548. It is necessary to observe, that every number is divisible by 1 and by itself, so that there is no number that has not at least two factors or divisors, the number itself and unity; but if a number have no other than these two it belongs to the class of numbers called prime numbers. With the exception of those, all numbers have other divisor besides unity and themselves, as may be seen from the subjoined table, wherein all its divisors are placed under each number, and the prime numbers marked with a P. Numbers Divisors 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Number of divisors PP P We must here observe, that O, or nothing, may be considered a number having the property of being divisible by all possible numbers, because by whatever number a0 is divided, the quotient must be 0; for the multiplication of any number by nothing produces nothing, hence Oa is 0. FRACTIONS. 549. When a number is said not to be divisible by another number, it only means that the quotient cannot be expressed by an integer number. For if we imagine a line of 7 feet in length, it is impossible to doubt that it may be divided into three equal parts, of the length of each whereof a notion may be formed. But as the quotient of 7 divided by 3 is not an integer number, we are thus led to the consideration of a particular species of numbers called fractions or broken numbers. If we have to divide 7 by 3 the quotient may be conceived and expressed by 3, placing the divisor under the dividend, and separating them by a stroke or line. 100 550. Generally, moreover, if the number a is to be divided by the number b, the quotient is, and this form of expression is called a fraction. In all fractions the lower number is called the denominator, and that above the line the numerator. In the above fraction of which is read seven thirds, 7 is the numerator and 3 the denominator. In reading fractions we call four fifths, 7 seven eighteenths, fifteen hundredths, and one half. 551. In order to become thoroughly acquainted with the nature of fractions it is proper to begin by considering the case of the numerator being equal to the denominator as Now as this is a representation of the quotient obtained by dividing a by a, it is evident it is once contained in it, that is, the quotient is exactly unity, hence is equal to 1; and for the same reason all the following fractions, 2, 3, 4, 5, 6, 7, &c., are equal to one another, each being equal to unity. It is evident, then, that fractions whose numerators are less than the denominators have a value less than unity, for if a number be divided by another which is greater, the result must necessarily be less than 1. Thus, if a line two feet long be cut into three parts, two of them will undoubtedly be shorter than a foot; it is evident, then, that is less than 1, for the same reason that the numerator 2 is less than the denominator 3. a 552. But if, on the contrary, the numerator be greater than the denominator, the value of the fraction is greater than unity. Thus is greater than 1, for is equal to together with. Now is exactly 1, consequently is equal to 1+, that is, to an integer and a third. So is equal to 13, or 14; is equal to 13; and 2 is equal to 23. Generally, if we divide the upper member by the lower, and add to the quotient a fraction whose numerator expresses the remainder and the divisor the denominator, we shall in other terms represent the fraction. For example, in the fraction f the quotient is 3 and the remainder S, hence f is the same as 39 553. Fractions, then, whose numerators are greater than their denominators, consist of two numbers; one of which is an integer, and the other a fractional number, in which the numerator is less than the denominator; and when fractions contain one or more integers, they are called improper fractions, to distinguish them from fractions properly so called, in which the numerator is less than the denominator, whence they are less than unity, or than an integer. There is another way of considering fractions, which may illustrate the sub ject. Thus, in the fraction it is evident that it is five times greater than This last 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 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 0, 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 I 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,, the double of it, must be greater, and so on. PROPERTIES OF FRACTIONS. a 3 a 2a 3a 4a 5 35 555. It has been seen that 3, 3, 1, §, &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, 1, 1, 7, 18, 13, &c. are equal, the common value being. So, to give another example, 4,,,, are all equal to 4. Hence, we arrive at the general conclusion that the fraction may be equally represented by the following expressions, each equal to viz. 72 &c. That this is the case we may see by substituting a certain letter c 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 36 will give 3a; and that in general e 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 mb' 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 ma ma fraction is equal to the fraction whatever the value of m. 5 6 556. 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 in the smallest numbers is more immediately understood. Thus the fraction, or one quarter, is more easily comprehended than &c. It therefore becomes a matter 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 we also learn that if they are divided by the same number their value is not altered. As 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 to ma mb mb 557. From the above, then, it is evident that to reduce a fraction to its least terms, we 360 we 180; this 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 48 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 1. 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 - 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: 12 gor nothing. 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 +3, 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 1, 3, 1, 1, . 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 4; 8 instead of; instead of ; instead of; and instead of §. 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 23, which is equal to 33 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 is now readily obtained, being d and their difference is evidently ad-bc ad bd Suppose the fractions and proposed, we have in their stead 49 and 3, 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 8, whence it is evident that is less than 3 by 7. 40 560. 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 be taken remain. Again, suppose is to be subtracted from 2, we may either write or 1, 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 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 2, 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 2} for the product sought. Likewise multiplied by 5 gives or 33, that is 34. 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. So that taken twice gives 1, taken 4 times gives 3. 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 6 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 3 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, &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. a bc a ac bc' 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 3 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, 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 by c, the result whereof is, after which we divide by d, which gives From this arises the rule for multiplying fractions, which is, to multiply the numerators and denominators separately. Thus 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 by we have only to divide 7 by 9 which is . So we shall have in 1 3 times, in T00 7 times. 49 100 ad bd bc ; db a 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 to be divided by 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. ad bc' 569. If this rule be applied to the division of by 7 we have 3 or 14, and by gives 18 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 frac tion To 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 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 multiplied by, which produces 37. of 16 is the same as 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 +13. Further, divided by + makes-, or -1; and - divided by - makes +28, or 3, that is, 31. 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. 576. Square numbers, therefore, are found by multiplying the root by itself. Thus, 1 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. 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 {}, |