the square root of 12. Notwithstanding this, we are not to assert that the square root of 12 is absolutely and in itself indeterminate; it only follows from what has been said, that this root, though it necessarily has a determinate magnitude, cannot be expressed by fractions.

128. There is, therefore, a sort of numbers which cannot be assigned by fractions, and which are nevertheless determinate quantities; the square root of 12 furnishes an example. We call this new species of numbers, irrational numbers; they occur whenever we endeavour 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, multiplied by itself, would produce 2, is an irrational quantity. These numbers are also called surd quantities, or incommensurables.

129. These irrational quantities, though they cannot be expressed by fractions, are nevertheless magnitudes, of which we may form an accurate idea. For however concealed the square root of 12, for example, may appear, we are not ignorant, that it must be a number which, when multiplied by itself, would exactly produce 12; and this property is sufficient to give us an idea of the number, since it is in our power to approximate its value continually.

130. As we are therefore sufficiently acquainted with the nature of the irrational numbers, under our present consideration, a particular sign has been agreed on, to express the square roots of all numbers that are not perfect squares. This sign is written thus, and is read square root. Thus, 12 respresents the square root of 12, or the number which, multiplied by itself, produces 12. So, v represents the square root of 2; 3 that of 3; that of, and in general, ✔a represents the square root of the number a. Whenever, therefore, we would express the square root of a number which is not a square, we need only make use of the mark ✔ by placing it before the number.

131. The explanation which we have given of irrational numbers will readily enable us to apply to them the known methods of calculation. For knowing that the square root of 2, multiplied by itself, must produce 2; we know also, that the multiplication √2 by √2 must necessarily produce 2; that, in the same manner, the multiplication of 3 by 3 must give 3; that 5 by 5 makes 5; that √ by makes; and, in general, that a multiplied by ✔a pro

duces a.

132. But when it is required to multiply ✔ã by √ɔ̃ the product will be found to be ab; because we have shown before, that if a square has two or more factors, its root must be composed of the Wherefore we find the square root of the ab, by multiplying the square root of a or va, by the square root of bor. It is evident from this, that if b were equal to a, we should have aa for the product of a by b. Now vaa is evidently a, since a a is the square of a.

roots of those factors. product ab, which is

133. In division, if it were required to dividea, for example, by, we obtain ; and in this instance the irrationality may van

a

b

ish in the quotient. Thus, having to divide √18 by ✔8, the quotient is, which is reduced to, and consequently to, because is the square of .

134. When the number, before which we have placed the radical sign, is itself a square, its root is expressed in the usual way. Thus is the same as 2; √9 the same as 3; √36 the same as 6; and 12 the same as, or 31. In these instances the irrationality is only apparent, and vanishes of course.

135. It is easy also to multiply irrational numbers by ordinary numbers. For example, 2 multiplied by 5 makes 2√5, and 3 make 32. In the second example, however, as 3 is

times equal to 9, we may also express 3 times 2 by 9 times √2, or by 18. So 2 va is the same as 4a, and 3a the same as 9 a. And, in general, ba has the same value as the square root of b b a, or Vabb; whence we infer reciprocally, that when the number which is preceded by the radical sign contains a square, we may take the root of that square and put it before the sign, as we should do in writing ba instead of abb. After this, the following reductions will be easily understood:

136. Division is founded on the same principles. va divided by

gives Va

a

or

In the same manner,

vb, gives

137. There is nothing in particular to be observed with respect to the addition and subtraction of such quantities, because we only connect them by the signs + and —. For example, 2 added to 3 is written 2 + √3; and √3 subtracted from 5 is written √5- √3.

138. We may observe lastly, that in order to distinguish irrational numbers, we call all other numbers, both integral and fractional, rational numbers.

So that, whenever we speak of rational numbers, we understand integers or fractions.

CHAPTER XIII.

Of Impossible or Imaginary Quantities, which arise from the same

source.

139. We have already seen that the squares of numbers, negative as well as positive, are always positive, or affected with the sign +; having shown that a multiplied by a gives a a, the same as the product of a by + a. Wherefore, in the preceding chapter, we supposed that all the numbers, of which it was required to extract the square roots, were positive.

140. When it is required, therefore, to extract the root of a negative number, a very great difficulty arises; since there is no assign

duce square,

4;

2, because the

4.

able number, the square of which would be a negative quantity. Suppose, for example, that we wished to extract the root of we require such a number, as when multiplied by itself, would pro4; now this number is neither + 2 nor both of +2 and of— 2, is + 4, and not 141. We must therefore conclude, that the square root of a negative number cannot be either a positive number, or a negative number, since the squares of negative numbers also take the sign plus. Consequently the root in question must belong to an entirely distinct species of numbers; since it cannot be ranked either among positive or among negative numbers.

142. Now, we before remarked, that positive numbers are all greater than nothing, or 0, and that negative numbers are all less than nothing, or 0; so that whatever exceeds 0, is expressed by positive numbers, and whatever is less than 0, is expressed by negative numbers. The square roots of negative numbers, therefore, are neither greater nor less than nothing. We cannot say however, that they are 0; for O multiplied by 0 produces 0, and consequently does not give a negative number.

143. Now, since all numbers, which it is possible to conceive, are either greater or less than 0, or are 0 itself, it is evident that we cannot rank the square root of a negative number amongst possible numbers, and we must therefore say that it is an impossible quantity. In this manner we are led to the idea of numbers which from their nature are impossible. These numbers are usually called imaginary quantities, because they exist merely in the imagination.

144. All such expressions, as −1, v=2, v=3, 4, &c., are consequently impossible, or imaginary numbers, since they represent roots of negative quantities; and of such numbers we may truly assert, that they are neither nothing, nor greater than nothing, nor less than nothing; which necessarily constitutes them imaginary, or impossible.

145. But notwithstanding all this, these numbers present themselves to the mind; they exist in our imagination, and we still have a sufficient idea of them; since we know that by number, which multiplied by itself, produces 4. also, nothing prevents us from making use of these imaginary numbers, and employing them in calculation.

4 is meant a For this reason

146. The first idea that occurs on the present subject is, that the square of 3, for example, or the product of 3 by √3,

must be 3; that the product of T by
in general, that by multiplying
square ofa, we obtain -α.

1 is -1; and,

—a by ✔a, or by taking the

1,

147. Now, asa is equal to + a multiplied by — 1, and as the square root of a product is found by multiplying together the roots of its factors, it follows that the root of a multiplied by ora, is equal to a multiplied by 1. Now a is a possible or real number, consequently the whole impossibility of an imaginary quantity may be always reduced to 1. For this reason, ✔4 is equal to √ multiplied by 1, and equal to 2 √1,

on account of being equal to 2. For the same reason, ✔ 9 is reduced to 9 X √1, or 3 √√1; and

16 is equal to 4 — 1.

148. Moreover, as a multiplied by √б makes ab, we shall have 6 for the value of 2 multiplied by

3; and 4, or 2, for the value of the product of 1 by 4. We see, therefore, that two imaginary numbers, multiplied together, produce a real, or possible one.

But, on the contrary, a possible number, multiplied by an impossible number, gives always an imaginary product: thus,

+5 gives 15.

3 by

149. It is the same with regard to division; for va divided by it is evident that 4 divided by I will make

✓ making √

✔4, or 2; that √3 divided by √3 will give √1; and that

This

150. We have before observed, that the square root of any number has always two values, one positive and the other negative; that ✔, for example, is both + 2 and 2, and that in general, we must take Va as well as + va for the square root of a. remark applies also to imaginary numbers; the square root of is botha anda; but we must not confound the signs + and -, which are before the radical sign, with the sign which comes after it.

a

151. It remains for us to remove any doubt which may be entertained concerning the utility of the numbers of which we have been speaking; for those numbers being impossible, it would not be surprising if any one should think them entirely useless, and the subject only of idle speculation. This, however, is not the case.

The cal