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This last

ject. Thus, in the fraction it is evident that it is five times greater than fraction expresses one of the 10 parts into which 1 may be divided, and that in taking five of those parts we have the value of the fraction f

554. It is from this mode of considering a fraction that the terms numerator and denominator are derived; that is to say, the lower number expresses or denotes the number of parts into which the integer is divided, and is therefore called the denominator, the upper number, or that above the line numbers the quantity of those parts, and is thence called the numerator. It follows, then, that as the denominator is increased the smaller the parts become, as in,,,,,,, and so on; and it is evident that if the integer be divided into two parts, each of those parts is greater than if it had been divided into eight.

In this division of the integer it is impossible to increase the denominator so that the fraction shall be reduced to nothing; for into whatever number of parts unity may be divided, however small they be, they still preserve some definite magnitude. Indeed, to whatever extent we continue the series of fractions just named, they will always represent a certain quantity. From this has arisen the expression that the denominator must be infinitely great, or infinite, to reduce the fraction to 0, or nothing, which in this case means nothing more than that it is impossible to reach the end of the series of the fractions in question. This idea is expressed by the use of the sign ∞, which indicates a number infinitely great, and we may therefore say that is really nothing, because a fraction can only be lessened to nothing when the denominator has been increased to infinity. This, moreover, leads us to another view of the matter, which is important. The fraction, as we have seen, represents the quotient resulting from the division of 1 by ∞. Now, if 1 be divided by or O, the divisor will be again, and a new idea of affinity is thus obtained, arising from the division of 1 by 0; and thus we are justified in saying that 1 divided by O expresses, or a number infinitely great. From this, moreover, we learn that a number infinitely great is susceptible of increase, for having seen that denotes a number infinitely great, 3, the double of it, must be greater, and so on.

10

PROPERTIES OF FRACTIONS.

с

a

a 2a 3a 4a

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555. It has been seen that, 3, 4, 3, &c. are equal to 1, and thence equal to one another; the same equality obtains in the fractions,,,, &c., which, from what has been said, it is obvious are each equal to 2, and to one another, so the fractions,,, are, from their common value, being 3 each, equal to one another. In the same way, a fraction may be represented in an infinity of ways by multiplying the numerator and denominator by the same number, be that number what it may; thus,,, 7, 8, &c. are equal, the common value being. So, to give another example,,,, are all equal to 4. Hence, we arrive at the general conclusion that the fraction may be equally represented by the That this is the case we following expressions, each equal to viz. 283616 &c. may see by substituting a certain letter e for the fraction which letter we will consider as representing the quotient of a divided by b; recollecting, then, that the multiplication of the quotient e by the divisor b must give the dividend; for by the hypothesis, as c multiplied by b gives a, it is evident that e multiplied by 26 must give 2a, that e multiplied by 3b will give 3a; and that in general c multiplied by mb (m representing any given number) must give ma. The converse brings us to the division of a by b, in which, if we divide the product ma by mb one of the factors, the quotient is equal to c, the other factor. But ma divided by mb gives also the fraction which is therefore equal to c, which was the matter to be proved; for c was assumed as the value of the fraction and hence this whatever the value of m fraction is equal to the fraction mb

ma

ma
mb

5

556. The infinite forms in which fractions may be represented, so as to express the same value, has been before shown; and it is obvious, that of those forms, that which is given Thus the fraction, or one in the smallest numbers is more immediately understood. It therefore becomes a matter quarter, is more easily comprehended than if y, &c. of convenience to express a fraction in the least possible numbers, or in its least terms. This is a problem not difficult of resolution when we recollect that all fractions retain their value if the numerator and denominator are multiplied by the same number, from which As we also learn that if they are divided by the same number their value is not altered. an example in the general expression if both numerator and denominator be divided by the number m, we obtain the fraction which has before been seen to be equal

ma

to mb

ma

557. From the above, then, it is evident that to reduce a fraction to its least terms, we

48

we

have only to find a number which will divide the numerator and denominator, and this number is called a common divisor, which if we can find, the fraction may be reduced to a lower form; but if we cannot find such a number, and unity is the only common divisor that can be found, the fraction is already in its simplest form. Thus, taking the fraction may immediately perceive that 2 will divide both the terms, whereof the result is ; this result is again divisible by 2, by which the fraction is reduced to 3, in which we again find 2 as a common divisor, and the result of that is 3. In this we may perceive that, as 2 will no longer divide the terms, another number must be sought, and by trial that number will be seen to be 3, by using which we obtain the fraction, the simplest expression to which it can be reduced, for 1 is the only common divisor of the numbers 4 and 15, and division by unity will not reduce those numbers. This property of the invariable value of fractions leads to the conclusion that in the addition and subtraction of them, the operations are performed with difficulty, unless they are reduced to expressions wherein the denominators are equal. And here it will be useful to observe that all integers are capable of being represented by fractions; for it is manifest that 9 and are the same, 9 divided by 1 giving a quotient of 9; which last number may also be equally represented by 18, 36, 72, 14, &c. &c.

ADDITION AND SUBTRACTION OF FRACTIONS.

558. When the denominators of fractions are equal they are easily added to and subtracted from one another: thus, + is equal to or 1, and -3 is equal to or In this case, either for addition or subtraction, it is only necessary to change the numerators and place the common denominator under the result, thus:

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If fractions have not the same denominator, they must, for the purpose in question, be changed into others that are in that condition. For an example, let us take the fractions and; it is evident that is the same as 3, and that is equivalent to ; the fractions for adding together therefore become +, whose sum is. If the latter is to be subtracted from the former, or, in other words, to be united by the sign -, as -, we shall have -, or

559. It often becomes necessary to reduce a number of fractions to a common denominator: thus, suppose we have the fractions, bbbt We have here only to find a number divisible by all the denominators of those fractions. In the above case, that number will, by trial, be seen to be 60, which therefore will be the common denominator. Substituting this, we shall have 38 instead of; 40 instead of 3; 15 instead of ; 8 instead of; and 8 instead of 3. The addition of all these fractions thus becomes simple enough, for we have only to add the numerators together, and place under that sum the common denominator, that is to say, we shall have 213, which is equal to 333 or 3 Thus, all that is necessary is to change two fractions whose denominators are unequal into two others whose denominators are equal. For the performance of this generally, if and be the fractions, first multiply both the terms of the first fraction by d, and we shall have bd equal to ; then multiply both the terms of the second fractions by b, and we have its equivalent value in whereby also the two denominators are become equal. The sum of these fractions

bc

bd'

ad+bc
9
bd

and their difference is evidently

ad-bc
bd·

ad

is now readily obtained, being Suppose the fractions and proposed, we have in their stead 49 and 93, whereof the sum is 103, and the difference 3. It is by the method just mentioned that we are enabled to ascertain which is the greater and which the less; thus, in the two fractions and, it is evident that the last is smaller than the first, for, reduced to the same denominator, the first is, and the second 29, whence it is evident that is less than 3 by 7.

560. To subtract a fraction from an integer, it is only necessary to change one of its units into a fraction having the same denominator as that which is to be subtracted: thus to subtract from 1 we write instead of 1, from which if 3 be taken remain. Again, suppose is to be subtracted from 2, we may either write or 14, from which subtracted leave or 14. It is only necessary to divide the numerator by the denominator, to see how many integers it contains. We have nearly the same operation to perform in adding numbers composed of integers and fractions; thus, let it be proposed to add 51 to 3, then taking and, or, which is the same, and 3, their sum is ; the sum total, therefore, will be 88.

MULTIPLICATION AND DIVISION OF FRACTIONS.

561. For the multiplication of a fraction by an integer, or whole number, the rule is to multiply the numerator only by the given number, the denominator remaining unchanged:

thus

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But when it can be done, it is preferable to divide the denominator by the integer, inasmuch as the operation is shortened by it; for example, in multiplying by 3, by the rule above given, we have 24, which is reduced then to, and, lastly, to 23 But if the numerator remain and the denominator is divided by the integer, we have at once or 23 for the product sought. Likewise multiplied by 5 gives 14 or 33, that is 31.

ac

562. Generally, then, the product of the multiplication of a fraction by e is ", and it is to be observed that when the integer is exactly equal to the denominator, the product must be equal to the numerator.

a

So that

taken twice gives 1,

taken thrice gives 2,

taken 4 times gives 3.

a

and, generally, if the fraction be multiplied by the number b, the product, as has already been seen, must be b, for as represents a quotient resulting from the division of the dividend a by the divisor b, and since we have seen that the quotient multiplied by the divisor will give the dividend, it is evident that multiplied by b must produce a. We are next to consider how a fraction can be divided by an integer before proceeding to the multiplication of fractions by fractions. It is evident, if I have to divide the fraction by 3, the result is, and that the quotient of divided by 4 is the rule is therefore to divide the numerator by the integer, and leave the denominator unchanged. divided by 2 gives

divided by 7 gives 2, &c. &c.

Thus

563. The rule is easily applied if the numerator be divisible by the number proposed; as this is not always the case, it is to be observed that a fraction may be transformed into an infinite number of similar expressions, in some of which the numerator might be divided by the given integer. Thus, for example, to divide by 2, we may change the fraction into , in which the numerator may be divided by 2, and the quotient is therefore .

564. In general, to divide the fraction by c, it is changed into, and then dividing the numerator ac by c, write for the quotient sought.

bc

a

565. Hence, when a fraction is to be divided by an integer c, it is necessary merely to multiply the denominator by that number, leaving the numerator as it is. Thus, divided by gives, and divided by 6 gives. When, however, the numerator is divisible by the integer, the operation is still simpler. Thus, divided by 3 would give according to the first given rule, but by this last rule we at once obtain, an expression equivalent to, but more simple than,

с

566. We now perceive, then, in what way one fraction may be multiplied by another

Here means that c is to be divided by d, and on this principle we must first multiply

ac

ас

by c, the result whereof is after which we divide by d, which gives ba

From this arises the rule for multiplying fractions, which is, to multiply the numerators and denominators separately. Thus

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567. We are now to see how one fraction may be divided by another. And, first, it is to be observed, that if the two fractions have similar denominators, the division is performed only with respect to the numerators, for it is manifest that are as many times contained in as 3 in 9, that is, three times; and in the same manner in order to divide have only to divide 7 by 9 which is 7. So we shall have in 3 times,

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ad to be divided by db
bc

bd

568. If the denominators of the fractions are not equal, they must, by the method before given, be reduced to a common denominator. Thus, if the fraction is to be divided by we have ; whence it becomes evident that the quotient will be represented simply by the division of ad by be or Hence the following rule: multiply the numerator of the dividend by the denominator of the divisor, and the denominator of the dividend by the numeartor of the divisor, the first product will be the numerator of the quotient and the second its denominator.

150

210

ad

bc'

569. If this rule be applied to the division of by 7 we have or 14, and by { gives or §.

570. There is a rule which operates the same results, and is more easily recollected; it is, to invert the fraction which is the divisor, that is, place the denominator for the numerator and the numerator for the denominator; then multiply the numerators together for a new numerator, and the denominators for a new denominator, and the product will be the quotient sought. Thus, divided by is the same as multiplied by, which make §. Also divided by is the same as multiplied by, which is ; that is, in general terms, to divide by the fraction is the same as to multiply by † or 2, that division by is the same as multiplication by or by 3.

571. Thus, the number 100 divided by is 200, and 1000 divided by will give 3000. So also if 1 be divided by the quotient would be 1000; and 1 divided by gives 100,000. This view is useful in enabling us to conceive that, when any number is divided by O, the result must be a number infinitely great; for the division of 1 by the small fraction Toooooooo gives for a quotient 1,000,000,000.

572. As every number, when divided by itself, produces unity, a fraction divided by itself must also give 1 for a quotient. For to divide by , we must, by the rule, multiply by, by which we obtain 3, or 1 ; and if it be required to divide by, we multiply

b

ab
ab

by; now the product is equal to 1.

a

573. There remains to explain an expression in frequent use, - such, for instance, as the half of this signifies that we must multiply by which is So, if it be required

to know the value of of, they are to be multiplied together, which produces; and of is the same as multiplied by, which produces 7. T6 574. We have, in a previous section, laid down for integers the signs of + and —, and the same rule holds with regard to fractions. Thus multiplied by makes ; and -multiplied by - gives +18. Further, divided by + makes, or -11; and - divided by makes +28, or 3, that is, S.

SQUARE NUMBERS.

575. If a number be multiplied by itself, the product is called a square, in relation to which the number itself is called a square root. Thus, if we multiply 12 by 12, the product 144 is a square whose root is 12. The origin of this term is borrowed from geometry, by which we learn that the contents of a square are found by multiplying its side by itself.

Thus, 1

576. Square numbers, therefore, are found by multiplying the root by itself. is the square of 1; since 1 multiplied by 1 makes 1. So 25 is the square of 5, and 64 the square of 8. 7, also, is the root of 49, and 9 is the root of 81. Beginning with the squares of natural numbers, we subjoin a small table, in the first line whereof the roots or numbers are ranged, and on the second their squares.

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577. A singular property will be immediately perceived in this table, which is, that in the series of square numbers, if the preceding one be subtracted from that following, the remainders always increase by 2, forming the following series

3, 5, 7, 9, 11, 13, 15, 17, 19, 21, &c.,

which is that of the odd numbers.

578. The squares of fractions are found in the same manner as those of whole numbers, that is, by multiplying any given fraction by itself; thus the square of is

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Hence we have only to divide the square of the numerator by that of the denominator, and the fraction expressing that division is the square of the given fraction. Thus is the square of, and, reciprocally, is the root of

579. If the square of a mixed number, or one that is composed of an integer and a fraction, be sought, no more is necessary than to reduce it to a single fraction, and then take the square of that fraction. Thus, to find the square of 21, it must first be expressed by the fraction; and, taking its square, we have, or 5 for the value of the square of 21. And so of any similar numbers. The squares of the numbers between 3 and 4, sup. posing them to increase by one fourth, are as follow:

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From this small tabular view it may be inferred that if a root contain a fraction, its square also contains one. Thus, let the root be 1, its square is 3, or 1, that is, rather more than half as great again as the integer 1.

580. Generally, when the root is a the square root must be aa; if the root be 2a the square will be 4aa; from which it is evident that by doubling the root the square becomes 4 times greater; for if the root be 4a, the square is 16aa. If the root be ab, the square is aabb; if abc, the square is aabbcc.

581. Thus, then, if the root be composed of more factors than one, their squares must be multiplied together; and, reciprocally, if a square be composed of more than one factor whereof each is a square, it is only necessary to multiply the roots of these squares to obtain the complete square of the root proposed. Thus, as 5184 is equal to 9 x 16 x 36, the square root of it is 3 x 4 x 6, or 72; and 72, it will be seen, is the true square root of 5184; for 72 x 72 gives 5184.

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582. Here we must for a moment stop to see how the signs + and tions: and, first, it cannot be doubted that if the root is a positive quantity, that is, with the sign+before it, its square must be a positive quantity; for+by+makes + : thus, the square of a will be +aa. So, also, if the root be a negative number, as -a, the square will still be positive, for it is +aa; from which it follows that of +a, as well as -a, the square is +aa; hence every square has two roots, one positive and the other negative. For example, the square of 16 is both +4 and -4, because -4 multiplied by -4 gives 16, as well as +4 by +4.

SQUARE ROOTS, AND THE IRRATIONAL NUMBERS THAT RESULT FROM THEM.

583. In the last section it has been seen that the square root of any number is but one whose square is equal to the given number, and that to those roots the positive or negative sign may be prefixed; so that if we could remember a sufficient number of squares, their roots would at the same time present themselves to our mind. Thus, if 144 were the given number, we should at once recollect that its square root is 12. 584. For the same reason fractions would be easily managed; for we at once see that is the square root of, inasmuch as we have only to take the square root of the numerator and that of the denominator to be convinced of it.

223

If we have to deal with a mixed number, we have only to put it in the shape of a single fraction: for example, 12] is equivalent to 19; and we see by inspection that or 34 must be the square root of 12. But when the given number is not a square, as, for example, 12, it is not possible to extract its square root, that is, to find a number multiplied by itself whose product is 12. It is, however, clear that the square root of 12 is greater than 3; for 3 x 3 produces only 9; and it must be less than 4, because 4 x 4 produces 16, which is greater than 12. From the table just given we may see that the square of 3 or is 12}; hence the root must be less than 33. We may, however, come nearer to this root by comparing it with 37; for the square of 37, or of 3, is 270, or 123, a fraction only greater by than the root required. Now, as 3 and 37, are both greater than the root of 12, it might be possible to add to 3 a fraction a little less than 75, precisely such that the square of such sum should be exactly equal to 12. Trying, therefore, with 34, being a little less than 7, we have 33, equal to 44, whose square is $75, and consequently less than 12 by 12; because 12 may be expressed by 588. Hence we perceive that 33 is less and 3, is greater than the root required. Trying a number, 3, which is a little greater than 34, but less than 37, its equivalent is 3, and it will have for its square ; and as 12 reduced to the same denominator is 14, we thus find that 3 is as yet less by than the root of 12. If for the fraction, which is a little greater, be substituted, we have the square of 3, equal to 2025; and 12 reduced to the same denominator, or multiplied by 169, equal to 2023; so that 3 is yet too small, though only by whilst 37 has been found too great. From this it is evident that whatever fraction be joined to 3, the

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