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Now we have already observed, that all positive numbers are greater than 0, and that all negative numbers are less than 0. The square roots of negative numbers, therefore, are neither greater nor less than O, and yet they cannot be said to be exactly equal to O, since 0 multiplied by O produces O, and consequently does not produce a negative number.

Since, then, every number of which we can form any conception must be either greater or less than 0, or must exactly amount to 0, or nothing, it is evident that the square root of a negative number cannot be ranked among possible numbers, and therefore that every such root must be termed an impossible or imaginary number, so called, because it exists only in imagination.

161.

CHAP. XVI.

OF CUBIC NUMBERS.

WHEN a number is multiplied twice into itself, or, which is the same thing, when the square of a number is multiplied by the number itself, the product obtained is called a cube, or a cubic number. Thus, the cube of a is aaa or a3, which is evidently obtained by multiplying first a into itself, and then the product a by a again.

The following is a table of the cubes of the natural numbers:

Numbers 1 2 3 4 5 6 7 8 9

10

Cubes 1 8 27 64 125 216 343 512 729 1000

162. If

162. If we observe the differences of these cubes, as we have done in the square numbers, by subtracting each cube from that which follows it, we shall obtain the following series,

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

In which we may remark this regularity, that if we take the respective differences of these numbers, we shall obtain a series, of which each term increases by 6; viz.

12, 18, 24, 30, 36, 42, 48, 54.

163. After the definition that we have given of a cube, we shall find no difficulty in obtaining the cubes of fractional numbers; for we have only to pursue the same course as has been pointed out in square numbers; viz. to take the cubes of the numerator and denominator of the fraction. Thus is the cube of; of; 7 of 4, &c.

164. If it be required to find the cube of a mixed number, we must reduce it first to an improper fraction, and then proceed in the same manner. For example, the cube of 14 of is or 3. The cube of 14 or 4 is 25, or 18, &c. off for

165. Since aaa or a3 is the cube of a, the cube of ab will be aaa bbb, or ab. By which we perceive, that if a number has two or more factors, the cube of it is found by multiplying together the cubes of the factors. For example, as 12 is composed of the factors 3×4, by multiplying the cube of 3, which is 27, by the cube of 4, which is 64, we shall obtain 1728, which is the cube of 12. In like manner the cube of 2 a is 8a3, of 3b, 27b3, &c.

.

166. With regard to the signs + and -, it is clear, first, that the cube of a positive number, as +a, must itself be positive, as+a3. But if it be required to find the cube of a negative number, as -a, we see first that the square will bea, and that being multiplied by -a, will give —a3. From which we deduce this rule, that the cube of a negative number will also be negative; a rule which differs from that of square numbers, which are always positive. Thus the cube of -2 is 8, of -3, -27, and so on.

CHAP. XVII.

OF CUBE ROOTS, AND IRRATIONAL NUMBERS
DERIVED FROM THEM.

167. SINCE, in the manner above shown, we are enabled to find the cube of any given number, so also we are enabled, when a number be given, to find one, which, being multiplied twice into itself, will produce the number proposed; and this number is called, with reference to the other, the cube root, so that the cube root of any number is a number such, that its cube will be equal to the given number.

168. When, therefore, the number proposed is a real cube, it is easy to find its cube root, as we can perceive from the examples in the preceding chapter. Thus we know that the cube root of 8 is 2; of 27, 3; of 64, 4, and so on; and in like manner that the cube root of 27 is -3; and of -125, -5, &c.

169. Again, if the number proposed be a fraction, as, the cube root will be; and that of will be ; and lastly, that the cube root of a mixed number, as 21, is or 13; since 21 is the same as $4.

170. But if the number proposed is not a real cube, its cube root can no longer be expressed, either by a whole number or a fraction. For example, 43 not being a cubic number, it is impossible to assign any number, either integral or fractional, whose cube will be exactly equal to 43. We are enabled, however, to state that the cube root of this number must be greater than 3, since we have already seen that the cube of 3 amounts only to 27; and also that it must be less than 4, since the cube of 4 is 64. We know therefore

that

that the cube root sought must be some intermediate number between 3 and 4.

171. If, therefore, since the cube root of 43 is greater than 3, we were to add to 3 a fraction, it is certain that we must approximate nearer to the true value of this root, although we are unable to assign a number which will express its precise value; since the cube of a mixed number can never be exactly equal to an integer, such as 43. If, for instance, we suppose 3 or to be the cube root of 43, the error will amount to, for the cube of is 49427.

172. It is clear, therefore, that the cube root of 43 can never be exactly expressed, either by an integer or a fraction. At the same time, since we have a distinct idea of the magnitude of this root, it is sufficient to represent it by the sign 3, placed before the number, and which is pronounced cube root. Thus 43 signifies the cube root of 43, or such a number that being multiplied twice into itself will exactly produce 43.

173. If it be proposed to multiply one cube root, as a, by another, as /b, the product will be ab; since we know that the cube root of the product ab is found by multiplying together the cube roots of the factors; and in like manner it is evident that if a be divided by the 3/6, the quotient

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174. From this it will be easily understood that 2 3/a is the same with 3/8a, since 2 is the cube root of 8; that 3/a is the same as 3/27 a, and in general that b3/a is the same as ab3. Thus, therefore, reciprocally, if the number under the radical sign has a factor, which is a cubic number, we can extract its root, and place it before the radical sign. For example,

3

a=4a

3/64a = 3/64 ×
/125a/125 x /a5/a
/16 = 3/8 × 3/2=23/2.

F

175. When

175. When the number proposed is negative, its cube root is not subject to the same difficulties that occurred in treating of square roots; for since the cubes of negative numbers are negative, the cube roots of negative numbers will also be negative. Thus, -8-2; 3-27-9. Thus also

-12 is the same thing as - 3/12, and 3-a may generally be expressed -Va. We are therefore not led here to those impossible or imaginary quantities, by the consideration of negative cube roots, as we met with in considering the square roots of negative numbers.

CHAP. XVIII.

OF THE SQUARES OF COMPOUND QUANTITIES.

176. We have already seen that in order to find the square of any quantity, we have only to multiply the quantity into itself, and the product will be the square required.

In the application of this rule to compound quantities, the square of a+b will be found in the following manner: a+b

a+b

a2+ab

+ab +b2

a2+2ab+b2

177. So that when the root consists of two terms added together, as a+b, the square consists of, 1st, the squares of each term, viz. a2 & b2; and 2dly, twice the product of the two terms, viz. 2ab. Now let us suppose a 10, and b=3, and let it be required to find the square of 10+ 3, or 13. We shall have 100+60 +9, or 169, for the square required.

178. We

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