148. Moreover, as a multiplied by ✔ makes ✔ab, we shall have б for value of √2 multiplied by √=s; 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 im possible number, gives always an imaginary product : thus, √3 by 5 gives 15. 149. It is the same with regard to division; for a divided by making , it is evident that 4 divided by 1 will make, or 2; that s divided by vs will give and that 1 divided by 1 gives +1 , or 1; because 1 is equal to F1. a 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 +va 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 useless, 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 67 8 9 10 18 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, 36, 42, 48, 54, 60; in which the terms, it is evident, increase always by 6. 8 154. After the definition we have given of a cube, it will not be difficult to find the cube of fractional numbers; is the cube ofis 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 14, we must take that of 3, which is 37, or 3 and 3. So the cube of 14, or of the single fraction 5, is 125, or 181 ; of 3, or of is 2127, or 3481. 13 21 and the cube 156. Since a a a is the cube of a, that of a b will be a a abbb; 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 × 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 a a, 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 respect, therefore, it is not the same with cubic numbers as with squares, since the latter are always positive: whereas the cube of 1 is 1, that of 2 is 8, that of 3 is - 27, and - so on. - 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 3; and that of 27 is Further, if the proposed number be a fraction, as, the cube root of it must be ; and that of is 4. Lastly, the cube root of a mixed number 21 must be, or 14: because 219 is equal to $4. 160. But if the proposed number be not a cube, its cube root cannot be expressed either in integers, or in fractional numbers. For example, 43 is not a cubic number; I say therefore that it is impossible to assign any number, either integer or fractional, whose cube shall be exactly 43. We may however affirm, that the cube root of that number is greater than 3, since the cube of 3 is only 27; and less than 4, because the cube of 4 is 64. We know therefore, that the cube root required is necessarily contained between the numbers 3 and 4. 161. Since the cube root of 43 is greater than 3, if we add a fraction to 3, it is certain that we may approximate still nearer and nearer to the true value of this root: but we can never assign the number which expresses that value exactly; because the cube of a mixed number can never be perfectly equal to an integer, such as 43. If we were to suppose, for example, 31, or to be the cube root required, the error would be ; for the cube of is only 33, or 427. 162. This therefore shews, that the cube root of 43 cannot be expressed in any way, either by integers or by fractions. However we have a distinct idea of the magnitude of this root; 3 which induces us to use, in order to represent it, the sign which we place before the proposed number, and which is read cube root, to distinguish it from the square root, which is often called 3 simply the root. Thus 43 means the cube root of 43, that is to say, the number whose cube is 43, or which, multiplied twice by itself, produces 43. 163. It is evident also, that such expressions cannot belong to rational quantities, and that they rather form a particular species of irrational quantities. They have nothing in common with square roots, and it is not possible to express such a cube root by a square root; as, for example, by √12; for the square of 12 being 12, its cube will be 12 √12, consequently still irrational, and such cannot be equal to 43. 164. If the proposed number be a real cube, our expressions 3 3 3 become rational; VI is equal to 1; √ is equal to 2; √27 is 3 equal to 3; and, generally, Va aa is equal to a. 3 3 165. If it were proposed to multiply one cube root, aby another, 3 √, the product must be ab; for we know that the cube root of a product a b is found by multiplying together the cube roots of the factors (156). Hence, also, if we divide vã by Vĩ, the qua tient will be 3 3 3 166. We further perceive, that 2 va is equal to 8a, because 3 3 3 2 is equivalent to 8; that 3 is equal to √27a, and ba is equal to Vabbb. So, reciprocally, if the number under the radical sign has a factor which is a cube, we may make it disappear by placing its cube root before the sign. For example, instead Hence 16 is equal to 2 √, because 16 is equal to 8 × 2. 167. When a 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, it follows that the cube roots of negative numbers are only negative. Thus, 8 is equal to -2, and 27 to 3 It follows also, that√12 is the same as may be expressed by -Va. Whence we see, that the sign —, when it is found after the sign of the cube root, might also have been placed before it. We are not therefore here led to impossible, or imaginary numbers, as we were in considering the square roots of negative numbers. |