108. But when the fractions have not equal denominators, we must have recourse to the method already mentioned for reducing them to a common denominator. Let there be, for example, the

now evident, that the quotient must be represented simply by the

ad

division of ad by be; which gives bc

Hence the following rule: Multiply the numerator of the dividend by the denominator of the divisor, and the denominator of the dividend by the numerator of the divisor; the first product will be the numerator of the quotient, and the second will be its denominator. 109. Applying this rule to the division of by, we shall have the quotient; the division of will give or or 1 and ; and by will give 15, or 5.

by

110. This rule for division is often represented in a manner more easily remembered, as follows: Invert the fraction which is the divisor, so that the denominator may be in the place of the numerator, and the latter be written under the line; then multiply the fraction, which is the dividend by this inverted fraction, and the product will be the quotient sought. Thus divided by is the same as multiplied by, which makes, or 1. Also divided by is the same as multiplied by, which is 15; or 25 divided by gives the same as multipled by g, the product of which is 5%, or §.

5.

We see then, in general, that to divide by the fraction, is the same as to multiply by, or 2; that division by amounts to multiplication by, or by 3, &c.

111. The number 100 divided by will give 200; and 1000 divided by will give 3000. Further, if it were required to divide 1 by, the quotient would be 1000; and dividing 1 by yo the quotient is 100000. This enables us to conceive that, when number is divided by 0, the result must be a number infinitely great; for even the division of 1 by the small fraction o gives for the quotient the very great number 1000000000.

any

112. Every number when divided by itself producing unity, it is evident that a fraction divided by itself must also give 1 for the quotient. The same follows from our rule: for, in order to divide

by, we must multiply by, and we obtain 12, or 1; and if it

a

b

be required to divide by, we multiply by; now the product

113. We have still to explain an expression which is frequently used. It may be asked, for example, what is the half of; this means that we must multiply by. So likewise, if the value of of were required, we should multiply by, which produces ; and of is the same as multiplied by, which produces

114. Lastly, we must here observe the same rules with respect to the signs and, that we before laid down for integers. Thus multiplied by - makes; and

multiplied by makes;

CHAPTER XI.

Of Square Numbers.

115. THE product of a number, when multiplied by itself, is called a square; and for this reason, the number, considered in relation to such a product, is called a square root.

For example, when we multiply 12 by 12, the product 144 is a square, of which the root is 12.

This term is derived from geometry, which teaches us that the contents of a square are found by multiplying its side by itself.

116. Square numbers are found therefore by multiplication; that is to say, by multiplying the root by itself. Thus 1 is the square of 1, since 1 multiplied by 1 makes 1; likewise, 4 is the square of 2; and 9 the square of 3; 2 also is the root of 4, and 3 is the root of 9.

We shall begin by considering the squares of natural numbers, and shall first give the following small table, on the first line of which several numbers, or roots are placed, and on the second their squares.

Numbers 1 2 3 4 5 6 7 8 9 10 11 12 13

Squares 14 9162536496481 100 121 144 169

117. It will be readily perceived, that the series of square numbers thus arranged has a singular property; namely, that if each of them be subtracted from that which immediately follows, the remainders always increase by 2, and form this series :

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

118. The squares of fractions are found in the same manner, by multiplying any given fraction by itself. For example, the square ofis,

We have only, therefore, to divide the square of the numerator by the square of the denominator, and the fraction, which expresses that division must be the square of the given fraction. Thus, is the square of; and reciprocally, is the root of 3.

119. When the square of a mixed number, or a number composed of an integer and a fraction, is required, we have only to reduce it to a single fraction, and then to take the square of that fraction. Let it be required, for example, to find the square of 21; we first express this number by, and taking the square of that fraction, we have 25, or 61, for the value of the square of 21. So to obtain the square of 31, we say 31 is equal to 3; therefore its The squares of the num

square is equal to 19, or to 10 and.

13

bers between 3 and 4, supposing them to increase by one fourth, are as follows:

From this small table we may infer, that if a root contain a fraction, its square also contains one. Let the root, for example, be 1; its 1/2 square is 2, or 2; that is to say, a little greater than

the integer 2.

120. Let us proceed to general expressions. When the root is a, the square must be a a; if the root be 2 a, the square is 4 a a; which shows that by doubling the root, the square becomes 4 times greater. So if the root be 3 a, the square is 9 a a; and if the root Eul. Alg.

5

be 4 a, the square is 16 a a. But if the root be ab, the square is a abb; and if the root be a b c, the square is a a b b c c.

121. Thus when the root is composed of two or more factors, we multiply their squares together; and reciprocally, if a square be composed of two or more factors, of which each is a square, we have only to multiply together the roots of those squares, to obtain the complete root of the square proposed. Thus, as 2304 is equal to 4 X 16 X 36, the square root of it is 2 X 4 X 6, or 48; and 48 is found to be the true square root of 2304, because 48 x 48 gives 2304.

122. Let us now consider what rule is to be observed with regard to the signs and -. First, it is evident that if the root has the sign, that is to say, is a positive number, its square must necessarily be a positive number also, because+by+makes +: the square of a will be + aa. But if the root be a negative number, asa, the square is still positive, for it is + a a; we may therefore conclude, that+a a is the square both of +a, and of a, and that consequently every square has two roots, one positive and the other negative. The square root of 25, for example, is both + 5 and 5, because 5 multiplied by -5 gives 25, as well

as + 5 by + 5.

CHAPTER XII.

Of Square Roots, and of Irrational Numbers resulting from them.

123. WHAT We have said in the preceding chapter is chiefly this: that the square root of a given number is nothing but a number whose square is equal to the given number; and that we may put before these roots either the positive or the negative sign.

124. So that when a square number is given, provided we retain in our memory a sufficient number of square numbers, it is easy to find its root. If 196, for example, be the given number, we know that its square root is 14.

Fractions likewise are easily managed; it is evident, for example, that is the square root of 25. To be convinced of this, we have

only to take the square root of the numerator, and that of the denominator.

If the number proposed be a mixed number, as 12, we reduce it to a single fraction, which here is, and we immediately perceive that, or 34, must be the square root of 121.

125. But when the given number is not a square, as 12, for example, it is not possible to extract its square root; or to find a number, which, multiplied by itself, will give the product 12. We know, however, that the square root of 12 must be greater than 3, because 3 × 3 produces only 9: and less than 4, because 4 × 4 produces 16, which is more than 12. We know also, that this root is less than 3; for we have seen that the square of 31, or is 121. Lastly, we may approach still nearer to this with 3; for the square of 3, or of that this fraction is still greater than the root required; but very little greater, as the difference of the two squares is only

root, by comparing it is 27, or 125, so

2704

126. We may suppose that as 3 and 37 are numbers greater than the root of 12, it might be possible to add to 3 a fraction a little less than, and precisely such that the square of the sum would be equal to 12.

576

Let us therefore try with 3, since is a little less than 7. Now 3 is equal to 24, the square of which is 5, and consequently less by 1 than 12, which may be expressed by 5. It is therefore proved that 3 is less, and that 3, is greater than the root required. Let us then try a number a little greater than 32, but yet less than 3, for example, 35. This number, which is equal to, has for its square. Now, by reducing 12 to this denominator, we obtain 1452; which shows that 35 is still less than the root of 12, viz. by T Let us therefore substitute for the fraction, which is a little greater, and see what will be the result of the comparison of the square of 3 with the proposed number 12. The square of 3, is 2025; now 12 reduced to the same denominator is 03; so that 3, is still too small, though only by, whilst 37 has been found too great.

2028

127. It is evident, therefore, that whatever fraction be joined to 3, the square of that sum must always contain a fraction, and can never be exactly equal to the integer 12. Thus, although we know that the square root of 12 is greater than 3, and less than 3,73, yet we are unable to assign an intermediate fraction between these two, which, at the same time, if added to 3, would express exactly