.

  • �; the successive terms of this series, after the first, are alternately positive and negative, and consist of fractions with numerators I and denominators continually increasing.

  • So persistently does the human ear rebel against the division of the tetrachord into two greater tones and a leimma or hemitone, as represented by the fractions 9, 9, 26, that, centuries before the possibility of reconciling the demands of the ear with those of exact science was satisfactorily demonstrated, the Aristoxenian school advocated the use of an empirical scale, sounding pleasant to the sense, in preference to an unpleasing tonality founded upon immutable proportions.

  • 2 We have given the fractions in the order in which they occur in the modern system.

  • Benzene is manufactured from the low-boiling fractions of the coal-tar distillate.

  • The fractions are agitated with strong sulphuric acid, and then washed with a caustic soda solution.

  • STYROLENE, C 6 H 5 CH:CH 2, also known as phenylethylene or vinylbenzene, an aromatic hydrocarbon found to the extent of 1 to 4% in storax; it also occurs with crude xylene in coal tar fractions.

  • Generally the components of a mixture will be vaporized in the order of their boiling-points; consequently if the condensates or "fractions" corresponding to definite ranges of temperature be separately collected, it is obvious that a more or less partial separation of the components will be effected.

  • For the purpose of collecting the distillates in fractions, many forms of receivers have been devised.

  • Dephlegmation of the vapours arising from such mixtures as coaltar fractions, petroleum and the "wash" of the spirit industry, is very important, and many types of apparatus are employed in order to effect a separation of the vapours.

  • Both the fractions into which they were divided by the Nerbudda river laid claim to antiquity: while the northern, however, did not trace their origin further back than the period of the early Mahommedan kings of Delhi, the southern fraction not only claimed an earlier and purer descent, but adhered also with greater strictness to the rules of their profession.

  • If each of the fractions (3) is put equal to i/4h, it is readily found, from the first property of the normal state, that, of the s molecules of the first kind, a number sal (h3m3 /13)e hm (u2+v2+w2)dudvdw (4) Velocities.

  • He studied at Berlin University, where he obtained the degree of doctor of philosophy in 1825, his thesis being an analytical discussion of the theory of fractions.

  • This was arranged for by a movable leaf carrying the sighting V, worked by means of a mill-headed screw provided with a scale in degrees and fractions to the same radius as the elevation scale, and an arrowb head for reading.

  • The Army of the North was to concentrate in three fractions - around Solre, Beaumont and Philippeville - as close to Charleroi as was practicable; and he arranged to screen the initial movements of the troops as much as possible, so as to prevent the allies from discovering in time that their centre was aimed at.

  • The time between the breaks could be measured in seconds by the clock signals, and in fractions of a second by the tuning-fork record.

  • The frequency ratios in the diatonic scale are all expressible either as fractions, with i, 2, 3 or 5 as numerator and denominator, or as products of such fractions; and it may be shown that for a given note the numerator and denominator are smaller than any other numbers which would give us a note in the immediate neighbourhood.

  • Thus, whilst the detachment was still disposed as a series of rearguards, the foremost fractions of it stood to fight on the Yalu, against odds of four to one.

  • On the side of the defence, each colonel had been left to retire as best he could, and thus certain fractions of the retreating Russians encountered Inouye's advancing troops and were destroyed after a most gallant resistance.

  • Calling in the brigade detached to the assistance of Nozu as well as all other available fractions of his scattered army, he himself attacked 1 The 5th division of the 2nd Army had been sent to join the 10th as the latter approached Hsimucheng.

  • The accuracy of a meter is tested by drawing calibration curves showing the percentage departure from absolute accuracy in its reading for various decimal fractions of full load.

  • was to divide his territories among his sons, whereby Poland was partitioned into no fewer than four, and ultimately into as many as eight, principalities, many of which (Silesia and Great Poland, for instance) in process of time split up into still smaller fractions all of them more or less bitterly hostile to each other.

  • 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.

  • (c) the standard gallon (and multiples and fractions of it), declared to contain 10 lb of water at 62° F., being in volume 277.274 cub.

  • (arithmetic) elementary lessons on the notation of decimal fractions.

  • 3 Besides his connexion with logarithms and improvements in the method of prosthaphaeresis, Byrgius has a share in the invention of decimal fractions.

  • Logistic or Proportional Logarithms. - The old name for what are now called ratios or fractions are logistic numbers, so that a table of log (a/x) where x is the argument and a a constant is called a table of logistic or proportional logarithms; and since log (a/x) =log a-log x it is clear that the tabular results differ from those given in an ordinary table of logarithms only by the subtraction of a constant and a change of sign.

  • be proper fractions, and the value of every one of the interminate continued fractions l!1 a2.

  • Thus, to take the latter one, if we suppose that of two editors of equal competence A requires a probability of four-fifths to admit a reading into his text and B a probability of three-fifths only, then in all the cases in which the probability lies between these two fractions B will be right seven times to A's three, while outside these limits there will be no difference between them.

  • Which Gives The Series Of Approximating Fractions, I 7 8 31 132 163 &C.

  • 2 an embolismic The result so obtained would in general be more accurate than the Jewish calculation, from which it may differ a day, as fractions of a day do not enter alike in these computations.

  • Recorde published several works upon mathematical subjects, chiefly in the form of dialogue between master and scholar, viz.: - The Grounde of Artes, teachinge the Worke and Practise of Arithmeticke, both in whole numbers and fractions (1540); The Pathway to Knowledge, containing the First Principles of Geometry.

  • Decimal fractions had been employed for the extraction of square roots.

  • The point separating the integers from the decimal fractions seems to be the invention of Bartholomaeus Pitiscus, in whose trigonometrical tables (1612) it occurs and it was accepted by John Napier in his logarithmic papers (1614 and 1619).

  • African mines the diamonds are not only crystals of various weights from fractions of a carat to 150 carats, but also occur as microscopic crystals disseminated through the blue ground.

  • The size of the angle between the median planes of two consecutive leaves in an alternate arrangement is their divergence; and it is expressed in fractions of the circumference of the axis which is supposed to be a circle.

  • Fractions except * were all primary, i.e.

  • with the numerator unity: in order to express such an idea as ~ the Egyptians were obliged to reduce it to a series of primary fractions through double fractions 1~+~1rt~r+1~w+ 1~ 4(1+

  • rabies and rule to facilitate the employment of fractions.

  • FLUORANTHENE, C15H10, also known as idryl, a hydrocarbon occurring with phenanthrene, pyrene, diphenyl, and other substances in "Stupp" fat (the fat obtained in working up the mercury ores in Idria), and also in the higher boiling fractions of the coal tar distillate.

  • Fractions of the day are disregarded to avoid dispute, though sometimes the law will consider fractions, as where it is necessary to show the first of two acts.

  • The law pays no regard to fractions of a day except to prevent injustice.

  • In the recording acts relating to real property, fractions of a day are of the utmost importance, and all deeds, mortgages and other instruments affecting the property, take precedence in the order in which they were filed for record.

  • The Aµ from C to F being taken as unity in each case then the AA's for the other regions of the spectrum are expressed in fractions A A (C to F) and are given under the asterisks.

  • Soc., 1904, 26, p. 922) divided the product into three fractions according to their volatility.

  • Should B or C, or both, be at once inconveniently large, and orime, then, instead of the exact ratio B/C some ratio anproxlmat:ng to that ratio, and capable of resolution into convenient factors, is to be found by the method of continued fractions.

  • Then, if possible, B and, C themselves are to be resolved each into rnI factors (counting 1 as a factor), which factors, or multiples of them, shall be not less than t nor greater than 6t; or if B and C contain inconveniently large prime factors, an approximate velocity ratio, found by the method of continued fractions, is to be substituted for B/C as before.

  • Let a denote the arc of contact expressed in turns and fractions of a turn; then O=62832a 6

  • Or, if n a,27- = a/6.2832 be the number of turns or fractions of a turn in a second, g 0.8165 ft.

  • At first it was only levied at irregular intervals; afterwards, in 378 B.C., it became a permanent tax based on elaborate valuation under which the richer members paid on a larger quota of their capital; in the case of the wealthiest class the taxable quota was taken as one-fifth, smaller fractions being adopted for those belonging to the other divisions.

  • The iron ore raised in the various countries, and in the most productive counties, is here shown: The number of furnaces in blast (fractions showing the proportion of the year furnaces were in blast) was: in England 298162, Wales 19,; Scotland 852, total 403 i '.

  • CONTINUED FRACTIONS.

  • If b 2 /a 2, 3 /a 3 ..., the component fractions, as they are called, recur, either from the commencement or from some fixed term, the continued fraction is said to be recurring or periodic. It is obvious that every terminating continued fraction reduces to a commensurable number.

  • a2 a3 a 4 Ia2 la3 la4 The terminating continued fractions b 2 b2 b 3 b, b, b4 al, a1 + ?2, al-1 7:2+Z' al' + a2 reduced to the forms ala2a3+b2a3+ b2a1 a2 a2a3 b3 ala2a3a4+b2a3a4+b3a,a4+b4a,a2 +b2b4 a2a,a4+a4b3+a2b4 ' are called the successive convergents to the general continued fraction.

  • Simple Continued Fractions.

  • The numerators and denominators of the successive convergents obey the law p n g n _ l - pn-1qn = (- O n, from which it follows at once that every convergent is in its lowest terms. The other principal properties of the convergents are The odd convergents form an increasing series of rational fractions continually approaching to the value of the whole continued fraction; the even convergents form a decreasing series having the same property.

  • The chief practical use of the simple continued fraction is that by means of it we can obtain rational fractions which approximate to any quantity, and we can also estimate the error of our b4 as a4 b5 approximation.

  • For the application of continued fractions to the problem " To find the fraction, whose denominator does not exceed a given integer D, which shall most closely approximate (by excess or defect, as may be assigned) to a given number commensurable or incommensurable," the reader is referred to G.

  • Chrystal's Algebra, where also may be found details of the application of continued fractions to such interesting and important problems as the recurrence of eclipses and the rectification of the calendar.

  • Lagrange used simple continued fractions to approximate to the solutions of numerical equations; thus, if an equation has a root between two integers a and a+1, put x=a+I/y and form the equation in y; if the equation in y has a root between b and b+i, put y = b + I /z, and so on.

  • The solution in integers of the indeterminate equation ax+by=c may be effected by means of continued fractions.

  • An interesting application of continued fractions to establish a unique correspondence between the elements of an aggregate of m dimensions and an aggregate of n dimensions is given by G.

  • Applications of simple continued fractions to the theory of numbers, as, for example, to prove the theorem that a divisor of the sum of two squares is itself the sum of two squares, may be found in J.

  • Recurring Simple Continued Fractions.

  • The Evaluation of Continued Fractions.

  • The Incommensurability of Infinite Continued Fractions.

  • The Conversion of Series and Products into Continued Fractions.

  • where a (a+I)y (a+n-I)(y+n-2) =y, (33= (Y+ I)(y+2),..., 132n-1 - (y+2n-3)(7+272-2)' (32= y-a, N4 = 2(y+I-a) ?..., N2 - 71(7-I-a) Y(' y + I) (7+2) (7+3)' (y+2n-2)(y+2nFrom this we may express several of the elementary series as continued fractions; thus taking a= I, 7=2, and putting x for -x, have log (I +x) = x I 2 x I 2 x 2 2 x 2 2 X 3 2 x 32x we +2 + 3 +4+5+ 6-I-7 +.

  • Ascending Continued Fractions.

  • The invention of continued fractions is ascribed generally to Pietro Antonia Cataldi, an Italian mathematician who died in 1626.

  • - For the further history of continued fractions we may refer the reader to two papers by Gunther and A.

  • For the application of continued fractions to the theory of O o O b 3 O O a3 b4 O - I a 4 b5 b2 irrational numbers there is P. Bachmann's Vorlesungen fiber die Natur der Irrationalzahnen (1892).

  • For the application of continued fractions to the theory of lenses, see R.

  • 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.

  • Fractions

  • Historical Development of Fractions and Decimals

  • Continued Fractions

  • Thus the fractions must be reduced to a common denominator.

  • This denominator must, if the fractions are in their lowest terms (§ 54), be a multiple of each of the denominators; it is usually most convenient that it should be their L.C.M.

  • we must regard the ratio of a to b as being the same as the ratio of c to d, if the fractions 12 b and d are equal.

  • Complex Fractions.

  • - In order to deal, by way of comparison or addition or subtraction, with fractions which have different denominators, it is necessary to reduce them to a common denominator.

  • To avoid this difficulty, in practical life, it is usual to confine our operations to fractions which have a certain standard denominator.

  • The modern method is to deal with fractions which have ioo as denominator; such fractions are called percentages.

  • Hence, to find a percentage of a percentage, we multiply the two numbers, put o's in front if necessary to make up four figures (not counting fractions), and prefix the point.

  • Decimal Fractions.

  • These two fractions are equal to each other, and also to 1530.

  • A fraction written in this way is called a decimal fraction; or we might define a decimal fraction as a fraction having a power of To for its denominator, there being a special notation for writing such fractions.

  • Fractions other than decimal fractions are usually called vulgar fractions.

  • The fractions used in ancient times were mainly of two kinds: unitfractions, i.e.

  • fractions representing aliquot parts (§ 103), and fractions with a definite denominator.

  • The Egyptians as a rule used only unit-fractions, other fractions being expressed as the sum of unit-fractions.

  • The Greeks originally used unit-fractions, like the Egyptians; later they introduced the sexagesimal fractions of the Babylonians, extending the system to four or more successive subdivisions of the unit representing a degree.

  • They also, but apparently still later and only occasionally, used fractions of the modern kind.

  • In the sexagesimal system the numerators of the successive fractions (the denominators of which were the successive powers of 60) were followed by', ", "', ", the denominator not being written.

  • Since represented 60, and o was the next letter, the latter appears to have been used to denote absence of one of the fractions; but it is not clear that our present sign for zero was actually derived from this.

  • In the case of fractions of the more general kind, the numerator was written first with ', and then the denominator, followed by ", was written twice.

  • The Romans commonly used fractions with denominator 12; these were described as unciae (ounces), being twelfths of the as (pound).

  • Hindu treatises on arithmetic show the use of fractions, containing a power of io as denominator, as early as the beginning of the 6th century A.D.

  • Even where the decimal notation would seem to arise naturally, as in the case of approximate extraction of a square root, the portion which might have been expressed as a decimal was converted into sexagesimal fractions.

  • Fractions of Concrete Quantities.

  • - The British systems of coinage, weights, lengths, &c., afford many examples of the use of fractions.

  • Also most fractions cannot be expressed exactly as decimals; and this is also the case for surds and logarithms, as well as for the numbers expressing certain ratios which arise out of geometrical relations.

  • For multiplication by a proper fraction or a decimal, it is sometimes convenient, especially when we are dealing with mixed quantities, to convert the multiplier into the sum or difference of a number of fractions, each of which has i as its numerator.

  • The fractions should generally be chosen so that each part of the product may be obtained from an earlier part by a comparatively simple division.

  • in the form L(a+f), where f is a fraction (or the sum of several fractions); we then say that the cost, being nXL(a-l-f), is equal to (a+f)XLn, and apply the method of compound practice, i.e.

  • The theory of continued fractions gives a method of expressing a number, in certain cases, as a continued product.

  • Any exact fraction can be expressed as a continued fraction, and there are methods for expressing as continued fractions certain other numbers, e.g.

  • square roots, whose values cannot be expressed exactly as fractions.

  • That grantee, the tenant-in-chief, has the right to demand from his sub-tenants, to whom he has given out fractions of his estate, the same dues that the king exacts from himself.

  • As the process of the partition of lands continued, the fractions grew smaller and smaller, and many of the tenants-in-chief were ere long very small and unimportant persons.

  • 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.

  • Enneper, and the expression of continued fractions as determinants by Jacobi, V.

  • Most of the elements are small numerical fractions: e, the eccentricity of the moon's orbit, about 0.055; e', the eccentricity of the earth's orbit, about o 017: y, the sine of half the inclination of the moon's orbit, about 0.046; m, the ratio of the mean motions of the moon and earth, about 0.075.

  • A divided cylinder is fixed to the turning knob, which thus makes it possible to measure fractions of the revolution.

  • With the help of this scale the total revolutions of the screw can be read; fractions of the revolution can be read from the divided cylinder d.

  • As fractions of intervals can only be estimated in this method, a measurement with such an eyepiece scale can of course not be as exact as with a screw micrometer ocular.

  • It was urged that studies should focus on the fine and ultrafine fractions of the ambient aerosol and on effects on sensitive individuals.

  • algebraic manipulation, fractions.

  • All the crude extracts and the fractions exhibited a very good level of broad spectrum antibacterial activity.

  • It only take fractions of a second for a child to get severely burnt.

  • decimal fractions.

  • denominators of the two fractions.

  • elute with 14 ml of EB into 7x2 ml fractions.

  • Finally the protein was eluted with 15 MLS of eluted with 15 MLS of Elute buffer with 2 ml fractions collected into eppendorf tubes.

  • fractions of millimeters and seconds, she is quite inconsolable.

  • Procedure: The eluted fractions from the Ni-affinity Histrap columns were loaded on the gel filtration column in GF buffer at 0.80 ml/min.

  • Adding and subtracting Before adding or subtracting fractions, they must be transformed so that they all have a common denominator.

  • His defining moment was when he showed his teacher how to multiply mixed fractions.

  • Algebra 2 Pages 66 to 69 Cancelation and four rules with algebraic fractions.

  • Effect of enalapril on survival in patients with reduced left ventricular ejection fractions and congestive heart failure.

  • C. Series, ' The modular surface and continued fractions ' J London Math.

  • Elute with 14 ml of EB into 7x2 ml fractions.

  • All teachers have heard the old gripes, why should I have to learn fractions?

  • only hyphenate compass points and fractions if they make a compound, eg southeast England, two-thirds full.

  • Use of symbols, basic algebraic manipulation, fractions.

  • methanol extract suggesting that the active compounds might be found in the polar fractions.

  • multiply mixed fractions.

  • Dundee One surgeon performing open radical prostatectomy; radical radiotherapy - external beam 50 gray in 20 fractions; brachytherapy referred to Edinburgh.

  • Mathematical Expressions use the solidus whenever possible in preference to built-up fractions.

  • square roots of rational fractions.

  • subtract fractions by writing them with a common denominator.

  • T1 t2 t3 translation vector in fractions of ROTATING cell edge.

  • The sum of the twin fractions must be 1.0 Twin Data stored by CRYSTALS For a twinned crystal the following equation holds.

  • You also want to look for whey that contains the greatest amounts of those important whey protein fractions.

  • The symmetrical separation of the edges is produced and measured by a single screw; the fractions of a revolution of the screw are obtained by an index attached to one end of the screw, reading on a dial divided into loo equal parts.

  • To Napier seems to be due the first use of the decimal point in arithmetic. Decimal fractions were first introduced by Stevinus in his tract La Disme, published in 1585, but he used cumbrous exponents (numbers enclosed in circles) to distinguish the different denominations, primes, seconds, thirds, &c. Thus, for example, he would have written 123.456 as 123@4050603.

  • In the Rabdologia Napier gives an "Admonitio pro Decimali Arithmetica," in which he commends the fractions of Stevinus and gives an example of their use, the division of 861094 by 43 2.

  • This single instance of the use of the decimal point in the midst of an arithmetical process, if it stood alone, would not suffice to establish a claim for its introduction, as the real introducer of the decimal point is the person who first saw that a point or line as separator was all that was required to distinguish between the integers and fractions, and used it as a permanent notation and not merely in the course of performing an arithmetical operation.

  • Napier thus had complete command over decimal fractions and the use of the decimal point.

  • 2), with a float plate on its upper side, carrying three indicating dials, recording respectively fractions, units and tens of miles (up to a hundred).

  • For forty years after the death of its founder it remained united under the authority of a series of grand khans chosen from among his descendants, and then it began to fall to pieces till the various fractions of it became independent khanates.

  • AS, the Roman unit of weight and measure, divided into 12 unciae (whence both "ounce" and "inch"); its fractions being deunx i 2, dextans, dodrans 4 i bes 3, septunx T7-2-, semis z, quincunx A, triens 3 i quadrans 4, sextans s, sescuncia $, uncia r i g.

  • Beneath are the official Liverpool quotations of " futures," as they appeared on the morning of the 19th of April 1906: A merican Deliveries, any port, basis of middling, good ordinary clause (the fractions are given in moths of a penny).

  • In order to separate the distillate into various fractions, and to remove as much of it as possible free from condensed steam, it is now usual to employ condensing appliances of special form with outlets for running off the different fractions.

  • Laurent generally agreed, except when the theory compelled the adoption of formulae containing fractions of atoms; in such cases he regarded the molecular weight as the weight occupying a volume equal to four unit weights of hydrogen.

  • If we express the pressure, volume and temperature as fractions of the critical constants, then, calling these fractions the " reduced " pressure, volume and temperature, and denoting them by 7r, 0 and 0 respectively, the characteristic equation becomes (7+3/0 2)(30-i) =80; which has the same form for all substances.

  • Obviously, therefore, liquids are comparable when the pressures, volumes and temperatures are equal fractions of the critical constants.

  • For the subjects of this heading see the articles DIFFERENTIAL EQUATIONS; FOURIER'S SERIES; CONTINUED FRACTIONS; FUNCTION; FUNCTION OF REAL VARIABLES; FUNCTION COMPLEX; GROUPS, THEORY OF; INFINITESIMAL CALCULUS; MAXIMA AND MINIMA; SERIES; SPHERICAL HARMONICS; TRIGONOMETRY; VARIATIONS, CALCULUS OF.

  • The amounts of Turkish gold, silver and debased coinage in circulation are approximately £T16,500,000, in gold, £T8,70o,000 (940,000,000 piastres at 108) in silver mejidies and fractions, and 200,000,000 piastres in beshlik and metallik.

  • The Liber abaci, which fills 459 printed pages, contains the most perfect methods of calculating with whole numbers and with fractions, practice, extraction of the square and cube roots, proportion, chain rule, finding of proportional parts, averages, progressions, even compound interest, just as in the completest mercantile arithmetics of our days.

  • Leonardo, making use of fractions of the sexagesimal scale, gives X = I° 221 7 42" i 33 iv 4v 40 vi, after having demonstrated, by a discussion founded on the 10th book of Euclid, that a solution by square roots is impossible.

  • We have to multiply a01; -alas+a2 by ao, -aif32+a2 and we obtain ao (3 - aoal(f31N2 +01133) +aoa2(SI+13) -i-a?31a2 - aIa2(31 + 02) + al, 131+02 = b, 131132 = b t'i +s2 = 2bob2, and clearing of fractions R 1,5 = (a o b 2 - a2 b o) 2 + (a i b o - aobi) (aib2 - a2b1).

  • In astronomical practice the masses of the planets are commonly expressed as fractions of the mass of the sun, the latter being taken as unity.

  • Unfortunately these eclipses are not sudden but slowly changing phenomena, so that they cannot be observed without an error of at least several seconds, and not infrequently important fractions of a minute.

  • In algebra he discovered the method of approximating to the real roots of an equation by means of continued fractions, and imagined a general process of solving algebraical equations of every degree.

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

  • The deductions follow directly from the definitions, and such mechanical processes as "clearing of fractions " find no place (� 21 (ii.)).

  • (ii.) Perhaps the worst thing we can do, from the point of view of intelligibility, is to " clear of fractions " by multiplying both sides by 6.

  • (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.

  • (ii.) The elements of the theory of numbers belong to arithmetic. In particular, the theorem that if n is a factor of a and of b it is also a factor of pa= qb, where p and q are any integers, is important in reference to the determination of greatest common divisor and to the elementary treatment of continued fractions.

  • (iii.) There are important theorems as to the relative value of fractions; e.g.

  • 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.

  • 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.

  • Thus, if we have an equation P=Q, where P and Q are numbers involving fractions, we can clear of fractions, not by multiplying P and Q by a number m, but by applying the equal multiples P and Q to a number m as unit.

  • (iii.) Another application of the method is to proving the law of formation of consecutive convergents to a continued fraction (see Continued Fractions).

  • There are extensions of the binomial theorem, by means of which approximate calculations can be made of fractions, surds, and powers of fractions and of surds; the main difference being that the number of terms which can be taken into account is unlimited, so that, although we may approach nearer and nearer to the true value, we never attain it exactly.

  • �; the successive terms of this series, after the first, are alternately positive and negative, and consist of fractions with numerators I and denominators continually increasing.

  • Continued fractions, one of the earliest examples of which is Lord Brouncker's expression for the ratio of the circumference to the diameter of a circle (see Circle), were elaborately discussed by John Wallis and Leonhard Euler; the convergency of series treated by Newton, Euler and the Bernoullis; the binomial theorem, due originally to Newton and subsequently expanded by Euler and others, was used by Joseph Louis Lagrange as the basis of his Calcul des Fonctions.

  • So persistently does the human ear rebel against the division of the tetrachord into two greater tones and a leimma or hemitone, as represented by the fractions 9, 9, 26, that, centuries before the possibility of reconciling the demands of the ear with those of exact science was satisfactorily demonstrated, the Aristoxenian school advocated the use of an empirical scale, sounding pleasant to the sense, in preference to an unpleasing tonality founded upon immutable proportions.

  • 2 We have given the fractions in the order in which they occur in the modern system.

  • Benzene is manufactured from the low-boiling fractions of the coal-tar distillate '(see' Coal-Tar).

  • The fractions are agitated with strong sulphuric acid, and then washed with a caustic soda solution.

  • STYROLENE, C 6 H 5 CH:CH 2, also known as phenylethylene or vinylbenzene, an aromatic hydrocarbon found to the extent of 1 to 4% in storax; it also occurs with crude xylene in coal tar fractions.

  • Generally the components of a mixture will be vaporized in the order of their boiling-points; consequently if the condensates or "fractions" corresponding to definite ranges of temperature be separately collected, it is obvious that a more or less partial separation of the components will be effected.

  • For the purpose of collecting the distillates in fractions, many forms of receivers have been devised.

  • Dephlegmation of the vapours arising from such mixtures as coaltar fractions, petroleum and the "wash" of the spirit industry, is very important, and many types of apparatus are employed in order to effect a separation of the vapours.

  • Both the fractions into which they were divided by the Nerbudda river laid claim to antiquity: while the northern, however, did not trace their origin further back than the period of the early Mahommedan kings of Delhi, the southern fraction not only claimed an earlier and purer descent, but adhered also with greater strictness to the rules of their profession.

  • Baker this would seem to point to the fact that the tellurium used was insufficiently purified, since his work showed that there was no difference between the first and last fractions (see Chem.

  • If each of the fractions (3) is put equal to i/4h, it is readily found, from the first property of the normal state, that, of the s molecules of the first kind, a number sal (h3m3 /13)e hm (u2+v2+w2)dudvdw (4) Velocities.

  • He studied at Berlin University, where he obtained the degree of doctor of philosophy in 1825, his thesis being an analytical discussion of the theory of fractions.

  • This was arranged for by a movable leaf carrying the sighting V, worked by means of a mill-headed screw provided with a scale in degrees and fractions to the same radius as the elevation scale, and an arrowb head for reading.

  • The Army of the North was to concentrate in three fractions - around Solre, Beaumont and Philippeville - as close to Charleroi as was practicable; and he arranged to screen the initial movements of the troops as much as possible, so as to prevent the allies from discovering in time that their centre was aimed at.

  • .; so that, marking off duodecimal fractions by commas, the area in the above case is 4 of 3, I, 8, 4, 8 X15, 3 X15, 3 sq.

  • The time between the breaks could be measured in seconds by the clock signals, and in fractions of a second by the tuning-fork record.

  • The frequency ratios in the diatonic scale are all expressible either as fractions, with i, 2, 3 or 5 as numerator and denominator, or as products of such fractions; and it may be shown that for a given note the numerator and denominator are smaller than any other numbers which would give us a note in the immediate neighbourhood.

  • Thus, whilst the detachment was still disposed as a series of rearguards, the foremost fractions of it stood to fight on the Yalu, against odds of four to one.

  • On the side of the defence, each colonel had been left to retire as best he could, and thus certain fractions of the retreating Russians encountered Inouye's advancing troops and were destroyed after a most gallant resistance.

  • Calling in the brigade detached to the assistance of Nozu as well as all other available fractions of his scattered army, he himself attacked 1 The 5th division of the 2nd Army had been sent to join the 10th as the latter approached Hsimucheng.

  • The accuracy of a meter is tested by drawing calibration curves showing the percentage departure from absolute accuracy in its reading for various decimal fractions of full load.

  • was to divide his territories among his sons, whereby Poland was partitioned into no fewer than four, and ultimately into as many as eight, principalities, many of which (Silesia and Great Poland, for instance) in process of time split up into still smaller fractions all of them more or less bitterly hostile to each other.

  • 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.

  • An essential accompaniment therefore of the potentiometer is a series of standard low resistances, say of o 1, o oi, o ooi ohm, and also a series of higher resistances divided into known fractions.

  • (c) the standard gallon (and multiples and fractions of it), declared to contain 10 lb of water at 62° F., being in volume 277.274 cub.

  • In Ptolemaic times the artaba (2336.), modified from the Persian, was general in Egypt, a working equivalent to the Attic metretes -- value 2 apet or 1/2 tama; medimnus=tama or 2 artabas, and fractions down to 1/400 artaba (35).

  • (arithmetic) elementary lessons on the notation of decimal fractions.

  • 3 Besides his connexion with logarithms and improvements in the method of prosthaphaeresis, Byrgius has a share in the invention of decimal fractions.

  • Logistic or Proportional Logarithms. - The old name for what are now called ratios or fractions are logistic numbers, so that a table of log (a/x) where x is the argument and a a constant is called a table of logistic or proportional logarithms; and since log (a/x) =log a-log x it is clear that the tabular results differ from those given in an ordinary table of logarithms only by the subtraction of a constant and a change of sign.

  • be proper fractions, and the value of every one of the interminate continued fractions l!1 a2.

  • Thus, to take the latter one, if we suppose that of two editors of equal competence A requires a probability of four-fifths to admit a reading into his text and B a probability of three-fifths only, then in all the cases in which the probability lies between these two fractions B will be right seven times to A's three, while outside these limits there will be no difference between them.

  • Which Gives The Series Of Approximating Fractions, I 7 8 31 132 163 &C.

  • 2 an embolismic The result so obtained would in general be more accurate than the Jewish calculation, from which it may differ a day, as fractions of a day do not enter alike in these computations.

  • Recorde published several works upon mathematical subjects, chiefly in the form of dialogue between master and scholar, viz.: - The Grounde of Artes, teachinge the Worke and Practise of Arithmeticke, both in whole numbers and fractions (1540); The Pathway to Knowledge, containing the First Principles of Geometry.

  • Decimal fractions had been employed for the extraction of square roots.

  • The point separating the integers from the decimal fractions seems to be the invention of Bartholomaeus Pitiscus, in whose trigonometrical tables (1612) it occurs and it was accepted by John Napier in his logarithmic papers (1614 and 1619).

  • 'NAPHTHALENE, C 1 oH 8, a hydrocarbon discovered in the "carbolic" and "heavy oil" fractions of the coal-tar distillate '(see Coal-Tar) in 1819 by A.

  • African mines the diamonds are not only crystals of various weights from fractions of a carat to 150 carats, but also occur as microscopic crystals disseminated through the blue ground.

  • The size of the angle between the median planes of two consecutive leaves in an alternate arrangement is their divergence; and it is expressed in fractions of the circumference of the axis which is supposed to be a circle.

  • Fractions except * were all primary, i.e.

  • with the numerator unity: in order to express such an idea as ~ the Egyptians were obliged to reduce it to a series of primary fractions through double fractions 1~+~1rt~r+1~w+ 1~ 4(1+

  • rabies and rule to facilitate the employment of fractions.

  • (b) Conversions of compound fractions (e.g.

  • FLUORANTHENE, C15H10, also known as idryl, a hydrocarbon occurring with phenanthrene, pyrene, diphenyl, and other substances in "Stupp" fat (the fat obtained in working up the mercury ores in Idria), and also in the higher boiling fractions of the coal tar distillate.

  • Fractions of the day are disregarded to avoid dispute, though sometimes the law will consider fractions, as where it is necessary to show the first of two acts.

  • The law pays no regard to fractions of a day except to prevent injustice.

  • In the recording acts relating to real property, fractions of a day are of the utmost importance, and all deeds, mortgages and other instruments affecting the property, take precedence in the order in which they were filed for record.

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