is $70, or 12,5; but this fraction is still greater than the root required, but very little so, as the difference between the squares is only.

147. Since therefore 34 and 37, are both greater than the square root, we may suppose that it would be possible to add to 3 a fraction less than, and precisely such that its square would be equal to 12.

Let us try then 32. Now 3 is the same as 2, the square of which is 576, and consequently less than 12 by, since 12 itself. It is proved therefore that 3 is less, and that 3 is greater, than the root sought. Let us then try a number a little greater than 3%, but less than 37; viz. 35. This number, which may be represented, has for its square 4. Now, by converting 12 into a fraction with the same denominator, we obtain 145, greater than 444 by TIT. It follows, therefore, that 3 is still too small for the 11 root required. Substituting, therefore, for the fraction, which is a little greater, and comparing the square of 3, viz. 2025 with the number 12 reduced to a fraction with the same denominator, viz. 20, we find 3 still too small, and at the same time 37, too great.

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T699

148. We can therefore easily understand that whatever fraction, be added 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 37, yet we are compelled to confess ourselves unable to assign an intermediate fraction between these two, which will exactly answer the purpose proposed. At the same time we are not to say that the square root of 12 is indeterminate in itself; it only follows from what has been shown, that this root, although it necessarily must have a determinate magnitude, cannot be expressed by fractions.

149. We

149. We are thus brought to the consideration of a particular kind of numbers, which, although they cannot be assigned by fractions, are still determinate quantities; as for instance the square root of 12. These numbers are called irrational numbers. They occur whenever we attempt to find the square root of a number which is not a square. Thus 2 not being a perfect square, the square root of 2, or the number which, when multiplied into itself, produces 2, is an irrational quantity. These numbers are also called surds, or incommensurable quantities.

150. These irrational quantities, although they cannot be expressed by fractions, are nevertheless magnitudes, of which we can form a very just idea; for, however indeterminate they may appear, as for instance the square root of 12, we still know that the number is such, that, when multiplied into itself, it will produce exactly 12. And this property is sufficient to give us an idea of the number, because we are enabled continually to approximate to its value.

151. As we are therefore sufficiently acquainted with the nature of irrational numbers, a particular sign is used to express the square roots of those numbers that are not perfect squares. This sign is the figure, and is read square root. Thus 12 signifies the square root of 12, 3 represents the square root of 3, &c.; and in general a signifies the square root of the number represented by the letter a.

152. The explanation that we have thus given of irrational numbers, or surds, enables us to apply to them the usual methods of calculation. For, knowing for instance that the square root of 2 multiplied into itself must produce exactly 2, we know also that √2 × √2=2:

√3 × √33:

and generally that √a × √ a=a.

153. When

153. When however it is required to multiply a by √b, we must write the product thus, ab; for since we have already shown that a square number contains two or more factors, its square root must contain the square roots of those factors. Thus, the square root of the product ab is

ab. It is clear, therefore, that if b be equal to a, the product of Na+ √b will be equal to a, Now √a2 is evidently a, because a2 is the square of a.

154. In division, if it were required to divide a by √b, the quotient must be written; and it may thus happen that in this quotient the irrational numbers will vanish. If, for example, we had to divide ✓18 by 8, the quotient

would be, which, divided by 2, would give√ viz.

155. When the number to which the radical sign is prefixed is itself a square number, the root is usually expressed by a rational number. Thus, 4 is the same as 2, 9 as 3, and 124 as, or 3, &c. We see, therefore, that in these cases the irrationality is only apparent.

156. It is easy also to multiply irrational quantities by ordinary numbers. For example, √5 multiplied by 2 would be expressed 25, and 3 times 2, 32. In the second example, however, since 3 is the same as √9, we might express 3√2 by √9 multiplied into √2; that is to say, √18. In like manner, 2a is the same as 4a, and 3a, as 9a. And in general ba is the same as the square root of bo multiplied into a or ab. From which we infer reciprocally, that when a number to which the radical sign is prefixed contains a square, we may extract the root of that square,

and

and place it before the radical sign, as we should do in writing ba instead of ab.

This will be more easily understood by considering the following examples:

157. The division of these quantities is founded upon the same principles. Thus a divided by the b is expressed

For

158. The addition and subtraction of quantities of this nature can only be performed by the signs + and -. example, 2 added to the 3 is expressed √2+ √√3, and

3 subtracted from 5, can only be written 5−√3.

It must be observed, however, that these remarks have reference only to those irrational quantities, of which the radical sign is the same in each: as, for instance, the multiplication, &c. of a square root by a square root, and a cube root by a cube root.

The general rules applicable to surds with different radical signs form the subject of succeeding chapters.

CHAP. XV.

OF IMPOSSIBLE OR IMAGINARY QUANTITIES.

WE 159. E have already seen that the squares of numbers, whether positive or negative, are always positive, or affected by the sign, since we have seen that a multiplied by -a gives the product +a, the same as +a by +a. We have therefore supposed, in the preceding chapter, that all the numbers, of which it was proposed to extract the square roots, were positive.

When therefore it is proposed to extract the square root of a negative number, there must arise a great difficulty, because there is no assignable number of which the square is negative. If it were proposed, for example, to extract the square root of —4, it would be necessary to find such a number as being multiplied into itself would give -4 for the product; now this number we know can be neither +2, nor -2, for the square of both these numbers is +4.

160. Since, therefore, the squares of all numbers, whether positive or negative, are themselves always positive, the square root of a negative number belongs to a distinct and particular species of numbers, since it cannot be ranked amongst any that have been as yet considered, either positive or negative.

Now