Oldalképek
PDF
ePub

whole year.

:

we

4

900

[ocr errors]
[ocr errors]
[ocr errors]

to

:

20

£

lation, we find that B will receive £96, 42., and C £74. certain rate per hundred pounds per annum, or for the Another method of com

£
£
.£

The hundred pounds, for shortness, is putation would consist in

1000 370

1

called cent., a contraction of centum (Latin), a hundred. ascertaining how many

20

If the money is lent for a less term than a year, then shillings per pound the

it becomes a question in arithmetic what is the proporeffects would yield. In

1000)7400(7s. 4 d. į

tion of interest chargeable. According to the existing this case state the

7000

| laws (1849), £5 per cent. is the highest legal interest terms as annexed. An

400

that is chargeable on money lent in any other form swer, 78. 4fd. }, or a

12

than by a bill or promissory-note. The interest, comfraction more than seven 1000)4800(4 monly called discount, that may be taken for advances shillings and fourpence

4000

on bills or promissory-notes is left unlimited. In gethree farthings in the

neral it varies from about 4 to 6 per cent., according

800 pound. By allotting this

to the state of the money market or trustworthiness rate of dividend to each

of the borrower. £5 being equal to 100 shillings, creditor, their respective

1000)3200(3

the interest on £l for a year at 5 per cent. is conseshares would be liquidated

3000

quently ls.; and if the interest be £2, 10s., or 2 per as above. Instead of di

} cent., the charge will of course be 6d. per £l. This is viding 7400 by 1000, the same result will be found if so very simple a matter of calculation, that interests 74 be divided by 10.

of 5 or 2 per cent. per annum may in most instances Questions in arith- Men.

Men.

be calculated mentally, or at least with a few figures. metic arise in which

16 45

20

When the interest is 3, 3, 4, or 4., per cent., and for sixth term is

28

35

a less period than a year, the calculation is more required, bearing 128

700

complex, and will require to be wrought as a question proportion 32

45

in Rule of Three, or the amount may be determined by five terms already 448

3500

an appeal to Interest Tables. known. This, which

2800
£ s. d.

Interest on broken periods is not calculated by caleninvolves what is called Double Rule of

418)31500(70 6 3

dar months, but for days—the exact number of days 3136

from the day of lending to the day of paying; and thereThree, is exemplified as

fore the calculation of the number of days is an imporfollows:-If I give 16

140

tant preliminary in the transaction. Bankers and mermen £45 for 28 days'

chants, to save the trouble of calculation, appeal to a work, what must I give,

448)2800(6 table which shows the number of days from one day at the same rate, to 20

2688

to another in the different months of the year. men for 35 days' work?

112

The following is an example of the mode of working The answer, according to

12

a question of interest for a whole year. What is the the plan of working here

£ £ 448)1344(3

amount of interest payable on shown, is £70, 6s. 3d.

1314
100 : 45 :: 649 £649, at 4} per £100 or per cent.

44 per annum
Practice and Mental Arithmetic.

2596 Under these names are comprehended short practical

Or shorter, as under-methods of working arithmetical questions, partly or

100)2920-10(29 4 1} £ wholly by the mind, or by the jotting of a few figures.

200

649 Ordinary business questions are seldom solved by the

920

45 Rule of Three, but principally by Practice or Mental Arithmetic. The kind of questions most commonly

900

2596 occurring are computations of the aggregate value of

20

324-10 a certain number of articles at a certain price, and the

20

29,20-10 adding of the whole together to find the sum-total. The

'100)41004

20 plan usually adopted is to calculate the value of any

400

4,10 number of articles by the nearest round sum, and then

10

12 to apply the difference. For instance, a lady is buying

12

1,20

= £29 4 1} a gown from a haberdasher; she has received 15 yards at 5d. per yard; the haberdasher tells her in an

100)120(1 instant she has to pay 6s. 104d. He knows it is so by

100 saying internally to himself—15 yards at 6d. would be

$ 7s.6d.; then if I take 15 halfpence, that is, 7£d., from

When the period conthe 7s. 6d., I find that 68. 104d. will remain.

235 10

sists of less than a year Another principle followed in this practical arith

7

(365 days), multiply the metic is to work by aliquot parts. By remembering that a penny is the 12th of a shilling, or the 240th of a

1648 10

principal by the num

ber of days, and by pound; that Os. 8d. is the third of a pound; that 3s. 4d.

twice the rate, and diis the sixth of a pound, and so on, we are able to save

6594

vide by 73000. (We much of the ordinary figuring. As an illustration, let

6

s. d. get this 73000 by mul. it be required to find the value of 3567 articles at 3s. 4d. 73000) 39564 0 (10 10,5 15 tiplying 365 by 2 and each. By the usual rules of arithmetic, this question

20

by 100). For example would be performed by multiplying the 3567 by 40

791280

-What is the interest (there being 40 pence in 3s. 4d.), and then dividing

730000

of £235, 10s. for 28 by 12 to bring it into shillings, and by 20 to bring it

61280

days, at 3 per cent. per into pounds. The practical method is much shorter;

annum ? Here, for con

12 3s. 4d. being the sixth of a pound, if we

6)3567

venience, we begin by divide 3567 by 6, we at once obtain the

735360

multiplying by 7 and by amount thus

£594, 108.
730000

4, instead of 28. The

6 is twice the inteInterest.

rest, 3. Interest is an allowance for the use of money, paid When partial payments are made, it is necessary to by the borrower to the lender. The amount of allow- deduct them from the principal, care being taken not ance on £100 is called the rate of interest. The amount to confuse principal with interest. The following is a of money lent is styled the principal. In the United rule to follow in such a case:-Calculate the interest Kingdom, it is customary to reckon the interest at al on the principal up to the time at which the first par

324-10 £ s. d.

£ s.

0

5360 * 3006

[ocr errors]

580

Evolution.

1

2

tial payment is made, and add it to the principal; from a second layer of dice on the first, and the number will this sum subtract the money paid, and the remainder be doubled, or 288 dice; and if we thus go on adding is a new principal; compute the interest on this prin- layer above layer till we have 12 layers, the number cipal from the time of the first payment up to the time of dice will be finally 12 times 144, or 1728 dice; in of the second payment, add it to the latter principal; other words, we shall have formed a cube consisting of from the sum subtract the second sum paid, and the 1728 solid inches. Such is solid measure. remainder is again a new principal; and continue this Solid measure is computed arithmetically, by multiprocess till the last payment.

plying the length by the breadth, and the product by

the thickness. Civil engineers, who require to calcuMensuration.

late the solid contents of masses of earth, with a view Mensuration refers to the measurement of objects, to excavation, resort to this simple rule; it is likewise and is of three kinds--lineal, or measuring by mere

29

followed by builders in reference to length; superficial, which respects breadth as well as

20

walls, plumbers to cisterns, and other length; and solid, which includes length, breadth, and

artificers. The following is a questhickness. In the United Kingdom, the foot of twelve

580

tion not unlikely to occur. Required inches is the common standard of measurement. A

14

the contents of a cistern 9 yards 2 draper measures cloth with a rod of three feet or one 2320

feet in length, 6 yards 2 feet in yard, and workmen usually measure the dimensions of

breadth, and 4 yards 2 feet in depth. walls, or the superficies of apartments, by a ribbon 27)8120 Analyse these dimensions—9 yards 2 marked in feet and inches. In common usage, the inch

300 y. 20 f. feet are 29 feet; 6 yards 2 feet are 20 is divided into eighths and sixteenths.

feet. Multiply the 29 by 20, and then As lineal measurement requires no explanation, we multiply by 14 (14 feet being equal to 4 yards 2 feet); pass to a consideration of superficial measurement, or next divide by 27, which is the number of solid feet in that of both length and breadth. A superficial foot, a solid yard. The question is thus wrought in the which is the basis of this kind of measurement, is margin. The answer is seen to be 300 yards 20 feet. either a square of a foot in length and a foot in breadth -in other words, a foot each way—or it is any dimension in which the length multiplied by the breadth will form The extraction or discovery of the square and cube a foot. For example, the surface of a piece of wood, roots of numbers, forms a department of arithmetic 2 feet in length and 6 inches in breadth, is a superficial | called Evolution, and is useful in some kinds of meafoot. A superficial foot is generally called a square surement. In the following table, the squares, cubes, foot, and is a superficies consisting of 12 times 12, or and fourth and fifth powers of the nine units are given. 144 square inches. Sometimes the term square feet is The square of any number, it will be observed, is confounded with that of feet square, which is quite a gained by multiplying the number by itself; the cube, different thing. A piece of cloth said to measure six by multiplying the square by the number; the biquasquare feet, consists of six squares of a foot each; but dratic by multiplying the cube by the number; and a piece said to measure six feet square would be six the sursolid by multiplying the biquadratic by the feet along each side, and comprise thirty-six squares number :of a foot each. Inattention to these distinctions has Root, or ist power, often led to awkward errors and disputes.

Square, or 2d power, 14 The method of finding the superficial contents of any Cube, or 3d power, oblong surface, is to multiply the length by the breadth; | Biquad., or 4th power, 1 16 81 256 625 1296 2401 but other points require attention in the calculation? Sursolid, or 5th power, 1 32 243 1024 3125 7776 16807 32768 39019 To arrive at exactness, the inch is reckoned to contain The powers of numbers are usually expressed by 12 seconds or parts, each second contains 12 thirds, adding a small figure to them; thus 32 signifies the and each third contains 12 fourths. Feet multiplied second power or square of 3—that is, 3 x 3 = 9; 33 sig. by feet, give feet; feet multiplied by inches, give inches; nifies the third power or cube of 3, or 3 x 3 x 3 = 27. feet multiplied by seconds, give seconds; inches multi- The small figure thus added is called an inder or erplied by inches, give seconds; inches multiplied by ponent, because it indicates or exposes to view the seconds, give thirds; and seconds multiplied by seconds, powers of the quantity to which it is affixed. The give fourths.

square root of any small quantity may easily be ascerRule for working questions—1. Write the multiplier tained by means of the multiplication table; for inunder the multiplicand, feet under feet, inches under stance, 6 is at once seen to be the square root of 36, inches, seconds under seconds, &c. 2. Multiply each because 6 x 6 = 36; but when larger numbers occur, denomination of the length by the feet of the breadth, it is requisite to have recourse to another process. beginning at the lowest, and place each product under Supposing it were required to find the length of the that denomination of the multiplicand from which it side of a square which contains 59,049 square inches arises, always carrying 1 for every 12. 3. Multiply by on its surface, it is evident that it cannot at once occur the inches, and set each product one place farther to to the mind what number, by being multiplied into the right hand. 4. Multiply by the seconds or parts, itself, produces this quantity. It is therefore requisite and set each product another place towards the right to find the square of one part of the number, and then hand. 5. Proceed in this manner with all the rest of the square of the other part. To effect this, the whole the denominations, and their sum will be the answer. number is divided into parts, by a dot being placed

Escample.-Multiply 6 feet 3 inches by 3 feet 2 over each alternate figure, beginning at the unit. The inches. In working, we begin by multiply

reason for thus separating it into periods of two figures ing the 3 inches by 3, and then the 6 feet

each, is because the square of a single figure never

3 also by the same 3 below it; this gives 18

consists of more than two places, the square of a numfeet 9 inches. This makes 18, 9. We now

ber of two figures of not more than four places, &c. inultiply by the 2 inches, placing the 6 one

18

After the whole number has been thus divided, the remove to a side. By then multiplying the

root whose square comes nearest to the first period 6 by 2, we have 12 inches or 1 foot, and 19 96 must be placed in the quotient, and its square subtracted setting down the one below the 18, we add

from the first period. Thus 5 being the first period, up. The answer is 19 feet 9 inches and 6 seconds. Questions of this kind may also be wrought by decimals.

2, as being the root of 4, is placed 59049(243

in the quotient, and the square of 2 To measure the solid contents of an object, a diffe

(2 x 2 = 4) is subtracted from 5, berent process is pursued. Suppose we take a piece of 44)190 cause 5 here forms the first period, wood measuring square foot, and cover it with dice, 176

The next period (90) is then annexed each die an inch square and an inch high, the wood 483)1449 to the remainder for a new dividend. will be covered with exactly 144 dice. Let us now put 1449 The root is doubled for a divisor, and

3 9 27

4 5 6 16 25 36 64 125 216

7 49 343

8 9 64 SI 512 729 4096 6561

1

Ft. In.
6
3 2

9
0

1

6

4

after ascertaining how often it is contained in the 8 is the required quotient, indicating how often 16 is dividend, omitting its last figure, the result (4) is contained in 4096-namely, 256 times. placed both in the quotient and also in the unit's place The indices or exponents, 1, 2, 3, 4, &c. might, howof the divisor : the whole number (44) thus formed, ever, denote the powers of any other number or ratio. must be multiplied by 4, and the product must be sub- Every different ratio or geometrical progression gives tracted from the dividend. The same process must be a different system of logarithms. Soon after the invenrepeated until there is no remainder, and the quotient tion of logarithms by Lord Napier, it occurred to will contain the root sought, which in this case is 243. Briggs, then professor of geometry at Oxford, that a The rule for extracting the cube root is equally tedious, system whose base or ratio is 10 is preferable to all and indeed almost too complex for practical purposes. others, on account of its being analogous to the general The methods of extracting the roots of higher powers method of notation. In A.D. 1624, Briggs published are so complicated, that they are usually omitted in the tables of logarithms which are now in common use. arithmetical works. The 4th, or biquadratic root, may In this system, 10 being the ratio or multiplier, the be ascertained by extracting the square root twice, terms may be thus expressed because it is the square root of the square root. The

1 10 100 1000 10,000, &c. 6th root is the square root of the cube root, or the cube 100 101 102 103 104 &c. root of the square root.

The exponents, 1, 2, 3, 4, are, as was shown in the The 8th root, being the square root of the biquad- previous table, the logarithms of the opposite numbers, ratic root, may be found by extracting the square and might therefore have been written thusroot three times. In the same way, the 9th root may

1 10 100 1000

10,000, &c. be called the cube root of the cube root, and may be

0 1 2 3

4 &c. found by extracting the cube root twice. But the The logarithms of all interinediate numbers, such as common arithmetical operations furnish no method of those between 1 and 10, 10 and 100, &c. are found by obtaining the 5th, 7th, 10th, 13th, 14th, 15th, 17th, ascertaining the geometrical mean proportions between 19th, 20th, and most other roots; and this part of any two numbers, and likewise the corresponding aritharithmetic continued, therefore, involved in practical metical means between the indices of those numbers. difficulties, until the invention of logarithms.

In every system, 0 is the logarithm of 1. Hence

the logarithm of any number between 1 and 10 must LOGARITHMS.

be less than a whole number, and therefore a decimal Logarithms (from two Greek words, signifying the fraction ; and the logarithm of any number between 10 number of the ratios or proportions) is a branch of and 100 must be one with a fraction. In the same way, arithmetic of comparatively modern invention, the dis- the logarithm of any number between 100 and 1000 coverer having been John Napier of Merchiston, near must be 2 and a fraction, and so on, through the rest of Edinburgh, in the early part of the seventeenth century. the series. The integers, 0, 1, 2, 3, &c. to the left of The principles upon which logarithms are founded, may the decimals in logarithms, are called the characteristics perhaps be rendered familiar by the following illustra- of those logarithms. Thus 0 is the characteristic of all tion :--Supposing that 1 is the first term of a geome- numbers between 1 and 10; 1 is the characteristic of trical progression, and that the ratio or multiplier is all between 10 and 100; 2 that of all between 100 and 2, the terms stand in the following manner :

1000, &c. In tables of logarithms the characteristics are generally omitted. [For a complete and accurate

collection of MATHEMATICAL TABLES, the reader is resquare of the ratio.

ferred to CHAMBERS'S EDUCATIONAL COURSE.] 4th power of the ratio.

The method of using a table of logarithms is the fol5th power of the ratio.

lowing :-To ascertain, in the most common tables, the 6th power of the ratio.

logarithm of a number less than 10,000, it is merely All this might, however, have been conveyed 20 necessary to find the first three figures of the number far more concisely by substituting signs for 2 in the left-hand column, and the fourth figure in the words, thus :

horizontal line at the top; then opposite to the former, It is evident that this might have been ex- 94 and under the latter, is the required logarithm. If pressed still more concisely by omitting the 32 25 the given number consist of five figures, find, as before, number signifying the ratio (2) throughout, 64 26 | the logarithm belonging to the first four figures ; the retaining only the indices or exponents, they alone be- difference between it and the next logarithm may be ing sufficient to indicate the degree of power to which ascertained from the margin ; this difference is to be the ratio is raised in each term. Exponents thus placed multiplied by the fifth figure of the given number, and in order, opposite to a series of numbers in geometrical one figure cut off from the right of the product; and progression, are, as we have said, called Logarithms, the other figures of this product being added to the or literally, number of the ratio.

preceding logarithm, and the index 3 prefixed, the The most tedious sums in multiplication can, by sum will be the required logarithm. means of logarithms, be solved simply by addition. For The process of finding the natural number of any instance, if it were required to multiply 256 by 32, it given logarithm is precisely the converse of that just would merely be requisite to add the logarithms stand- described. The given logarithm must be sought in the ing opposite to those two numbers--their sum (13) table ; and against it, in the left-hand column, will be stands opposite the product required, namely, 8192:- found the natural number. It has already been shown Num. Log. Num.

that tables of logarithms afford an easy method of find. ing the product in multiplication, and the quotients in division, by means of simple addition and subtraction. Logarithms are equally applicable to the solution of questions in Proportion or the Rule of Three, it being merely requisite to add the logarithms of the second

and third terms together, and from the sum to subtract Again, to multiply 128 by 16, we take the 7th + 4th, or the logarithm of the first : the remainder is the logaIlth, power of 2, because 128 is the 7th and 16 the 4th rithm of the fourth term. power of 2; opposite to the sum of 7 and 4 (11) is 2048, A number may, by means of logarithms, be raised equal to the result of the multiplication of the two to any required power, by multiplying the logarithm numbers. This also affords a quick and easy method of of the number by the index of the power. Thus to dividing one number by another: thus if it be required square any number, multiply its logarithm by 2; to to divide 4096 by 16, it is only requisite to ascertain cube a number, multiply its logarithm by 3; and so on. the difference between the logarithm of these two num- To extract the root of any given number, the logabers, which in this case being 8, the figure opposite to rithm of the number must be divided by the proposed

1 is the first term.
2 ratio,
4
8 cube of the ratio.
16
32
64

1
2
4 ... 22
8 23
16

Log.

1 2

0
1
2
3

8 16 32 64

128 256 512 1024 2048 4096 8192

7 8 9 10 11 12 13

5 6

:

Wwwww

a

index, and the quotient is the logarithm of the required | be divided, with a small line between, in the form of a root. Thus to find the square roothach 19, 10,35 erdy fraction; thus denotes that a is divided by b.

, is 2.2278867 by 2; opposite to the quotient will be found the num- It has been shown in Arithmetic that the powers of ber 13, which is the square root of 169, because quantities are denoted by a small figure, called the 13 x 13 = 169. [To those who desire a thorough exponent or index of the power. Thus a x a, or the knowledge of ARITHMETIC, we refer to the complete square of a, is expressed by a°; 6 x 6 x b, or the cube and accessible treatise, by Mr Marr, published in of b, is expressed by 63, &c. The cube of a + b, or CHAMBERS'S EDUCATIONAL COURSE.]

of (a + b) (a + b) is expressed thus : (a + b)3.

The roots of quantities are represented by the sign

N, with the proper index afixed; thus Va, or, more ALGEBRA.

simply, Va, expresses the square root of a; Va the We have seen that logarithmic tables may be used cube root of a; Va + b represents the 4th or biquadas a substitute for many lengthened operations in arith- ratic root of a + b. Fractional indices are also fremetic. It is evident that the value of all methods of quently used to denote the roots of quantities, thus: computation lies in their brevity: Algebra must be at is the square root of a. considered as one of the most important departments of mathematical science, on account of the extreme

at is the cube root of a. rapidity and certainty with which it enables us to de- at is the 4th root of a, &c. termine the most involved and intricate questions. The term algebra is of Arabic origin, and has a reference

Again, al is the cube root of a’, or of the square of a. to the resolution and composition of quantities. In the

ad is the square root of a', or of the cube of a. manner in which it is applied, it embodies a method of

as is the 5th root of a. performing calculations by means of various signs and When two or more letters or quantities are conabbreviations, which are used instead of words and nected together by signs, the combination is called an phrases, so that it may be called the system of sym- algebraic expression, and each letter or quantity is bols. Although it is a science of calculation, yet its called a term. operations must not be confounded with those of arith- Quantities of one term are called simple quantities; metic. All calculations in arithmetic refer to some as a, 2a, 36, &c. particular individual question, whereas those of algebra A quantity of two terms, as b+c, is called a binomial. refer to a whole class of questions. One great advan- When a binomial expresses the difference between tage in algebra is, that all the steps of any particular two quantities, it is called a residual, as a - - b. course of reasoning are, by means of symbols, placed A quantity consisting of 3, 4, or many terms, are at once before the eye; so that the mind, being unim- called respectively trinomials, quadrinomials, multinopeded in its operations, proceeds uninterruptedly from mials. one step of reasoning to another, until the solution of The sign placed between two quantities shows, as the question is attained.

in arithmetic, the equality of those quantities. Symbols are used to represent not only the known, When quantities are connected by this sign, the exbut also the unknown quantities. The present custom pression is called an equation : thus, 2 + 4 = 6, is an is to represent all known quantities by the first letters equation; as also, a +0=(- f. of the alphabet, as a, b, c, &c. and the unknown quan- The symbol 7 or < is called that of inequality, it tities by the last letters, x, y, z.

being placed between two quantities, of which one is The symbols used in arithmetic to denote addition, greater than another; the open part of the symbol is subtraction, &c. belong properly to algebra. Thus always turned towards the greater quantity: thus, the sign + plus, denotes that one quantity is to be a b denotes a to be greater than b; and c <d added to another, and is called the positive or addi- denotes d to be greater than c. The sign of difference tive sign ; all numbers to which it is prefixed are is ~, only used when it is uncertain which of two called positive. The sign -- minus, denotes that one quantities is the greater; thus, e ~ f denotes the difquantity is to be subtracted from another; it is called ference between e and f when it is uncertain which is the negative or subtractive sign, and all quantities to the greater. which it is prefixed are called negative. If neither + The word therefore, or consequently, often occurring nor be prefixed to a quantity, then the sign + plus in algebraical reasoning, the symbol .. has been chosen is understood.

to represent it: thus the sentence, Therefore a + b The general sign to denote that one number is to be is equal to c + d, is thus expressed in algebra, multiplied by another, is x; but it often occurs that :.a + b = c + d. one letter has to be multiplied by another, and this is Like quantities are such as consist of the same letter represented by placing those letters one after the other, or letters, or powers of letters: thus, 6a and 2 a are generally according to the order in which they stand in like quantities, and also 4 abc and 9 abc. Unlike quan. the alphabet; thus a multiplied by b, is expressed by tities are such as consist of different letters; as, 4a, ab. The multiplication of quantities consisting of more b, 6 ax“, 4 cd, which are all unlike quantities. than one term, as for instance a + b by c + d, may be represented by anyolc of the following methods :

Addition. a + bxc + d, or a + b.c + d, or (a + b) (c + d). The operation of addition in arithmetic consists, as has The bar drawn over a + b and c + d, which in the been shown, simply in joining or adding several quantitwo first examples marks them as distinct quantities, is ties together : thus, 4 + 8 + 7 + 6 = 25. This same called a vinculum, but brackets or parentheses, as in process is always used in algebra, whenever like the last example, are now in more frequent use. tities with like signs are required to be added: thus,

When a letter is multiplied by any given number, it 2a+3 a +Ga= 11 a; and — 76-46-66=-176. is usual to prefix that number to the letter. Thus But as it often happens that like quantities which are twice a, three times b, four times c, six times x, &c. to be added together have unlike signs, addition has in are expressed thus: 20, 36, 40, 6.x ; and the numbers algebra a far more extended signification than in arith2, 3, 4, 6, thus prefixed, are called the coefficients of the metic. Thus to add 7 a + 4 a to 8a - 3 a, it is evi. letters before which they stand.

dent that, after 7 a + 4 a + 8 a have been added accord. The sign • between two numbers shows, as in ing to the usual method, 3 a must be subtracted. Hence arithmetic, that the former of those numbers is to be the general rule for the addition of like quantities divided by the latter; thus a = b, means that a is to with unlike signs is to add first the coefficients of the be divided by b. It is, however, more usual to place positive terms, and then to add those of the negative the number to be dirided above that by which it is to l terms; the less sum inust be subtracted from the

quan

:

6

a X

greater, and to this difference the sign of the greater When like quantities are to be subtracted from each must be annexed, with the common letter or letters. other, it is usual to place them in two rows, the one

Thus let it be required to add 7 a - 3a+4 a+5 a - above the other ; the signs of the quantities to be sub6a -- 2a and I a; 25 a will be found the sum of the tracted must, for the reason above adduced, be conpositive terms, and lla that of the negative; 11 a being ceived to be changed; and the several quantities must the less number, must therefore be subtracted from be added, as shown in the following examples :25 a, the greater, leaving a remainder of 14 a, which is

From 5 ax + 7 xy - 2 y the required amount. The reason of this mode of pro

Take 3y + 3 ax – 6 ry cedure may be shown by a simple illustration :- As a

Remainder, 2 ax + 13 xy 5 y letter may represent any quantity, let a represent £5; and suppose that a gentleman has in one bank 7 a, or

Multiplication. seven five pounds; in another bank 4 a, and in another

The multiplication of two quantities is performed 9 a, or nine five pounds : let us suppose, too, that his by multiplying, as in arithmetic, the coefficients of the tailor's bill is 6 a, or six five pounds; his baker's 3 a, quantities, and then prefixing the proper sign and or three five pounds; and his butcher's 2 a, or two annexing letters: thus the product of 3 a, multiplied five pounds ; it is evident that, to ascertain how much by 5 b, is 15 ab, and 7 a x 4 ab = 28 a*b. money is really his own, he must first compute the

When the signs of both quantities are alike, the sign whole value of his property dispersed in the different + is to be prefixed; but when unlike, the sign – must banks, then the amount of the bills of his creditors, be prefixed, which may be thus shown at one view :and then find the difference between the two sums by

1. + multiplied by + produces + subtracting the less from the greater. When the

2. — multiplied by produces + aggregate of the positive terms is equal to that of the

3. + multiplied by - produces negative ones, the sum of the two quantities will be

4. — multiplied by + produces equal to 0: thus, if a man possesses £2000, and owes £2000, it is evident that when his debts have been de- Hence the technical rule generally given is, that like ducted from his property, nothing will remain. Unlike numbers produce plus +, and unlike produce minus — quantities can only be added by collecting them in one This, however, is not perfectly true when more than two line, and prefixing the proper sign of each ; thus the quantities are to be successively multiplied; because, sum of 3 a + 2b + 40 -- 2 d can only be rendered although the product of an even number of negative 3a + 2b + 4c-2d; this will be evident by reflecting quantities is positive, yet the product of an odd numthat different letters in the same algebraical expression ber of negative quantities is always negative; thus, always represent different quantities, which cannot of

-bxd= abd course be added into one sum unless their precise

and -ax-bx d x-e = abde value be known. Thus the addition of a and b cannot be represented by 2 a or 2 b, because that would indices must be added ; thus, a2 x a3

When the same letter occurs in both quantities, the imply that a is equal to b, which it is not necessarily; In the multiplication of compound quantities, it is usual

= aaaaa = 25. neither could it be represented by ab, because ab denotes the multiplication of the two quantities; the only to commence from the left-hand figure; the multiplimethod, then, of expressing these sums is thus, a + 6. cation, for instance, of 8 ab 4 ac + x by 2 a, is thus When like and unlike quantities are mixed together, performed :

8 ab 4 ac + x as in the following example, the like quantities must

2 a first be collected together according to the method above described, and all unlike quantities must be

16 aob 8 a'c + 2 ax annexed in order :

To multiply two compound quantities, each term of 9a + 5 xy

the one must, as in arithmetic, be multiplied by each 10x + 2 ry

term of the other; these particular or partial products

must be added according to the rules of addition, and 5 ar — 6 ax + lly

their sum will give the whole product, as shown the xy 4 a + 9 as

following instance :
2 ay + 120

Multiply 3 a + 86
- 10 y
3 xy + 13 ay

By

b
3 a
5 xy + y

3 a2 + 8 ab

- 3 ab--- 8 62

Product, 3 a+ 5 ab 8 6* When two like quantities, having like signs, are to be subtracted the one from the other, the process is

Division. precisely the same as that already described in Arith- The operations of division being in algebra, as in metic: thus, 3 a subtracted from 7 a, leaves as a re- arithmetic, merely the converse of those of multiplicamainder 4 a. From 8 a + 5 a take 6 a + 2 a, and the tion, the same rules respecting signs apply in both. remainder will be 2 a + 3 a, or 5 a.

Thus, 6 ab”, divided by 2b, is equal to 3 ab, But supposing it were required to sul act 6a-4a

8 cza from 9 a, it is evident that some other process must be

And — 8 cx? = 4X, or adopted ; because if 6 a be subtracted from 9 a, the In division, all letters common to both quantities must proposed operation will not be performed; for it is not be omitted in the quotient; and when the same letters 6 a, but 6a- - 4 a, that is, 2 a, which is required to be subtracted from 9 a; 6 a subtracted from 9 a leaves 3 a, letter in the divisor must be subtracted from that in

occur in both with different indices, the index of the which is 4. a less than would result from subtracting the dividend ; thus, 2 a from 9a; but if to 3 a we add the other term

abx namely, 4 ——the sum will be the remainder sought,

abx ab, or

and

ab because 3 a + 4a=7 a; and if 2 a be subtracted from 9 a, which is just the same question in another form,

6 25 6 a5 2 a', or

3 a? for 6 a - 4 a is = 2 a, the remainder is just 7 a, as before. So if a - b is to be subtracted from c, the

When the exponent of any letter in the divisor exremainder would be c - a + b, and for the same rea

It may therefore be given as a general rule, ceeds that of the same letter in the dividend, the latter that all the signs of a quantity which is required to quotient will be in the form of a fraction ; thus,

exponent must be subtracted from the former, and the be subtracted from another, must be changed : thus

12 a3x2 when 4 *—3 y is subtracted from 7 a + 5%, the re

3 a2

- 12 a'.'; 8 a.x5 mainder is written thus, 7 a + 56 -- 4 x + 3 y.

8 ay

8 xy

3 x

7 ay

5

2 a

a

8 a.c

Subtraction.

[ocr errors]

4 x

[ocr errors]
[ocr errors]

son,

8 a25

2.23

« ElőzőTovább »