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 shewn 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 assignable number, the square of which would be a negative quantity. Suppose, for example, that we wished to extract the root of -4; we require such a number, as when multiplied by itself, would produce-4; now this number is neither + 2 nor -2, because the square, both of +2 and of— 2, is +4, and not

4.

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 can

not say however, that they are 0; for 0 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 F1, No 2, No 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 is meant a number which, multiplied by itself, produces -4. For this reason also, nothing prevents us from making use of these imaginary numbers, and employing them in calculation.

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 1 by 1 is -1; and, in general, that by multiplying √ɑ by √-a, or by taking ofa, we obtain

the

-

·a.

square 147. Now, as a 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 1, or ✔a, is equal to a multiplied by 1. Now is a possible or real number, consequently the whole impossibility of an imaginary quantity may be always reduced to V=1. For this reason, 4 is equal to ✔=1, and equal to 2 √1, on account of √

multiplied by being equal to 2. For the same reason, 9 is reduced to ✔9 × √1, or 3 v=1; 16 is equal to 4 √−1.

and
Eul. Alg.

6

shall have for value of 2 multiplied by 2, for the value of the product of 1 by

makes ab, we

Es; and 4, or

4. We see, there

fore, 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, by 5 gives

15.

3

149. It is the same with regard to division; for ✅ā divided

by making v making

i

it is evident that 4 divided by will

make √4, or 2; that s divided by 3 will give ✔;

and that 1 divided by √=1 gives

equal to

1.

or

1; because 1 is

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 — as well as + for the square root of a. This remark applies also to imaginary numbers; the square root of a is both +√a and —√a; but we must not confound the signs + and —, which are before the radical sign ✔, with the sign which comes after it.

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 uscless, and the subject only of idle speculation. This however is not the case. The calculation of imaginary quantities is of the greatest importance: questions frequently arise, of which we cannot immediately say, whether they include any thing real and possible, or not. Now, when the solution of such a question leads to imaginary numbers, we are certain that what is required is impossible.*

*This is followed in the original by an example intended to illustrate what is here said. It is omitted by the Editor, as it implies a degree of acquaintance with the subject, which the learner cannot be supposed to possess at this stage of his progress.

CHAPTER XIV.

Of Cubic Numbers.

152. WHEN a number has been multiplied twice by itself, or, which is the same thing, when the square of a number has been multiplied once more by that number, we obtain a product which is called a cube, or a cubic number. Thus, the cube of a is a a a, since it is the product obtained by multiplying a by itself, or by a, and that square aa again by a.

The cubes of the natural numbers therefore succeed each other in the following order.

Numbers.

Cubes. 1

2 3 4 5 6 7 8 9 10

8 27 64 125 216 343 512 729 1000

153. If we consider the differences of these cubes, as we did those of the squares, by subtracting each cube from that which comes after it, we shall obtain the following series of numbers:

7, 19, 37, 61, 91, 127, 169, 217, 271.

At first we do not observe any regularity in them; but if we take the respective differences of these numbers, we find the following series:

12, 18, 24, 30, S6, 42, 48, 54, 60;

in which the terms, it is evident, increase always by 6.

27

154. After the definition we have given of a cube, it will not be difficult to find the cube of fractional numbers; is the cube of; is the cube of; and is the cube of . In the same manner, we have only to take the cube of the numerator and that of the denominator separately, and we shall have as the cube of, for instance, 27.

155. If it be required to find the cube of a mixed number, we must first reduce it to a single fraction, and then proceed in the manner that has been described. To find, for example, the cube of 1, we must take that of 3, which is 37, or 3 and 3. So the cube of 14. or of the single fraction, is 15, or 181; and the cube of 94, or of 13 is 2127, or $421.

156. Since a a a is the cube of a, that of a b will be a a ab bb; whence we see, that if a number has two or more factors, we may find its cube by multiplying together the cubes of those factors. For example, as 12 is equal to 3 x 4, we multiply the cube of 3, which is 27, by the cube of 4, which is 64, and we obtain 1728, for the cube of 12. Further, the cube of 2 a is 8 a a a, and consequently 8 times greater than the cube of a: and likewise, the cube of 3 a is 27 a aa, that is to say, 27 times greater than the cube of a.

157. Let us attend here also to the signs + and -. It is evident that the cube of a positive number + a must also be positive, that is +aaa. But if it be required to cube a negative number-a, it is found by first taking the square, which is +aa, and then multiplying, according to the rule, this square bya, which gives for the cube required-a a a. In this respeci, therefore, it is not the same with cubic numbers as with squares, since the latter are always positive: whereas the cube 8, that of — 3 is — 27, and

of

1 is

so on.

1, that of 2 is

-

CHAPTER XV.

Of Cube Roots, and of irrational numbers resulting from them.

158. As we can, in the manner already explained, find the cube of a given number, so, when a number is proposed, we may also reciprocally find a number, which, multiplied twice by itself, will produce that number. The number here sought is called, with relation to the other, the cube root. So that the cube root of a given number is the number whose cube is equal to that given number.

159. It is easy therefore to determine the cube root, when the number proposed is a real cube, such as the examples in the last chapter. For we easily perceive that the cube root of 1 is 1; that of 8 is 2; that of 27 is 3; that of 64 is 4, and so on. And in the same manner, the cube root of 27 is and that of - 125 is -5.

3;