ARITHMETIC is the science that treats of numbers, and of the methods of computing by means of them.
In introducing the subject, and endeavouring to trace the progress of the science, there appear to be three points that call for particular notice, viz., the conception of number, the representation of numbers either by words or graphically by characters, and the principles and modes of computation.
1. The primary conceptions of numbers are necessarily of a very crude kind. The child attains the notion slowly by experience, and the ability that even adults have to apprehend the significance of numbers with precision is restricted to an extremely narrow area. This is still more the case among uncivilised races, some of which do not appear to be able to count beyond 3, or 4, or 5, or are at least believed to have no words in their vocabularies that express larger numbers. It is to be remembered that the knowledge that is acquired regarding numbers through experience and culture is not of numbers absolutely or in the abstract, but rests almost entirely on a perception of the relations which numbers bear to each other.
The power to form a direct and immediate conception of numbers is very limited; but the relative magnitudes of numbers, large as well as small, can be expressed with the utmost accuracy, and so as to .be clearly understood. The system of notation in common use, whereby we express not merely numbers, but parts of numbers, supplies us with means of comparing arithmetically together either the greatest or the minutest magnitudes, to which there is absolutely no limit. The proportion, for instance, that the circumference of a circle bears to the diameter, though it cannot be stated with arithmetical exactness, has been calculated to upwards of 200 decimal places,—a nicety for which the vast dimensions that science discloses in the physical universe furnish no means of comparison whatever. For let it be supposed that a circle were described, with a point on the earth’s surface as centre, so as to extend beyond the most distant star that can be discerned by the most powerful existing telescope, that the radius of the circle were known, and that the circumference were computed from it, it does not appear that an error in the thirtieth decimal place would be of such magnitude that the keenest vision, aided by the most powerful microscope, could detect it.
In all systems of number, with the exception, perhaps, of the very rudest, numerical conceptions are aided by the introduction, usually at a very early stage, of methods of grouping. In nearly every case the methods adopted connect themselves with the number of the fingers, either of one or, more usually, of the two hands. Having reached 5 or 10, the reckoner proceeds by adding to these the prior numbers; and when a second 5 or 10 is reached, a new word or sign is employed, the significance or derivation of which is generally well marked in the name or form it bears. Similarly when (say) five fives or ten tens are reached, a fresh start is made. These processes of grouping are of great importance, conveying clearer conceptions than could otherwise be obtained of the relative magnitudes of numbers. An additional evidence of the value of the principle is to be found in certain numerical combinations which are not additive, as such combinations mostly are, but subtractive. Thus, if a conception be formed of 10, and again of 20, 19 and even 18 will connect themselves more readily with the latter than with the former, and so we have such forms as duodeviginti and undeviginti; and there is little doubt that similar considerations, in combination with a regard for brevity of expression, have led to the use of the subtractive forms IX., XL., &c.
2. There are two ways in which numbers are represented, —either in words or by particular characters or symbols. It is with the latter that arithmetic has more especially to do, the numerous important and interesting questions that relate to word-numerals falling rather within the domain of language. It is only in so far as they arise out of the systems of grouping already referred to, suggesting in their formation such processes as addition or multiplication, that these call for any notice here. Dr Tylor {Primitive Culture, chap, vii.) gives a variety of remarkable collocations of this sort, as well as of descriptive numerals, that are in use among different races and tribes.
There is also a common use of characters to represent numbers which does not properly belong to this subject. The letters A, B, C, &c., or α, β, γ, or the like, the order of whose succession is fixed, are often employed to indicate numerical order, either singly or in combination with numerals proper. This is not, however, an arithmetical use of the characters; they are merely ordinals, and cannot furnish a basis for calculation.
The origin of the various characters which have been employed to indicate numbers proper, and by means of which computations can be effected, is in most cases alphabetical. In the Hebrew and Greek notations, for instance, the letters were taken in the main in their alphabetical order, being divided into three groups, of which the first represented units, the second tens, and the third hundreds; and very similar Systeme have been in use among other nations. As the Hebrew alphabet contained no more than twenty-two letters, the numbers from 500 to 900 were represented by five final forms, and sometimes by additive combinations with ∏, 400. The following is the usual Hebrew notation, numbers exceeding ten being made up by placing the larger numeral characters first :— [table]
The ordinary Greek notation has not merely a general resemblance of structure to the Hebrew, but corresponds to it closely, character for character, up to 80. The Greek alphabet consisting of twenty-four letters, three additional characters were introduced. The first, for 6, occupying the place of 1, is ς- (named στaυ), which was afterwards used as a contraction for στ; the other3, for 90 and 900, were named κόππα and σαμπι respectively, and wτitten? or and ft). The notation is thus as follows :— [table]
To distinguish the numeral letters an accent was written after the last,—thus, 38 was λη, while thousands were indicated by writing the accent below,—thus z δ was 4000. The letter M (for μυpioi) increased the numeral 10,000 times. Fractions were separated by a space from the integers they were affixed to, and the denominator was written like an index or power in our notation,—thus, μγ θ ιa was 43 1 ¾ r .
In addition to this another entirely different notation is found in Greek inscriptions. It more nearly resembles the Roman system, the numeral characters (except the first) being the initials of the numeral names, and being repeated till the number to be signified was expressed. I represents, 1, ∏ (πAτ∈) 5, A (δtκα) 10, Η (cκaτo'v, the aspirate being written as an H) 100, X (χίλιοι) 1000, M (∕ιυpioι) 10,000. Characters enclosed within three lines (forming a ∏) are thereby multiplied by 5. To give an example, in this notation ∣ II ∣ HH ]~∆1 ΔΠIII represents 768.
The Roman system, with which we are still familiar nearly in its completeness, employs the letters of the alphabet, but-is not based upon alphabetical order. Many attempts have been made to account for the symbols, and the complete solution of the problem of their origination is perhaps unattainable. Sir John Leslie, following some writers of the 16th and 17th centuries (see in particular the Cursus Mathematicus of Dechâles, vol. i. 1674, 2d. ed. 1690), advocated the opinion that, one line or stroke being taken to represent the unit, when ten of these were set down, a stroke would be drawn across them in a slanting direction to cancel them, and the unit stroke and cancelling stroke would thus give the form X for 10; that a repetition of this proceeding would give a third stroke when 100 was reached, and the three might take the form C or C; and that, similarly, the combination of four strokes would give M for 1000. This explanation is perhaps too ingenious. It has the merit of accounting for the X (the crux of the method), which may have been introduced in some such way. But one does not readily see Iιow the CZ could be formed from the X, or the M from the Q i and it appears far more likely that the signs for 100 and 1000 are merely the initial letters of Centum and Mille, — all the more that the very ancient notation noticed above as found in Greek inscriptions has evidently an origin of this kind. In any case V, L» and D appear to be respectively the halves of X, the angular C (C), and the rounded M (co). The ancient forms of D and M, viz., IO and CIO, have ceased to be familiar. By an extension of this style of characters, IOO denoted 5000; CCIOO, 100,000; CCCIOOO, 1,000,000, &c. To represent two, three, <fcc., millions, the CCCIOOO was repeated the required number of times.
The Roman notation employs fewer characters than the Greek, and makes greater use of combinations. One is repeated up to four; to the new character for five, ones are repeated up to nine; ones are added to the ten character; at fifteen the five enters, and so at twenty-five, &c.; for the tens up to forty the ten is repeated, and so on,—the symbols I, V, X, L, C, D expressing all numbers by regular combinations up to M, a thousand. The subtractive collocations, IV, IX, XL, XC, now the ordinary, were originally alternative forms, as were also the rarer combinations IIX for 8, XIIX for 18, XXC for 80, <tc. To the extent of these subtractive forms, the values of the characters depended on their position, a smaller number being added to a greater when it followed, and subtracted from it when it preceded it. This element of position is, however, an irregular and exceptional one; and, instead of being of such advantage as the local value of the Arabic notation, is rather a hindrance in calculation.
Far superior to all the ancient systems, and indeed to every other system that exists, is the arithmetical notation that is in common use. The Arabic numerals, aa they are called, are ten in number, nine of them representing the first nine numbers, and the tenth, the cipher or zero, indicating the absence or negation of numerical value. The significance of these figures or digits depends on their relative position, and the great merit of the system is due to this element of local value. Standing singly, the figures denote simply one, two, three, &c.; but in combinations of them every removal towards the left increases the value of the figure ten times. In 5673, for instance, the 3 denotes three, the 7 seven tens (70), the 6 six times ten tens (600), the 5 five times ten hundreds (5000). Should any of the series of tens be wanting, as in nine thousand and forty-eight, where hundreds do not occur, the place is supplied by a cipher, which throws back the digit that expresses thousands into its proper place, thus 9048. It is by this use of the cipher in supplying blanks, and so regulating the places of the significant digits, that the principle of local value is carried out,. See further the sections on notation and numeration below, p. 527.
The ordinary numerals are called Arabic, and it appears to have been through the Arabians that they were introduced into Europe; but they are now generally acknowledged to be of Indian origin. As may be imagined, they have passed through a great variety of forms, one of the earliest types of them being the Devaηagari, a species of Sanskrit numerals. In the early Indian treatise of Bhascara and Brahmegupta, translated by Mr Colebrooke (see article Algebra, p. 517 of vol. i,), as well as in the still earlier writings of Arya-Bhatta, there are both indications and illustrations of the use of the nine digits and the cipher; and the system is thus traced back to the 7th, and in all probability to the 5th century of the Christian era. Even then it was evidently no novelty, but is alluded to as holding an established position; and the Hindu writers nowhere lay claim to the invention of it, but constantly assign to it a supernatural origin. The method was known to the Arabians in the 9th century; and in the course of the 10th it seems to have come into general use among them, especially in their astronomical writings and tables. It was probably in the following century that the Arabs introduced the notation into Spain; but in regard to this we have no explicit information, and different accounts are given of the earliest instances of the use of the system in Europe. On the one hand, it is alleged that the figures first occur in a translation of Ptolemy, of the date 1136, while others maintain that they were introduced (about 1252) by means of the celebrated astronomical tables published by and named from Alphonso the Wise. That their use was known in Italy at the commencement of the 13th century, appears to be satisfactorily established, for there is no good reason to doubt the genuineness of the MS. writings of Leonardo of Pisa, copies of which have been found bearing the dates 1202 and 1220. Numerous other instances are given of the early use of the nine figures and the cipher, especially by astronomers, and in calendars. The great superiority of this to earlier modes of numerical expression became gradually apparent, and in course of time it came into almost universal use among civilised nations.
For a time there was, not unnaturally, considerable inexactness or confusion in the employment of the notation. In early writings such combinations are found, for example, as X2 for 12, 301 for 31, &c. In the latter case the law of local value is lost sight of, and the characters 30 are used as equivalent to thirty, irrespective of their position.
3. Calculation or computation by means of numerical characters is what is ordinarily regarded as the distinctive province of arithmetic, and the worth of a system of notation is to be estimated by the facilities it affords for the operations of reckoning. The methods in common use will be detailed, and the principles on which they depend briefly expounded, in subsequent sections of this article.
Computation of a comparatively rude kind was often carried on in ancient times, and is practised still in some countries, by what are called palpable methods,—as, for instance, by means of counters, or by balls strung on rods or running in grooves. Of this the abacus of the Romans (used too, but not so generally, among the Greeks), the swan-pan, still in constant use among the Chinese, &c., aro examples. (See Abacus, vol. i. p. 4.)
The methods that preceded the adoption of the Arabic numerals were all comparatively unwieldy, and very simple processes involved great labour. The notation of the Bomans, in particular, could adapt itself so ill to arithmetical operations, that nearly all their calculations had to be made by the abacus. One of the best and most manageable of the ancient systems is the Greek, though that too is very clumsy, as may be seen from an elementary example.
Let it be required to muliply 862 by 523, i.e., <Mβ , by φκγ'. The product of ω and φ is 400,000, i.e., ∕χ κ ; of ξ and φ, 30,000, or γ κ ; of β ω ⅛ ß and φ, 1000, or y α∙ Similarly the multipli- Φ κ y cation of ω by κ gives 16,000, or a M/ r; of V*y*μ ξ by κ, pσ' l of β by κ, μ; of ω by γ, βυ', a *,S^ of ξ by γ, ρπ; and of β by γ, These aσ μ may be arranged in some such order as in β v the margin, and summed. The result is pπ 5^
μ lt ε lt ωκ√, or 450,856.
∕x m ∈ m ωκy
The notation employed here extends to hundreds of millions. A scheme proposed by Archimedes, the most distinguished mathematician of antiquity, in his work entitled ⅛aμμhrη<s, Ar enamus, goes far beyond this. Taking the limit of the ordinary system, viz., a myriad of myriads, or 100,000,000, as the basis of a new scale, he treats of numbers formed by the continued product of that amount repeated eight times, extending to sixty-four places in our notation. These he divided into what we should call periods, of eight figures each, to which he gave the name of octades. He thus appears to have to some extent anticipated the modern method of grouping, though, from the want of knowledge of the principle of local value, the practical advance he made was unimportant. Apollonius of Perga, who flourished a little after the time of Archimedes, took the myriad as the basis of his system, and repeated the sign Mυ for each product of a myriad; thus λδMυ.Mυ.Mυ indicates what we would write as 34,000,000,000,000. But his most important contribution to arithmetical science was his grouping the tens in multiplication, so as to connect large numbers, as far as possible, with those we represent by the nine digits. He thus endeavoured, and with some measure of success, to remedy the defect of the general system arising from the want of apparent connection of such characters as λ and τ, for example, or π and ω, with what he called their bases, γ and η. It is chiefly in the commentaries of Eutocius on the works of Archimedes and Apollonius that the examples of the ancient Greek arithmetic which we possess are preserved.
The operations of the Greek arithmetic involving fractions were necessarily very complicated. A simpler system was introduced (by Ptolemy, according to his commentator Theon, but it appears to have existed before his time) in what are known as sexagesimals, which are precisely analogous to the duodecimals by which we sometimes calculate areas. The division of the circle into 360 degrees arose, no doubt, from 360 approximating the number of days in the year, and containing a large number of divisors. The radius of the circle, or side of the inscribed hexagon, subtended at the centre 60 of these degrees, and from this the degree was again divided into 60 parts called minutes, the minute into 60 seconds, <fcc., just as the foot is divided into 12 inches, the inch into 12 seconds, &c., in duodecimals. The sexagesimal system, though applied in the first instance to circular, was also employed in linear, measurements. As an example, the square of λζ δ v∈, i.e., of 37° 4' 55', will be found to be z ατoε δ ιδ ι κε, i.e., 1375° 4' 14" 10" ’ 25"". The numerals beyond 60 or £ being dispensed with in this system, the next letter, o, was used to denote zero; and it has been conjectured that this may have been the origin of the form of our cipher.
The most important step in the progress of modern arithmetic was the introduction of decimal fractions, and the extension of the Arabic notation to the expression of them. The first writer who advocated and exemplified the use of decimals was Simon Stevin de Bruges, better known as Stevinus, in a paper (Zα Disme) published about 1585; but he employed an awkward notation, and it was only after a considerable period that they assumed the form— entirely consistent with the notation of integers—with which we are familiar. From a form which Lord Napier employs in his Rabdologia (1617), the introduction of the decimal point (as a comma) has been ascribed to him, but apparently without sufficient evidence.
The following writers on arithmetic may be named, in addition to those already mentioned:—Diophantus, who flourished in or about the 4th century; Maximus Planudes, who died about 1350; Lucas Pacioli (de Burgo, or di Borgo), whose Summa de Arithmetica (1494) was the first work on algebra printed, and one of the earliest on arithmetic; Bishop Tonstall, whose De Arte Supputandi (1522) was the first work on the subject printed in English; Robert Recorde (died 1558), whose Grounds of Arles and Whetstone of Witte were arithmetical works of great value; Nicolo Tartaglia (died 1559); Michael Stiefel or Stifelius (1486-1567), said to have been the inventor of the signs + and Peter Ramus (1515-1572); Albert Girard (died 1634); William Oughtred (died 1660); Edward Cocker, whose Arithmetic (1st ed. 1677), a posthumous work, and probably a forgery, is of slight merit, though it passed through many editions; Kästner (Geschichte der Mathematik, 4 vols. 1796-1800); Montucla (Histoire des Mathématiques, voL i. 1799); De Morgan (Arithmetical Books, 1847). By far the best history of arithmetic is that of Dr George Peacock, late Dean of Ely, published in the Encyclopaedia Metropolitana.
Of recent works on arithmetic there is great abundance, and many of them are of great excellence. They usually contain statements of modes of operation, under the name of “Rules,” with a number of examples under each for practice; and not a few of them give in addition explanations of the rationale of the methods.
In the following sections the ordinary processes of arithmetical calculation and their commonest practical uses are briefly explained. The various methods of operation are given with greater or less detail, as has appeared necessary for the exposition of the principles on which the operations depend; and light is in many instances thrown on both processes and principles by illustrative examples. For further examples the reader is referred to the manuals above alluded to. The earlier sections, forming the greater part of the article, are occupied with numbers in the abstract, and the remainder with arithmetic in its practical applications.
I. Abstract Arithmetic.
1. Notation is the name usually given to the expressing of numbers by means of characters or figures.
The number ten is the basis or radix of the Arabic system of notation, and every number may be expressed in that notation by combinations of the ten digits, or numeral figures, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, the last (the cipher or zero) having no value except in combination. When several of these are placed together, every removal towards the left increases the value of a figure ten times. The figure placed furthest to the right has the same significance as when it stands alone, i.e., it represents units, the figure next to it denotes tens, the next hundreds, the next thousands; 8735 is therefore read as eight thousand seven hundred and thirty-five. With more than four figures we should have—still proceeding to the left—tens of thousands; then hundreds of thousands, and then thousands of thousands; but instead of this we use the term millions. A million of millions is a billion, and a million of billions a trillion. A digit followed by three figures thus expresses thousands; by six figures, millions; by twelve, billions; and so on. Beyond trillions we have—taking the figures in groups of six—quadrillions, quintillions, sextillions, septillions, octillions, nonillions, <fcc. But occasion is very seldom found to employ these terms, and indeed it is difficult to form any distinct or definite idea of even a billion.
The system of notation in ordinary use is remarkable at once for its simplicity and its completeness. The selection of ten as the basis or radix of the system (whence numbers expressed in this way are said to be in the denary scale of notation) is in all probability to be traced to primitive calculations by means of the ten fingers. Ten is well chosen, being neither too large nor too small, but twelve might, in some respects, have been found more convenient. All numbers can be expressed with twelve or any other number as basis, just as with ten.
2. Numeration is the art of reading figures employed to express numbers. The following table shows the places of the figures, as already described, up to billions :— [table]
The figures 8,607,034,740,952, for example, are read thus: —Eight billions, six hundred and seven thousand and thirty-four millions, seven hundred and forty thousand, nine hundred and fifty-two. Ciphers are passed over in reading, their function being to determine the proper position of the significant figures. When the number of figures exceeds four, it is usual to print them off as above in groups of three, by means of commas. At the odd commas, reckoning from the right hand, we always read thousands; at the even commas, millions, billions, &c.
3. Notation has, in addition to the general meaning already explained, a special signification as the converse of numeration, i.e., it is the art of representing by figures numbers that are given expressed in words.
4. Addition is the method of finding the sum of two or more given numbers, that is, the number to which they amount when taken together.
Suppose a column, added in the ordinary way, amounts to 34. The 4 is set down under the column, and the 3, representing 30, is added with (or, as usually expressed, “carried to ”) the next column.
5. Subtraction is the method of finding the difference between two given numbers, that is, the number by which the greater number exceeds the less, or the remainder after the less is taken from the greater.
In subtracting, say, 38 from 92, since 8 cannot be taken from 2, we “borrow” from the place of tens, i.e., the 2 is increased by 10, and 8 taken from 12 leaves 4. The 10 added (or “borrowed ”) has to be taken away again, and this is done by taking 1 from 9 in the place of tens. But 3 has also to be taken from this 9; so first of all the 1 and 3 are added together, i.e., 1 is “carried” to the 3, and 4 taken from 9 leaves 5, giving 54 as the complete remainder. The “borrowing” process depends on the consideration that when the same amount is added to both numbers their difference is unaltered—it being remembered that ten in the upper line corresponds to one a place towards the left in the lower line.
6. Multiplication is a method of finding the result produced by adding a given number taken a given number of times. The number to be repeated is called the multiplicand; the number expressing the times it is taken, the multiplier; and the result obtained, the product. The multiplicand and multiplier are also said to be factors of the product.
The multiplicand might be written down the required number of times, and the sum found by addition. But this tedious process is unnecessary when the numbers to be added are all the same We do not then require to pass from figure to figure, as when they are different. In adding six nines, for instance, we do not need to proceed step by step (as with 6 + 5 + 9 + 8 + 6 + 7 = 41), but know the result, 54, at once from a table of products which is committed to memory.
When two numbers are to be multiplied together, either may be taken as the multiplier. For it is evident that the four rows of dots, five in a row, in the margin, are the same as the five rows, four in a row—according as we take the rows horizontally or vertically; that is, 4 times 5 and 5 times 4 amount to the same number.
The process of multiplication by a single digit—by 8, for instance—is nothing but an abridgement of the operation of writing the multiplicand eight times and adding. When there are more digits than one in the multiplier, we arrange the successive products in the well-known fashion, because the multiplication has to be by 10 times the second digit (counting from the right), 100 times the third, and so on. Thus, in multiplying 92058 by 734, the 2 of the multiplicand is really 2000; the 3 of the multiplier, 30; and the product of these, 6, i.e., 60,000, falls by the ordinary process into its proper place.
The process of working may sometimes be considerably shortened by multiplying a product already obtained. Thus, to multiply by 568, we may multiply first by 8, and then take 7 times the result, obtaining the product by 56 at once. So with 549378, we may begin with 9, and multiply the 9 product by 6 for 54, and the 54 product by 7 for 378, observing to arrange the lines correctly, by placing under the right-hand figure of every multiplier the righthand figure of the corresponding product.
7. Division is the method of finding how often one given number contains another.
Of these two numbers, the former is called the dividend, and the latter the divisor. The number expressing the times that the first contains the second is called the quotient. When the number of times is not exact, the excess of the dividend over the divisor taken the greatest number of times that the dividend contains it exactly is called the remainder.
As multiplication is a short method of addition, division (which is the converse of multiplication) is an abridged subtraction. Were we to subtract the divisor from the dividend, subtract it again from the remainder, and continue the process till a remainder less than the divisor were obtained, the number of subtractions would give the required quotient. But this operation is greatly shortened by means of multiplication. Thus, if it is required to find how often 9 is contained in 49, remembering that 5 × 9 is 45, and that this is the nearest product not greater than 49, we have at once the quotient 5 and remainder 4.
In dividing, e.g., 167685 by 287, by the ordinary process of long division, we find (after, it may be, a trial or two) that 1676 is more than 5 times but less than 6 times 287. The first quotient figure is therefore 5, representing 500; and proceeding in the customary way, we take from the dividend 500 times the divisor, then 80 times, and then 4 times, with 77 over. The dividend, therefore, contains the divisor 584 times, with a remainder of 77.
When the divisor does not exceed 12, the operation is conducted mentally, and the quotient set down at once. This is called short division. If the divisor is made up of factors not greater than 12, short division may be employed, the factors being taken in succession. When there are two such divisors, the 6)4352^ remainder of the whole division is obtained 8)725—2
from the partial remainders, by multiplying 90—5
the second of these by the first divisor, and adding the first to the product. Thus, in the annexed example, division by 48 gives a remainder of 32, for it will be seen that 725 is 5 more than 8 times 90, and 4352 is 2 more than 6 times 725. Therefore, 4352 is 2 more than the sum of 48 times 90 and 6 times 5, i.e., it exceeds 48 times 90 by 6 × 5 + 2 = 32, or contains 48 90 times and 32 over.
Measures and Multiples of Numbers.— 8. A measure of any number is a number that divides it without a remainder. A multiple of any number is a number that it divides without a remainder. A common measure, or common multiple of several numbers, is a number which is a measure or a multiple of each of them. Thus, 3 is a common measure of 12, 18, and 24; 60 is a common multiple of 6, 10, and 15. A measure of a number is sometimes called a submultiple of it.
A prime number is a number which no other, except unity, divides without a remainder; as 2, 3, 5, 7, 11, 13, 17, &c. Numbers which are divisible by other numbers without remainder, that is, which can be resolved into factors, are called composite numbers; as 4, 6, 8, 9, 10, &c. Any factors into which a composite number can be divided are called its component parts. Numbers are said to be prime to each other when they have no common measure, as 15 and 28. The prime factors of a number are the prime numbers of which it is the continued product. Thus, 2, 3, 7 are the prime factors of 42; 2, 2, 3, 5, of 60.
9. To find the greatest common measure of two given numbers, the greater number is divided by the less; the former divisor is then divided by the remainder, and each successive divisor by the remainder obtained in dividing by it, till there is no remainder. The last divisor is the greatest common measure.
This depends on the two following principles :—(1.) If a number measures any other, it measures every multiple of that other; for obviously, since 7 measures 56, it also measures 12 times or 17 times 56; and, (2.) Every number that is a common measure of two others measures also their sum or their difference; for the sum or difference of, say, 13 times 8 and 22 times 8, must, it is evident·, be some multiple of 8.
Thus, to find the greatest common measure of 475 ar.d 589, dividing 589 by 475, we have the remainder 114; dividing 475 by 114, we have the remainder 19; and 114 divided by 19 leaves no remainder. Therefore, 19 is the greatest common measure. For any number that measures 589 and 475 will measure their difference, 114, and will, therefore, measure 456, which is a multiple of 114. Also, any common measure of 475 and 456 will measure their difference, 19. Therefore, no number greater than 19 can measure both 589 and 475. Again, 19 will measure both, for it measures 114, and therefore measures 456, a multiple of 114. Hence it measures 475, which is 456 + 19, and also 589, which is 475 + 114. Therefore, 19 measures both numbers, and since no greater number does so, it is their greatest common measure.
In seeking for measures or factors, it will be of advantage to attend to the following properties of numbers. (For the sake of brevity, we use “divisible” here for “divisible without remainder.”)
(1.) A number is divisible by 2, if the last digit is divisible by 2; by 4, if the last £wo digits are divisible by 4; and by 8, if the last three digits are divisible by 8. For, to take the last case, the figures preceding the last three are a multiple of 1000, which is a multiple of 8. If, then, the last three figures are divisible by 8, so is the whole number.
(2.) Similarly, and for precisely similar reasons, a number is divisible by 5, if its last digit is so; i.e., if the number ends with 5 or 0; by 25, if it ends with 25 or 75; and by 125, if it ends with 125, 625, 375, or 875.
(3.) A number is divisible by 9 when the sum of its digits is so divisible. This is a case of the important property that the division of a number by 9 produces the same remainder as the division of the sum of its digits by 9. Taking any number, e.g., 583, we see that it is made up of 58 tens and 3; that is, of 58 nines and 58 and 3. Again, 58 is 5 tens and 8; that is, 5 nines and 5 and 8. Thus, 583 is made up of 58 nines, 5 nines, and 5 + 8 + 3; that is, of nines + the sum of its digits. Therefore 583 and 5 + 8 + 3 must give the same remainder when divided by 9.
A proof of multiplication, by “casting out the nines,” depends on this property. If a number, made up, say, of nines and 7 over, be multiplied by another made up of nines and 5 over, their product must be nines and 35, that is, nines and 8 over; and unless this relation holds good, there must be an error in the multiplication.
(4.) Similarly, a number is divisible by 3, if the sum of its digits is so; for, if every number be made up of nines and the sum of its digits, it must be made of threes and the same sum.
(5.) A number is divisible by 11, if the sums of its alternate digits are equal, or if they differ by a multiple of 11. Take any number whose alternate digits are equal, or differ by a multiple of 11, as 8294. This is equal to the sum of 8000, 200, 90, and 4; that is, of 80 times 99, 80, twice 99, 2, 90, and 4. Leaving out the multiples of 99 (as being multiples of 11), we have 80 + 90 + 2 + 4. But as 2 + 4 and a multiple of 11 give 8 + 9, therefore, if 80 + 90 + 2 + 4 be divisible by 11, so must 80 + 90 + 8 + 9 be, and vice versa. But 88 + 99 is divisible by 11; so, therefore, is 80 + 90 + 2 + 4, and so also 8294.
If, in finding the greatest common measure by the method described, a divisor occurs containing a factor that evidently does not measure one of the numbers given, that factor may at once be omitted, since it can be no part of the common measure. Thus, in finding the greatest common measure of 59241 and 223014, we get 13950 as a divisor. Now, 50 divides this, i.e., 5 × 5 × 2, and neither 5 nor 2 measures the first of the given numbers. We can therefore reduce 13950 to 279 at once, whence we immediately find 93 to be the greatest common measure.
10. To find the least common multiple of two given numbers, divide either of the numbers by the greatest common measure of the two, and multiply the other number by the quotient. Thus, the greatest common measure of 30 and 48 is 6. Therefore the least common multiple is 48 X 5 or 30 × 8, t.e., 240. For the product 5 × 6 × 8 is evidently a common multiple of 30 (t.e., 5 × 6) and 48 {i.e., 6 X 8), and since 8 and 5 are prime to each other, this product must be the least that contains both 5 × 6 and 6χ8.
To find the least common multiple of any given numbers, arrange them in a line, and strike out any of them that measure any of the others; take any number that measures all or part of the remainder, and divide all that it measures by that number, setting down the quotients and the undivided numbers in a second line; proceed with this second line as with the first, and continue this process till a line is obtained of numbers prime to each other. The continued product of the divisors and the numbers in the last line is the least common multiple required. This depends on the principle just demonstrated, that a common factor of two or more numbers needs to be taken but once for the common multiple.
In finding the least common multiple of 42, 45, 50, 54, 60, 63, 70, 75, and 90, for instance, we may divide in succession by 5, 7, 5, and 6. It is evident that 45 may be omitted, since it will measure every multiple of 90. The division by 5 is virtually the substitution for 50, 60, 70, 75, and 90, of the product 5 × 10 × 12 × 14 × 15 × 18, which is manifestly a common multiple of them. Then 14 and 18 are omitted, being contained in 42 and 54. Next we substitute for 42 and 63 the product 7x6x9; omit 6 and 9 as being contained in 54; then take 5 × 2 × 3, instead of 10 and 15; omit 2 and 3, as measures of 12; and, lastly, substitute 6 × 9 × 2 for 54 and 12. The product of 5, 7, 5, 6, 9, 2, gives 18900 as the least common multiple.
If the divisors are all prime numbers, the result must be the least common multiple; and it is better to avoid using composite numbers as divisors, except when they measure all the numbers in the line. Had 10, for instance, been taken in the example as the first divisor, 7 5 would have remained in the second line, and the result obtained would have been five times too great. It is often found convenient to write the prime factors of the least common multiple, which is in the example 2×2×3×3×3×5×5×7.
Fractions.— 11. If unity be divided into any number of equal parts, one or more of these parts is called a fraction. If, for example, we divide unity into 7 equal parts, and take 5 of these, we shall obtain the fraction we speak of as five-sevenths. (This, and what follows, may be familiarly illustrated by taking any object and dividing it in the way described—a straight line, for instance, thus, I I I I I I I I φ )
There are two kinds of fractions— Vulgar Fractions, often spoken of simply as Fractions, and Decimal Fractions or Decimals.
A vulgar fraction is represented by two numbers, called the fcrnzs of the fraction, which are written, the one above and the other below a horizontal line; thus, the fraction already mentioned is written -Ç-· The number under the line indicates the number of equal parts into which unity is divided, and is called the denominator, x as showing the “denomination” (see § 32) of the fraction. The number above the line, indicating the number of those equal parts that the fraction consists of, is called the numerator.
The most usual definition of a fraction is that which is given above. But it may be also defined or regarded as one number divided by another, the numerator being the dividend, and the denominator the divisor. Thus, the fraction 4, which we have interpreted to mean 5 of the 7 parts into which unity is supposed to be divided, may also be regarded as the seventh part of 5 units. For, if each of 5 units be divided into 7 equal parts, there will be in all 35 of these parts, each of them equal to the seventh part of unity, and the seventh part of these 35 parts is 5 of them. That is, the seventh part of 5 units is the samo as 5 seventh parts of unity, or ∣ according to the former definition. It follows from this that, when there is a remainder after division, the quotient is completed by the addition of a fraction, of which the remainder is the numerator, and the divisor the denominator. For, let it be required to divide 76 by 9. Since 72 divided by 9 gives the quotient 8, and 4 divided by 9 gives ⅜, 76 divided by 9 must be 8⅜.
A proper fraction is one whose numerator is less than its denominator; as -⅛∙, ⅜⅜. An improper fraction is one whose numerator is not less than its denominator; as ⅜, ⅝. A proper fraction is evidently less, and an improper fraction not less, than unity. A mixed number is an integer or whole number with a fraction annexed to it; as 8⅜, 5⅛ The integer and fraction here are to be considered as added together. A simple fraction is a single fraction that has both its terms whole numbers; as ⅜, ⅜. A compound fraction is a fraction of another fraction, or of a whole or mixed number; as f of ⅜, ⅜ of 10∣. A complex fraction is one that has a fraction or mixed number for one of its terms or for both; as -⅛, ⅛, ⅛ A fraction is
4⅜ 4f 5
said to be in its lowest terms when the numerator and denominator have no common divisor.
12. If both terms of a fraction be multiplied, or both divided, by the same number, the value of the fraction will remain unchanged. For, if unity be divided into 7 equal parts, and again into 28 equal parts, it is evident that each of the former parts will be identical with 4 of the latter. 5 of the former will thus be equal to 20 of the latter; that is, y = ⅜⅜∙
13. We multiply or divide a fraction by any number, if we multiply or divide the numerator by it. For it is evident that four times ⅜ is ⅜, since unity is divided into 9 parts in both cases, and the number of parts taken in the one case is four times the number taken in the other. Conversely, dividing f by 4, we have ⅜.
To multiply or divide either term of a fraction is the same as to divide or multiply the other. If, for instance, we divide the denominator by 3, we divide unity into one-third of the number of parts, and each of these parts must thus be three times greater than before. The fraction is therefore multiplied by 3. Or otherwise, dividing the denominator of ⅜ by 3, we obtain f∙; and multiplying the numerator by 3, we get ⅜, which (§ 12) is the same as ⅜. Therefore, to divide a fraction, we may multiply the denominator,—a method which must be employed when the divisor is not a measure of the numerator.
14. To reduce a fraction to its lowest terms, divide both terms by their greatest common measure. The reason of this is evident from the definitions. The value of the fraction remains unchanged (§ 12). The results of operations in fractions should, with rare exceptions, be expressed in their lowest terms.
15. To reduce a mixed number to an improper fraction, the integer is multiplied by the denominator of the fraction, and the numerator added to the product. This gives the numerator of the required improper fraction, and the denominator of the given fraction is its denominator. Take, e.g., 4∣. If unity be divided into 8 equal parts, 4 units will make 32 of these parts, and the fraction ⅜ contains 5 of them; therefore 4∣ = ⅞L Conversely, to reduce an improper fraction to a whole or mixed number, divide the numerator by the denominator; the quotient is the integer required, and the remainder, if there is one, is the numerator of a fraction of which the given denominator is the denominator. Thus, ~ = 5∣; for, if unity be divided into 7 equal parts, 35 of these parts will be the same as 5 units; therefore 38 parts will be 5 units and 3 parts, i.e. , Ί -°7‘
16. To reduce a compound fraction to a simple one, multiply the numerators together for the numerator of the simple fraction, and the denominators together for its denominator. Thus, of ⅞ is equal to ⅜f. For, if we divide unity into 7 equal parts, and each of these again into 4, we shall have for the fraction 4 20 of these parts, ι.e,, 2,o 0 f u ∏ity. The fourth part of this is 5 of these parts, and therefore ⅜ is 15 of them, i.e., ⅜ of ∣ is ⅜f.
Mixed numbers must be put in the form of improper fractions before the multiplication. The reduction of the result to its lowest terms may be effected by removing before multiplication any factors that are common to the numerators and denominators.
In such compound expressions as where one number is spoken of as one-fourth greater or less than another, the fourth is always to be taken of that number with which the comparison is made. Thus, 30 is one-fifth more than 25 li.e., ⅜ of 25), but 25 is one-sixth less than 30 (i.e., ⅜ of 30).
17. In order to compare the values of fractions, or to add or subtract them, it is necessary to reduce them to others of equal value that have the same denominator. From the definition of a fraction we see at once that 4 is greater than ⅛, but we do not readily see whether ⅛ is greater or less than ∣. If, however, we take the equivalents of these fractions, ⅜⅜ and ⅛, unity being now divided into the same number of parts in both cases, we have like quantities to compare, and see that the former fraction is the greater. In practice it is usually the least common denominator that the fractions are compared by. To find this, we first find the least common multiple of the denominators (§ 10), then divide this by the denominator of each fraction separately, and multiply both terms of that fraction by the quotient.
Take, e.y., the fractions i⅜, ⅜⅞, f⅜, and ∣∣∙ The least common multiple of the denominators is 180, and dividing this by 15, 20, 36, and 45, we get 12, 9, 5, and 4. Multiplying both terms of the fractions in succession by these numbers, we have ⅛⅜, ⅜f⅜, ⅜⅜⅞, and ⅜⅜⅜. The value of the fractions has not been altered, and all have now the same denominator, 180, which we must obtain in each case, because we re-multiply the factors into which we resolved it. In practice, indeed, we merely divide 180 by 15, multiply 13 by the 12, and set down 180 under 156 at once.
18. To add fractions, reduce them to others having a common denominator, then add the numerators of these, and write the common denominator under the sum. Thus,
5 t 1 i 7 fi Ο i 15 I 4 2 — 10 7 T∏ P∩ pl·i <^ι^f ill π TP-
T2 + κ + ∙≈rσ - ττσ + τvσ + τmτ - ι^τσ∙ j - n eac ξ 01 tue re ’ duced fractions unity is divided into 120 equal parts, and the fractions are respectively equivalent to 50, 15, and 42 of these parts; therefore the sum of the fractions must be 50 +15 + 42 = 107 of them, i.e., ⅜⅞⅞.
Compound fractions must be reduced to simple ones before addition. Mixed numbers may be brought to improper fractions, and so added; but it is generally preferable to add the whole numbers and the fractions separately, and then add the two results. The operation may often be shortened by first adding any of the given fractions whose denominator is considerably less than that of all the fractions; thus, to add ∣⅞, ⅛ ⅜⅜, ∣-2∙, the sum of the first three may be found first, amounting to 1⅞⅜∙ The addition of ∣4 to this gi ve ≡ 2⅜⅞4r∙
19. To subtract fractions, reduce them to others having a common denominator, take the difference of the numerators so found, and write the common denominator under it. The principle is precisely the same as in addition.
Compound fractions must be simplified as in addition. With mixed numbers the subtraction will generally be best effected by treating the fractions separately, borrowing and carrying, if necessary, on the principle explained in § 5. In subtracting 47∣⅜ from 85⅜⅞, for instance, borrowing fff and carrying 1 gives 85⅜⅛ - 48f∣4 = 37⅜⅛ or 3747, 20. To multiply fractions, multiply the numerators together for the numerator of the product, and the denominators together for the denominator. Thus, in multiplying I by ∣, if we multiply ∣ by 7, we have (§ 13); but our multiplier, ⅞, is the ninth part of 7; we must therefore divide by 9, which gives (§ 13).
When there are several fractions to be multiplied continuously together, we proceed in the same way. Mixed numbers are reduced to improper fractions, and common factors may be struck out, precisely as in § 16. When an integer has to be multiplied by a fraction, we may convert the integer into a fraction by putting 1 as the denominator; or we may multiply the integer by the numerator of the fraction and divide by the denominator, since to multiply by ⅞ is, as has just been shown, to multiply by 7 and divide by 9. So, to multiply an integer by a mixed number, the common method is to multiply by the integer and the fraction of the latter separately, and add the results. In multiplying mixed numbers like 46J and 14⅜ together, instead of using improper fractions, we may take the four products 46×14, -j-×14, 46×⅜, and ⅛ × ⅜, and add them. The amount is 644 + 3⅜ + 15⅜ + -⅛ = 662∙⅛.
21. To divide fractions, invert the terms of the divisor (ι.e., interchange the positions of the numerator and denominator), and multiply the dividend by the inverted fraction; thus, ⅞÷⅞ = ⅛×τ = ⅜⅜∙ we divide ⅞ by 7, we have f s (§ 13). But since our divisor, ⅞, is the ninth part of 7, in dividing by 7 we divide by a number nine times too large. The true quotient must therefore be nine times -⅛, ie,, ⅜∣.
The common method of dividing an integer by a mixed number is a modification of this division. When, in dividing by 37⅛, for instance, we multiply both divisor and dividend by 5, and then divide, we really multiply the dividend by τ ∣- g -. Or the method may be explained on the principle (identical with that of § 12) that the multiplication of the divisor and dividend by the same number does not affect the quotient. As an instance of the division of a mixed number by an integer, let 3982⅜⅜ be divided by 54. The quotient is 73, with remainder 40∣-∣; and to obtain the complete quotient, this remainder must be divided by 54 (see § 11), giving ∙⅛⅜4×⅛= 1 i.e., 73 1 ⅛⅛ is the result of the division.
A complex fraction is reduced to a simple one by dividing the numerator by the denominator; thus,
6⅛ - 77 _9 33
10⅛ - 12 × 91 “52 ·
When one term only is fractional, it will be found convenient to multiply both terms of the complex fraction by the denominator that occurs in the fractional term; thus,
13∣ 13∣×4 55 5
22 “22 × 4 “22 x 4 “8 - Decimals.— 22. In the ordinary denary notation, a figure in combination with others has only the tenth part of-the value it would have if removed a place towards the left (§1); thus, in 374, the 3 signifies 3 times 100; the 7, 7 times 10; the 4, simply 4. By an extension of this notation we obtain a species of fractions that are often of very great use, especially for purposes of comparison. If we mark the place of units by a point put after it, and write other figures after the point, we can denote by the first of these figures One-tenth of the value it would have in the units’ place; by the second, one-hundredth part, and so on. In 374’691, then, the 6 is 6 times ⅛ i.e., 1 %; the 9, 9 times τ <⅛, or τ ⅞ ιy ; and the 1, rξ ⅛ 7y . Whence, by giving these fractions a common denominator and adding them, we have 374∙691 = 374 1 ‰ These decimal fractions or decimals, therefore, are fractions of which the numerator only is written, the denominator being the continued product of as many tens as there are decimal figures.
In addition, subtraction, multiplication, and division of decimals, the operations are, and from the structure of decimals must be, the same as the like operations with integers. The position of the decimal point in the results is the only thing that needs particular explanation.
Results, it may be mentioned here, are often expressed by decimals that are not exact, but approximate only, it being held sufficient to give the correct value to some assigned number of decimal places. An amount correct to four decimal places differs from the true amount by less than the ten-thousandth part of unity. See above, p. 524.
To reduce a decimal to a vulgar fraction, write the decimal as the numerator of the fraction, and set under it for the denominator 1 followed by as many ciphers as there are decimal places. This follows at once from the definition of a decimal.
To reduce a vulgar fraction to a decimal, annex ciphers to both terms of the fraction—the same number in both cases; divide both terms by the significant figures of the denominator, and then write the numerator as a decimal, pointing off as many decimal places (prefixing ciphers, if necessary) as there are ciphers in the denominator. (See also § 25, infra.} Thus, i ⅞ tr = 4 o o oxr = ιoo ε oo “ 'θθ75. Here the vulgar fractions retain their values unchanged (§ 12), and from the last the decimal is set down according to the definition.
It very often happens that in dividing as above we find there must always be a remainder. In this case, however, a remainder we had before may soon recur, giving the same figure or group of figures over and over again in the quotient; thus ⅜⅞ is found to be *47222; v 6 ⅛ is
•05729729 The recurring figures are distinguished by points placed over them, the above results being written •472 and - 05729. We shall return to the consideration of these recurring decimals at § 26.
23. To add or subtract decimals, write the numbers under each other, placing units under units, <tc., add or subtract as with integers, and place the decimal point under the points in the given numbers. This follows directly from the definition of decimals.
24. To multiply decimals, multiply as with integers, and point off as many decimal places as there are in both factors taken together. Thus, in 37 ’ 64 × - 082 = 3 - 0S648, the 4 and 2 in the factors are τ ⅜ σ and 10 2 00 ; their product is therefore ιoo 8 ooo > *'∙ β ∙> 'θθθθ8; and so with the others.
If an approximate product be sufficient, the multiplication may be shortened thus. Let it be required to find the product, say, of 4 - 273 and 6*859 correct to three place3- of decimals. Write the figures of the multiplier in reverse order, with the unit figure 4273 under the third decimal place of the multipli-cand. Begin each multiplication with the 25638 figure above the multiplier, adding what has 3418
to be carried from the right hand figure, and 214
observing to carry the nearest ten (i.e., for 36, 38
for instance, not 3, but 4); place the first 29-308 figures of each multiplication under each other; then add and point off three decimal places. It will be· seen that every figure of the product falls into its proper place, e.g., the 8 and 7, the 5 and 2, <fcc., each give thousandth parts.
25. To divide decimals, divide as with integers, and point off as many decimal places in the quotient as the dividend has more than the divisor. This follows from the dividend being the product of the divisor and quotient. Exs. 228∙956 ÷ 3∙64 = 62 9; 2∙28956 ÷ 36∙4 = ‘0629; 22895’6 ÷'364 = 62900. In the second example, the dividend having five decimal places and the divisor one, the quotient must have four; a cipher is therefore prefixed to the three figures. In cases like the third example, where the dividend has fewer decimal places than the divisor, the number has to be made up in the former by ciphers, and as far as the end of the ciphers making up that number the quotient must be an integer. The quotient may be pointed at any stage after as many decimal places of the dividend have been made use of as there are in the divisor, the first figure after this being always the first decimal. This is especially to be attended to when the division does not terminate, or when a few figures only are required; thus, 63 , 94 divided by ∙237 to two decimal places gives 269∙79; 10 divided by *264 gives 37’87.
The method of converting a vulgar fraction into a decimal, given in § 22, is in effect the division of the numerator by the denominator, the result being expressed as a decimal. Thus, τ ⅞⅛ = 83 ÷ 160 = *51875.
When the quotient is required to a given number of places only (as three in the example), the operation may be shortened by dropping the last figure of the divisor at each successive multiplication. But this must not be commenced till the figures required in the quotient are fewer than the figures in the divisor, and the carriage from the dropped figure is to be added in each instance. [table]
26. We have seen that the division by means of which vulgar fractions are converted into decimals (§§ 22, 25) will, in certain cases, always leave a remainder. If the fraction be in its lowest terms, there must always be a remainder whenever the denominator contains any other prime number as a factor besides 2 and 5. For in adding ciphers we multiply by tens, and we introduce no other factor. It often happens that we can speedily discover what the decimal must be, so as to be able to set down any number of figures without further actual division. Unless the division terminate, the same figures will recur sooner or later; and the same figures must recur before we use as many ciphers as there are units in the divisor. Supposing the divisor is 17, we can only have the numbers from 1 to 16 as remainders; the quotient must therefore repeat itself after all these remainders occur, if not earlier. All do occur in dividing by 17; thus, 1 ⅛=∙i 176470588235294.
The fraction τ 5 ^∙ must always be greater than the decimal *41666...., however far the latter be extended; but we can make the difference as small as we choose. Thus •416666 does not differ from ⅛ by the millionth part of a unit. The form of the decimal - 416 is used to show that the 6 is to be considered as repeated continually; and this being understood, we can say that the decimal is equal
Decimals of this kind are called Recurring Decimals. They are sometimes distinguished as Repeating or Circulating Decimals, according as one figure or more than one recur; and as Pure or Mixed, according as the recurring figures stand alone or are preceded by non-recurring decimals. Thus, T48 is a pure circulating decimal; T83 is a mixed repeating one.
27. To reduce a recurring decimal to a vulgar fraction, subtract the decimal figures that do not recur from the whole decimal; set down the remainder as the numerator of the fraction, and as many nines as there are recurring figures, followed by as many ciphers as there are nonrecurring figures, as the denominator.
The reason of proceeding in this way will appear if, taking any mixed recurring decimal, we multiply it by as many tens as there are decimal places, and again by as many tens as there are non-recurring decimals, and subtract the second product from the first. Take, e.g., *79054: 100000 times *79054 = 79054Ό54054054....
and 100 times ∙79054 = 79Ό54054054....
therefore 99900 times *79054 = 78975 whence *79054 = ⅞⅞‰≡, i.e., ⅜⅛
In the case of pure recurring decimals, we have no subtraction, and the denominator consists entirely of nines. Thus ∙3 = ⅜ = ⅛ j ∙27 = ⅜∣ = ⅛.
28. In practical arithmetic recurring decimals are little used, except in approximations. They can be added or subtracted readily to any number of places by extending all a place or two beyond what is required. When we have to multiply or divide by recurring decimals, we must convert them into vulgar fractions. They may themselves be multiplied or divided by integers or ordinary decimals, though in this case, too, it is often better to use vulgar fractions, especially when exact results are desired.
Powers and Roots of Numbers.— 29. When a given number is multiplied by itself, the product multiplied again by the number, and so on, the result obtained is called the second, third, fourth, &c., power of the number, according as it is the product of the number repeated two, three, four, &c., times. Thus, 7 × 7 × 7 = 343 is the third power of 7; 5×5×5×5 = 625, the fourth power of 5. The term “square” is nearly always used instead of “second power” {e.g., 81 is the square of 9), and “cube” frequently instead of “third power.” The power to which a number is raised is indicated by a small figure written over the number to the right; thus, 8 5 is the fifth power of 8.
The square root of a given number is the number which, when multiplied by itself, produces the given number. And so, in general, whatever power one number is of another, the same root is the second of the first. Thus, 7 is the third root, or cube root, of 343; 5 is the fourth root of 625; 2 is the fifth root of 32. The sign s ∕ (which is really an r, from radix, a root) prefixed to a number indicates a root of it. The simple sign stands for the square root; a figure is placed over it to denote other roots. Thus, √100 is the square root of 100; ^∕256, the fourth root of 256.
30. To extract the square root of a given number, divide it into periods of two figures, by putting a point over every second figure, commencing with that in the units’ place; set down as the first figure of the root, the largest number whose square does not exceed the first or left-hand period; place the square of this number under the first period, and subtract it from it; to the remainder annex the next period; place before this as a trial divisor twice the root figure; consider how often the former (omitting the right-hand figure) contains the latter, and set down the number that expresses this as the next figure of the root; place also this root-figure to the right of the trial divisor; multiply by it the divisor thus completed; subtract the product from the number formed of the former remainder and the period taken down; add another period to the remainder now found; then double the whole root for a new trial divisor, and proceed as before. [table]
The 58 of the example being 580000, the 49 must be 490000, and the root 700. So the 146 is 1460; the 921, 92100; the 6 is 60; and so on. This explains the principle of pointing in periods of two figures.
In place of doubling the whole root for every trial divisor, it will be sufficient to add to the preceding complete divisor its last figure; thus, 146 + 6 = 152. The 7 has been already doubled in 14, and this doubles the 6.
The reason of proceeding as above will appear from the composition of an ordinary product. Thus, 23 X 23 is the sum of the four products 20 × 20, 3 × 20, 20 × 3, and 3 × 3, i.e. , 20 2 , twice 20 × 3, and 3 × 3. Taking away, then, the square of 20, i.e., 400, the remainder must be the sum of 2 × 20 × 3, and 3 × 3, i.e., 3 times the sum of 2 × 20 and 3, which we obtain by the method adopted. [table]
If there is a remainder, ciphers may be taken down in pairs, and as many decimal places obtained as we please. In this case there must always be a remainder, since no unit multiplied by itself produces ciphers. After getting half the decimal places required, we may proceed by contracted division (§ 25).
To extract the square root of a vulgar fraction, we find the roots of the numerator and denominator separately. For, since £ × ∣∙ = τ s i , the square root of τ 0 s - must be If either term of the fraction is not a complete square, it should be reduced to a decimal. Thus, s ∕⅛ = s ∕ ’ 5 = x ∕0 - 50 = •7071067812 nearly.
31. To extract the cube root of a given number, point off the number from the units’ place into periods of three figures; write under the first period the greatest cube contained in it, subtract, and annex the next period to the remainder; then, regarding the root found as tens, multiply the square of it by 3 for a trial divisor, and divide by this for the next figureof the root; to the trial divisor add three times the product of the two root numbers (the first being tens), and also the square of the last root figure; multiply the sum by the last root figure; subtract the product from the number obtained by taking down last period; annex the next period to the remainder, and proceed as before. [table]
For demonstration of the reason of this process, see Algebra, voL i. p. 528. It depends on the form of the product obtained in raising a number to its third power. Thus, 24 x 24 x 24 is 24 times 20 2 + 2 × 20 χ 4 + 4 2 , which will be found to be 20 3 + 3 × 20 2 × 4 + 3 x 20 × 4 2 + 4 3 . And this is 20 3 added to 4 timee 3 × 20 2 + 3 × 20 × 4 + 4 2 , which agrees with the process described above.
II. Practical Arithmetic.
32. Having explained in the foregoing sections the various operations of arithmetic, we now proceed to consider them in their combinations and practical applications. What has been said up to this point refers to numbers merely as numbers, or numbers in the abstract. Now they are to be regarded as applied to particular things, or representing particular magnitudes. Numbers so regarded are called concrete; and we now treat of concrete as distinguished from abstract arithmetic. Concrete numbers frequently represent not so much number as quantity. To form a distinct and accurate idea of 5 lb of tea bought for 15s., it is not necessary to think of the tea as divided into five portions, or as paid for with fifteen pieces of money.
It would be found extremely (indeed intolerably) inconvenient to have to make all payments, great and small, by means of one particular species of coin, or to serve out all quantities of goods, using only one kind of weight or measurement. Various monies, weights, and measures are therefore in customary use, this or that being employed in each particular case according to circumstances. When these measures are of the same kind, differing only in the unit of one of them being so many times the unit of another, they are said to be of different denominations; as, pounds, shillings, pence; or again, yards, feet, inches.
In addition and subtraction, the quantities added and subtracted must be either abstract numbers or concrete quantities of the same kind. In multiplication, the multiplicand may be concrete, but the multiplier is regarded in the process as abstract. If 20 men, for example, receive £5 each, the £5 is not multiplied by 20 men, but taken 20 times, the number of times merely corresponding to the number of men. In division, when the dividend is concrete, the divisor may be abstract, giving a concrete quotient of the same kind as the dividend, or concrete giving an abstract quotient. Thus, £100 may be divided into 20 parts, giving £5 as quotient, or it may be divided into parts of £5 each, giving as quotient the abstract number 20, i.e., containing £5 20 times. A fraction is strictly abstract, though we often write £| and the like for of £1.
33. The following are the tables of monies, weights, and measures in common use :— [table]
On these tables we make the following remarks :—-
I. The guinea is 21 shillings, and the crown 5 shillings. Sales are often made in guineas, though the coin is not now used. Farthings are always written as fractions of a penny. 10∣<Z. ⅛. means tenpence three farthings and seven-eighths of a farthing.
III. Apothecaries' weight agrees with this, except that the ounce (℥) is divided into 8 drams (5), and the dram into 3 scruples (3) of 20 grains.
IV. In measuring cloth, the yard is divided into 4 quarters, the quarter into 4 nails of 2¼ inches.
V. To measure land, the surveyor’s chain of 100 links is used. The chain is 22 yards long, and 10 square chains make an acre.
VII. The gallon and lower denominations are liquid measures; the quart and those above it are for dry goods.
VIII. The year, strictly speaking, is 365∙24224 days. Every fourth year (leap year) has 366 days.
34. Reduction is the method of expressing quantities in a denomination lower or higher than that in which they are given.
To reduce a higher denomination to a lower, multiply in succession by the numbers which show the times that the unit of each denomination (beginning with that given) contains the unit of the one next below it till the denomination is reached to which the quantity is to be reduced. If quantities in intermediate denominations are given, add each as its denomination is reached.
To reduce a lower denomination to a higher, divide in succession by the numbers which show the times that the unit of each denomination (beginning with that given) is contained in the unit of the next above it, observing that the remainder after such division is of the denomination of the dividend.
Sometimes the two processes are combined; thus, in reducing guineas to crowns, we multiply by 21, obtaining shillings, and then divide by 5.
35. Compound Addition and Compound Subtraction are the addition and subtraction of quantities expressed in more than one denomination.
In Compound Addition, arrange the quantities according to their denominations, each under its proper heading; add the lowest, and reduce the sum by division to the next higher, setting down the remainder, and carrying the quotient; add the others, including carriage, in the same way. Suppose that several sums of money are added, and the farthings amount to 29, that is to 7⅛d., the {d. is set down and the 7d. carried to the pence column, and so in other cases.
In Compound Subtraction, arrange the quantities as in Compound Addition, placing the greater amount over the other, and subtract, beginning with the lowest denomination. If in any case the lower number exceeds that above it, increase the latter by as many as make one of the next higher denomination, and afterwards add one to that denomination in the lower line. In subtracting, e.g., 1 qr. 25 lb from 3 qr. 17 lb, since 25 cannot be taken from 17, a quarter, i.e., 28 fi>, is added to the 17, making 45; from this 25 is subtracted, leaving 20 δ), and the quarter “borrowed” is taken away again by being added to the 1 qr. The remainder is thus 1 qr. 20 ft.
Fractions of the lowest denomination are to be added or subtracted according to §§ 18, 19; thus, £72, 8s. 3⅛d. - £45, 17s. 6||d. = £26, 10s. 8|d.
36. Compound Multiplication is multiplication in which the multiplicand is expressed in more than one denomination.
When the multiplier does not exceed 12, multiply the different denominations by it, beginning with the lowest, and setting down and carrying as in Compound Addition. When the multiplier consists of several figures, multiply by each separately in the same way, taking them from right to left, and setting the result of each successive multiplication always one place further towards the left in 12 ∕f each denomination, and add the results as thus arranged. 254 10 7£= 4 times.
In the accompanying example 572 13 9∣ = 90 times, the above arrangement puts ^⅞T^S~8⅛ = 94 times, the product by 90 in the place of tens; and the sums to be reduced to higher denominations are 30 farthings, 104 pence, 148 shillings. If the multiplier be a composite number, we may multiply by the factors in succession. When the multiplier consists of two or more figures, the multiplication is often performed by the whole quantity at once.
37. Compound Division is division in which either the dividend or both dividend and divisor are expressed in more than one denomination.
(1.) To find the amount that a given amount contains a given number of times, divide the highest denomination by the given number, reduce the remainder to the next lower denomination, adding the corresponding term of the dividend; divide again, and proceed in the same way with the other denominations. The denominations of the quotient correspond to those of the dividend.
Let it be required, for instance, to divide £370,16s. l∣d, into 58 equal shares. As 58 shares of £6 each amount to £348, £6 is part of the quotient, and there remains £22, 16s. lfd. Reducing £22,16s. to shillings (456), we find that this gives 58 shares of 7s. each, with 50s. over. Similarly we obtain lOd. and 1 farthing, with 29 farthings over, which, since ⅜⅜ = ⅜, is just half a farthing for each of the 58 shares. The quotient then is £6, 7s. 10|d. ∣q.
(2.) To find the number of times that one given amount contains another, reduce both to the same single denomination, and then divide the one by the other.
To find, for example, how often 12s. 9∣d. is contained in £171, 13s. 9d., since 12s. 9|d. = 615 farthings, and £171, 13s. 9d. = 164820 farthings, the number of times the second amount contains the first must be 164820 ÷ 615, i,e., 268. It is to be observed that the quotient here is an abstract number.
38. In multiplying or dividing by fractions or mixed numbers, we follow the methods explained in §§ 20, 21. As an illustration, we give here the multiplication of £24, 5s. 9∣d. «q. by 62- 1 ⅞-, where we may note that in the division by 12 we have to divide 2∣ farthings, giving ™q. × ⅛ = Aq., and that in adding the fraction we have V + A = ⅛ 3 r+ τ κ A=2⅓q. [table]
39. Reduction of Fractions and Decimals.— To find the proper value of a fraction or a decimal of any denomination, multiply the fraction or the decimal by the numbers in succession that reduce the denomination to lower denominations.
Thus, to find the value of ⅜⅜ of a pound, it is manifest that this is ⅜⅜ of 20 shillings, t.e., ∣⅜ × 20s. = 12 1 ⅛8., or 12s. 3|d.; and so in other cases.
To find the value of a fraction of a quantity consisting of different denominations, we may either first reduce the quantity to one denomination, or we may multiply the compound quantity by the numerator, and divide by the denominator of the fraction; thus, ⅜ of £5, 7s. ll∣d. being the same as the ninth part x>f twice the amount, we may multiply by 2 and divide by 9, obtaining £1, 3s. ll∣d. ⅜q. as the result.
40. To reduce any amount to the fraction or the decimal of another denomination or amount, reduce both amounts to the same denomination, and write them as the terms of a fraction, the quantity of which the fraction is required being made the denominator. If the decimal is required, convert the vulgar fraction into a decimal.
Thus, to reduce 13s. l⅜d, to the fraction and also to the decimal of a pound, since a pound contains 480 halfpence, a halfpenny is τ ⅛ σ 0 f a pound, and therefore 13s. l∣d., ue., 315 halfpence, is <if∣⅜ = £|| = £-65625.
It is often sufficient to throw the expressions into the reduced form, without actually performing the reduction, and divide the numerator of the complex fraction so obtained by the denominator. Thus, to find the fraction of £4531. ⅛ a t 3. 0 f 46∣ guineas is, both terms may be expressed as shillings,—∣ × × W> and "H∏Γ × *Γ> whence, dividing the former by the latter, the required fraction is found to be ⅛⅛.
In reducing a compound quantity to a decimal of a higher denomination, it is generally best to proceed by successive divisions, beginning with the lowest denomination, and inserting the others as they occur, as in the accompanying example, where 13 cwt. 8 lb 12 oz. is found to be ∙65390625 of a ton. It is to be noted that the integers here are inserted from the data, and the decimals obtained by division. The form, though somewhat incongruous, is convenient. [table]
41. Practice is the name given to a method of calculating pπces from certain rates being aliquot parts (that is, exact measures) of other rates.
The basis of this species of calculation is generally £1, and the given price is broken up into portions, the first of which is an aliquot part of a pound, and each of the others an aliquot part of some one before it. By the method adopted we virtually multiply the different aliquot parts in succession by the number of articles given. The sum of the result is the total price.
42. Proportion.— When the first of four quantities is the same multiple or the same fraction of the second that the third is of the fourth, the first is said to have the same ratio to the second that the third has to the fourth, and the four quantities are said to be proportionals, or in proportion. Thus, since 30 is ⅜ of 36, and 50 is ⅛ of 60, the numbers 30, 36, 50, and 60 form a proportion.
The proportion is written 30: 36 = 50 :60, or, more commonly, 30: 36 :: 50: 60. This is read 30 is to 36 as 50 is to 60. It may also be written ⅜⅞ = ⅞⅞.
The ratio of two numbers is thus equivalent to the fraction that the one is of the other;. and a fraction may therefore be defined as the ratio of the numerator to the denominator.
Ratio is a mere abstract relation between two numbers, or between two concrete quantities of the same kind. The ratio of 4s. to 6s., or of 48d, to 72d., is not shillings or pence, but the abstract fraction ⅜.
If two quantities of one kind be proportional to two of another kind, when any three are given, the fourth can be found. If, for instance, we know the value of any quantity of goods, we can determine the value of any other given quantity, or the quantity that has any assigned value, —it being always understood that the rate is the same in both cases; that is, that were the quantity doubled, trebled, halved, &c., so also would the value be, or, in other words, that the quantity is proportional to the value. Questions of this sort occur with very great frequency in practical arithmetic. The three quantities given are usually arranged as the first three terms of a proportion, whence the fourth term is found.
The rule by which we proceed in such cases of Simple Proportion, as it is called, has often the name given to it of the Rule of Three. It is as follows:—Of the three quantities given, set that down for the third term which is of the same kind as what is required. Consider whether the amount to be found will be greater or less than this third term; if greater, make the greater of the two remaining quantities the second term, and the other the first term; but if less, put the less term second, and the greater first. Having thus arranged or “stated” the three terms of the proportion, multiply the second and third together, and divide the product by the first.
The first and second terms must be reduced to the same denomination, and it will often be convenient to reduce the third term to the lowest denomination contained in it.
Ex. 1. If 54 yards of cloth cost 63s., what will 30 yards cost at the same rate?
Stating by the rule, we have 54 yards: 30 yards :: 63s., whence the fourth term is = 35 s-
54
This process is to be explained by the consideration that, since the rate of price is the same for both quantities, the one price must be the same fraction of the other that the one quantity is of the other: that is, = ≈- ,-~-—-—3 , 3 30 yds. the price required ’
30 the number of shillings required or - = ——≡ i — . Multiplying each of these equal fractions by 63, we have fθ- × 63 = 35, the number of shillings required, as above.
The first and second terms are to be stated according to the rule, because their ratio is equal to that of the third and fourth, and must therefore be a proper or an improper fraction, according as the other is so. After the proportion is stated, and the terms reduced, any common factor may be removed from the firβt and second terms, or from the first and third; for, as will appear from the working above, this is virtually reducing a fractional expression to lower terms.
Ex. 2. A bankrupt, whose debts amount to £1275, pays 14s. 6d. in the pound. What do his creditors lose?
Since 14s. 6d. is paid, there is 5s. 6d. loss for every pound of debt, and the question is—If £1 give 5s. 6d. loss, what will £1275 give? The “stating,” therefore, is— £1: £1275 ::5s. 6d., and the result, £350, 12s. 6cL Particular care must be taken, when all the terms are money, as here, that the first and second be of the same kind. In this instance these terms are debt; and the third term, loss, corresponds to the term required.
Ex. 3. If 91 men could perform a piece of work in 78 days, in what time could 21 men do it, working at the same rate?
Here, if the number of men were doubled, trebled, halved, &c., the time required would be one-half, one-third, double, &c., the given time; or as the former is increased the latter is diminished in the same proportion, and vice versa. The time in this case is said to be inversely proportional to the number of men. We have then as equal .. 21 men 78 days ,. , . ., .. , r . ,, .
ratios — = 5 —.—3: that is, the “stating of
91 men days required the proportion is 21 men: 91 men :: 78 days, according to the rule, and the result 338 days. The process may alβo be explained thus:—The work will be 91 × 78 times what 91 × 78
1 man can do in 1 day, or ——— times what 21 men can ju1
91 × 78
do in 1 day, i.e., it will be ——— days’ work for 21 men. AL
43. Compound Proportion.— The ratio of two quantities frequently depends on a combination of other ratios. If, for instance, we have to compare the times required for building two walls, one twice the length, twice the height, and twice the thickness of the other, the men employed on the former being half the number employed on the other, and the day half the length, each of these separate conditions implies double the number of days. Each condition gives the ratio 1 :2, and the result must correspond to the product of all the ratios, that is, it is 1:32. This is an instance of what is called compound proportion.
In such caseβ set down for the third term the quantity which is of the same kind as that required. State each proportion as though it alone had to be considered, writing them under each other. Multiply the product of all the second terms by the third term, and divide the result by the product of the first terms.
Ex. If 36 men, working 10 hours a day, perform three-fifths of a piece of work in 17 days, how long must 25 men work daily to do the rest of it in 16 days?
Men, 25:36:: 10
Fifths of the work, 3: 2 Days, 16:17
36×2×17×10 51 , A . . ,
= — = 104 hours a day.
25×3×16 5 δ ,7
The length of the day will be greater the fewer the men and the fewer the days are, and less the less the work is; we therefore state as above.
44. It not unfrequently happens that ratios have to be compounded, or other reductions made, before we can state the proportion which will give the required result. We give an example or two of this.
Ex. 1. Though the length of my field is one-seventh greater than that of my neighbour’s, and its quality is one-ninth better, yet, as the breadth of mine is one-fourth less, his is worth five guineas more than mine. What is my field worth!
The length of his field is to that of mine as 1 to 14, i.e., as 7 to 8. The other ratios are 9 :10 and 4 :3 (see §16 ad fin.). Therefore the values of the fields are as 7 × 9 × 4 to 8 × 10 × 3, i.e., as 21 to 20, and the difference of these values being £5, 5s., we have the proportion 1: 20 :: £5. 5s., which gives £105 as the worth of my field.
Ex. 2. If 9 men or 15 women, working 10 hours a day, could reap a field in 8 days 6 hours, in how many days of 10⅛ hours each could 10 men and 12 women reap a field one-fourth larger 1
Since 15 women do as much as 9 men do in the same time, 12 women will do 7⅜ times a man’s work, for 15:12:: 9: 74. Therefore 10 men and ,~ 1 1κ op , ir⅜ *11J Ί er 1 j · ' lljî lv ∣∙Ov ilΓS.
12 women will do 17f times a mans . g work. From this the stating in the margin follows. The result is 56∣ hours, i,e. f 5 days 3∣ hours, the day being 10£ hours long.
Ex. 3. A dealer who has bought 9 oxen and 5 sheep for £186, 2s. 6d., would lose £2 by exchanging 2 oxen for 11 sheep. What is the price of an ox Î
The price of 2 oxen being the price of 11 sheep and £2 more, the price of 9 oxen will be (from the ratio 2:9) 49 ⅜ times the price of a sheep and £9 more. Hence from the data, 54∣ times the price of a sheep and £9 more will amount to £186, 2s. 6d.; i.e., 54∣ times the price of a sheep is £177, 2s. 6d., and therefore a sheep cost £3, 5s. Also, since the price of 2 oxen is that of 11 sheep and £2 more, 2 oxen cost £35, 15s. +£2, i.e., £37, 15s.; therefore 1 ox cost £18, 17s. 6d.
45. Proportional Parts.— To divide a number or quantity into parts in proportion to given numbers, state and work out the proportions,—As the sum of the given numbers is to each of them in succession, so is the number to be divided to the several parts required.
If, for example, a bankrupt owes A £580, B £935, C £675, and D £770, and his assets amount to £999, the stating 2960:580: £999 gives £195, 15s. as A’s share, and the others are found similarly.
Here, too, there may be a compound proportion, as when different sums are invested for different times. The division in those cases must be in proportion to the amounts invested, and also to the time; each amount is therefore to be multiplied by its time.
Applications of Proportion.— 46. In commercial and financial transactions frequent use is made of proportion; and very often, when it is not directly employed, computations are performed according to formulae or rules which rest on this as a basis. Advantage is very generally taken of the convenience of 100 as a standard of reference or comparison, proportional relations being stated as at such and such rates per cent. This occurs continually in the calculation of interest, discount, stock-exchange operations, &c., as well as in the expression of mercantile losses and gains.
47. Interest is the allowance given by the borrower to the lender for the use of money lent. It is usually computed at a rate agreed upon of so many pounds for every hundred lent for a year; this is called the rate per cent. The interest of £564, for instance, for
3 years 4 months at 3¾ per cent, per annum, is to be found by a compound proportion, the meaning being—If the interest of £100 for 1 year be £3|, what will the interest of £564 be for 3⅛ years? The result (£70, 10s.) may be obtained by the general rule based on this and similar proportions—Multiply the amount lent (called the principal) by the rate per cent, and by the number of years, and divide the product by 100. When the time is given in days, the fraction of a year is taken that the days amount to. [table]
Money is laid out at Compound Interest, when at the end of a year or other assigned period the interest that has accrued is not paid to the lender, but is put to interest along with the amount originally lent. Here the simple interest has to be computed for each successive year or period, and added to the principal or former amount.
48. Commission is the allowance paid to an agent for transacting commercial business, and usually bears a fixed proportion or percentage, as may be agreed on, to the amount of value involved in the transactions. Brokerage is the allowance paid to a broker for buying or selling shares in the public funds, or bargaining otherwise with reference to money investments. Insurance Premiums are payments in return for which the owner of the property insured is entitled to receive the assured value of it in the event of its being destroyed. In all these the rates are commonly stated at so much per cent., and the computation is similar to that for interest, but simpler, as the element of time does not enter into it.
49. Discount is a deduction allowed for a payment being made at a date prior to the time when the full amount is exigible. The true discount on a money payment due on the expiration of a certain time is the excess of the amount over its present value— the present value being the sum which, laid out at interest, would in the given time amount to the given sum. Suppose, for instance, it is required to find the true discount on £664 due 10 months hence at 4⅛ per cent. Here the interest of £100, being £4| in 12 months, will in 10 months be £3f, ».<?., £100 would amount to £103f in 10 months, and the discount of £103| due 10 months hence is therefore £3^. Hence the stating £103f: £664 ::£3f, which gives as the discount £24. The present value is seen from this to be £640. A banker or merchant, in discounting a bill, charges interest instead of discount, and would in this instance gain the interest of £24, since the £24, the true discount of £664, is evidently the interest of £640.
50. In calculations relating to mercantile Profit and Loss, which are also effected by proportion, it must be carefully remembered that the percentage of gain or loss is always reckoned on the buying price, unless the contrary is expressly stated. Thus, let it be required to find the cost of goods that are sold for £448, 17s. 6d. at a loss of 6^ per cent. Since goods that cost £100 are sold for £93|, we have the stating £93|: £448J ::£100, which gives £478, 16s. as the cost price. (a. m‘a.)
