Hence we have only to divide the square of the numerator by that of the denominator, and the fraction expressing that division is the square of the given fraction. Thus # is the square of , and, reciprocally, is the root of $3. 579. If the square of a mixed number, or one that is composed of an integer and a fraction, be sought, no more is necessary than to reduce it to a single fraction, and then take the square of that fraction. Thus, to find the square of 21, it must first be expressed by the fraction 4; and, taking its square, we have #, or 5 is for the value of the square of 24. And so of any similar numbers. The squares of the numbers between 3 and 4, supposing them to increase by one fourth, are as follow : — From this small tabular view it may be inferred that if a root contain a fraction, its square also contains one. Thus, let the root be 1 or its square is ##, or 1 for that is, rather more than half as great again as the integer 1. 580. Generally, when the root is a the square root must be aa ; if the root be 2a the square will be 4aa; from which it is evident that by doubling the root the square becomes 4 times greater; for if the root be 4a, the square is 16aa. If the root be ab, the square is aabb; if abc, the square is aabbc.c. 581. Thus, then, if the root be composed of more factors than one, their squares must be multiplied together; and, reciprocally, if a square be composed of more than one factor whereof each is a square, it is only necessary to multiply the roots of these squares to obtain the complete square of the root proposed. Thus, as 5184 is equal to 9 x 16 x 36, the square root of it is 3 x 4 x 6, or 72; and 72, it will be seen, is the true square root of 5184; for 72 x 72 gives 5184. 582. Here we must for a moment stop to see how the signs + and — affect our operations: and, first, it cannot be doubted that if the root is a positive quantity, that is, with the sign + before it, its square must be a positive quantity; for + by + makes + : thus, the square of + a will be + ad. So, also, if the root be a negative number, as —a, the square will still be positive, for it is + ad; from which it follows that of + a, as well as —a, the square is + aa ; hence every square has two roots, one positive and the other negative. For example, the square of 16 is both + 4 and —4, because —4 multiplied by –4 gives 16, as well as +4 by +4. squalir Roots, AND THE IRRAtionAL NUMBERs THAT REs ULT FROM THEM. 583. In the last section it has been seen that the square root of any number is but one whose square is equal to the given number, and that to those roots the positive or negative sign may be prefixed ; so that if we could remember a sufficient number of squares, their roots would at the same time present themselves to our mind. Thus, if 144 were the given number, we should at once recollect that its square root is 12. 584. For the same reason fractions would be easily managed ; for we at once see that # is the square root of {}, inasmuch as we have only to take the square root of the numerator and that of the denominator to be convinced of it. If we have to deal with a mixed number, we have only to put it in the shape of a single fraction: for example, 12! is equivalent to *; and we see by inspection that or 34 must be the square root of 12}. But when the given number is not a square, as, for example, 12, it is not possible to extract its square root, that is, to find a number multiplied by itself whose product is 12. It is, however, clear that the square root of 12 is greater than 3; for 3 × 3 produces only 9; and it must be less than 4, because 4 x 4 produces 16, which is greater than 12. From the table just given we may see that the square of 34 or ; is 12}; hence the root must be less than 3}. We may, however, come nearer to this root by comparing it with 37; ; for the square of 37, or of #, is o, or 12,33, a fraction only greater by 14; than the root required. Now, as o and 37, are both greater than the root of 12, it might be possible to add to 3 a fraction a little less than 7, precisely such that the square of such sum should be exactly equal to 12. Trying, therefore, with 3}, # being a little less than 7, we have 3}, equal to *, whose square is "o, and consequently less than 12 by #3; because 12 may be expressed by of Hence we perceive that 33 is less and 37, is greater than the root required. Trying a number, 3Å which is a little greater than 33, but less than 37, its equivalent is #, and it will have for its square 'o'; and as 12 reduced to the same denominator is of we thus find that 3i, is as yet less by or than the root of 12. If for # the fraction #, which is a little greater, be substituted, we have the square of 3 or equal to #; and 12 reduced to the same denominator, or multiplied by 169, equal to *; so that 3%, is yet too small, though only by so, whilst 3i, has been found too great. From this it is evident that whatever fraction be joined to 3, the square of that sum will always contain a fraction, and will not be equal exactly to the integer 12. For, knowing that the square root of 12 is greater than 3%, and less than 37, we are nevertheless unable to assign between the two an intermediate fraction, which, added to 3, precisely expresses the square root of 12. But it must not therefore be said that the square root of 12 cannot be absolutely determined, but only that it cannot be expressed by fractions. 585. We hence find that there exists a species of numbers which, though not expressible by fractions, are yet determinate quantities, and of this the square root of 12 furnishes an example. This species of numbers are termed irrational numbers, and occur as often as we attempt to find the square root of a number which is not a square. Thus, 2 not being a perfect square, its square root, or the number which, multiplied by itself, would produce 2, is an irrational quantity. Such numbers are also called surd quantities, or incommensurables ; and though they cannot be expressed by fractions, they are, nevertheless, magnitudes of which an accurate idea may be formed. In the case of the number 12, for example, though its square root is not apparent, we know that it is a number which, multiplied by itself, would exactly produce 12; and this is a property which, by the power of approximating to it, is enough to enable us to form some idea of it. 586. Having now obtained a distinct idea of the nature of these irrational numbers, we must introduce to the reader the use of the sign w (square root), which is used to express the square roots of all numbers that are not perfect squares. Thus w/12 signifies or represents the square root of 12, or that number which, multiplied by itself, produces 12. So v2 represents the square root of 2, wi that of 3, and, generally, Va represents the square root of the number a. If therefore, we have at any time to express the square root of a number, all that is necessary is, to prefix to it the sign V. This explanation of irrational numbers enables us to apply to them the known methods of calculation. For, inasmuch as the square root of 2 multiplied by itself must produce 2, we know that V2 x v2 will produce 2, and that v3 x v3 makes ; ; and so of any other number, and, generally, that va x va produces c. 587. When, however, it is required to multiply wa by Vb, the product is Vab, for it has been heretofore shown that when a square has two or more factors, its root is composed of the roots of those factors. Hence we find the square root of the product ab, which is vab, by multiplying the square root of a, or Va, by the square root of b, or Vb. And from this it is evident that if b were equal to a, Vaa would be the product of va by wb. Now, there can be no doubt that waa must be a, for aa is the square of a. 588. In division, if it be required to divide va by W b, the quotient must be 27. in which it may be, that the irrational number may vanish in the quotient. Thus, in the case of dividing / 18 by v's, the quotient is v', which is reduced to v3, and, consequently, to 3, # being the square of 3. 589. When the number to which the radical sign v is prefixed happens to be a square, the expression of the root follows the ordinary course. Thus, v.4 is equivalent to 2; wg is the same as 3; V81 the same as 9; and 12, the same as , or 3}; in which instances the irrationality is but apparent, and vanishes. 590. No difficulty occurs in multiplying irrational by ordinary numbers. Thus 2 multiplied by V5 produces 24/5, and 3 multiplied by V2 produces 3 v2. In the last instance, however, as 3 is equal to V.9, the expression is also 3 times V2 by v0 multiplied by V2 or by v18. In the same way of considering this matter, 2 va is the same as W4a, and 3 va is equivalent to W9a. Generally, b va is equivalent to the square root of bba or x/abb; and, reciprocally, when the number preceded by the radical sign contains a square, the root of the square may be prefixed to the sign, as in writing b Va instead of Vbba. From this it will be easy to comprehend the following expressions : — On the foregoing principles the operations of division are based, for va divided by vö must be 4 or vo, and thus — 591. It is unnecessary to follow this out in division and subtraction, because the numbers are merely connected by the signs + and —. For example, v2 added to V3 is expressed w/2 + v 3 ; and vô subtracted from v1.0 is written v 1.0–~/6. 592. For the purpose of distinguishing these numbers from all others not similarly circumstanced, the latter, as well integral as fractional, are denominated rational numbers; and thence, when we speak of rational numbers, it is to be understood that we speak of integers or fractions. IMpossiblE on IMAGINARY QUANTITIES. 593. The squares of numbers, whether negative or positive, as we have shown above, are always affected by the + or positive sign, for it has been seen that – a multiplied by — a produces + ad, in the same way as + a by + a produces the same result; and it was on this account that in the preceding section all the numbers whose roots were to be extracted were considered positive. If, however, the root of a negative number is to be extracted, a difficulty arises, because there is no assignable number whose square would be a negative quantity. If, for instance, we wanted the root of —4, we have to search for a number which, multiplied by itself, will produce –4. This number can be neither +2 nor -2, because the squares of both will be + 4, and not —4. Hence we must conclude that the square root of a negative number is neither positive nor negative, inasmuch as that the squares of negative numbers are affected by the sign +. The root must, therefore, belong to a species of numbers entirely distinct from all others, for it cannot be placed among either positive or negative numbers. 594. It has been observed that all positive numbers are greater than 0, and that all negative numbers are less than 0; hence whatever exceeds 0 is a positive number, and that which is less than o must be expressed by negative numbers. Thus the square roots of negative numbers are neither greater nor less than nothing. But they are not O, because the product of 0 multiplied by 0 is 0, and does not, therefore, produce a negative number. But as all conceivable numbers are greater or less than 0, or are 0 itself, the square root of a negative number cannot be ranked among possible numbers; hence it is said to be an impossible quantity; and it is this which leads us to an idea of numbers which are naturally impossible. They are usually called imaginary quantities, from their existing only in imagination. Such expressions, therefore, as V–1, V-2, W-3, V-4, &c. are impossible or imaginary numbers, because they represent roots of negative quantities; and of such numbers it may be said that they are neither nothing nor greater nor less than nothing; they are, therefore, imaginary or impossible. Though existing only in our imagination, we may form a sufficient idea of them, for we know that y – 4 expresses a number which, multiplied by itself, produces – 4. For this reason there is nothing to prevent, in calculation, the use of these imaginary numbers. 595. The most obvious idea on the above matter is, that the square of W —3, for instance, or the product of V-3 by V–3 will be –3; that the product of V-1 by W – 1 is – l ; and, in general, that by multiplying v – a by V—a we obtain -a. Now -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 v-a is equal to Va multiplied by V-1. But va is a possible or real number, consequently the whole impossibility of an imaginary quantity may be always reduced to w/–1. Thus v-4 is equal to v4 multiplied by v-1, and equal also to 2v — 1, for the V4 is equal to 2; and so also v-9 is reduced to v0 x v — 1 or 3 v-1, and similarly V-16 is equal 4 v-1. Thus, also, as va multiplied by vb produces vab, we have vb for the value of v-2 multiplied by v-3; and v4 or 2 for the value of the product of v–1 by v-4. Hence we see how two imaginary numbers multiplied together produce one which is real or possible. But, on the other hand, a possible number multiplied by an impossible one always produces an imaginary product: thus, v-3 by v 4-5 gives v-15. 596. The same species of results prevail in division; for, as va divided by vb makes ... it is clear that v-4 divided by v-1 will make v. 4 or 2, that was divided by v-3 gives v-1 ; and that I divided by v-1 gives vo or v-1, because 1 is equal to v 4-1. It has been already stated that the square root of a number has universally two values, one positive and the other negative; that v4, for example, is both + 2 and —2; and that, generally, -va as well as +va exhibit equally the square root of a. It is the same in the case of imaginary numbers, for the square root of —a is both + v —a and -v-a, but the signs + and – before the radical sign v must not be confounded with the signs that come after it. 597. However, on first view, it may seem idle speculation thus to dwell on impossible numbers, the calculation of imaginary quantities is of the greatest importance, for questions constantly arise wherein it is impossible to say whether anything real or possible is or is not included, and when the solution of such a question leads to imaginary quantities, we are certain that what is required is impossible. Thus, suppose it were required to divide the number 12 into two such parts that the product of them may be 40. In resolving this question by the ordinary rules we find, for the parts sought, 6 + v.–4 and 6–v–4, both imaginary numbers; hence we know that it is impossible to resolve the question. The difference is manifest in supposing the question had been to divide 12 into two parts whose product should produce 35, for it is evident that those parts must be 7 and 5. 598. A number twice multiplied by itself, or its square multiplied by the root, produces a cube or cubic number. Thus the cube of a is aaa, for it is the product of a multiplied by a, and that square aa again multiplied by a. The cubes of the natural numbers are placed in the subjoined table: — Analysing the differences of these cubes, as we did those of the squares, by subtracting each cube from that following, the following series of numbers occur : — 7, 19, 37, 61, 91, 127, 169, 217, 271, And in these there does not appear any regularity; but, taking the differences of these, we shall have the following series : — 12, 18, 24, 30, 36, 42, 48, 54, 60; On the inspection of which it will be seen that the terms increase regularly by 6. 599. From the definition of a cube the cubes of fractional numbers are easily found: thus, is the cube of , , is the cube of , and ; is the cube of 3. Thus, also, we have only to take the cube of the numerator and that of the denominator separately, and for the cube of we have so To find the cube of a mixed number it must be reduced, first to a single fraction, and the process is then conducted in a similar manner. Thus, to find the cube of 14 we must take the cube of 3, which is of or 13, and the cube of 14 is that of 3, or 7, or 3}. As aaa is the cube of a, that of ab will be aaabbb; from which we learn, that if a number has two or more factors, its cube may be found by multiplying together the cubes of those factors. For instance, as 12 is equal to 3 x 4, the cube of 3, which is 27, if multiplied by the cube of 4, which is 64, gives us 1728, the cube of 12. Again, the cube of 2d is 8aaa, that is to say, 8 times greater than the cube of a ; so the cube of 4a is 64aaa, that is to say, 64 times greater than the cube of a. 600. The cube of a positive number will, of course, be positive : thus, that of + a will be + ada; but the cube of a negative will be negative, for — a by — a gives + ad, and that again multiplied by — a gives —aaa. So that it is not the same as with squares, for these are always positive. cube Roots AND The in RAtion AL Nuxibens That Result frto M The M. 601. As we can, by the mode above given, find the cube of any given number, so may we find one which, multiplied twice by itself, will produce that number. With relation to the cube this is called the cube root, or that whose cube is equal to the given number. When the number proposed is a real cube the solution is easy enough. For there is no difficulty in seeing that the cube of 1 is 1, that that of 8 is 2, that of 4 is 64, and so on; and equally that the cube root of -27 is –3, and that of –216 is -6. Similarly, if the proposed number be a fraction, as 1%; the cube root is 3, and that of #, is . . And last, in the case of a mixed number, as 2,3, the cube root will be 3 or 13, because 2! is equal to #. 602. If, however, the proposed number be not a cube, its cube root cannot be expressed either in integers or fractional numbers. Thus, 43 is not a cube number; hence it is impossible to assign any number, integer or fractional, whose cube shall be exactly 43. We may, however, assert that the cube root of that number is greater than 3, for the cube of 3 is only 27, and less than 4, because the cube of 4 is 64. The cube root required lies, therefore, between 3 and 4. The cube root of 43 being greater than 3, by adding a fraction to 3 we may approach nearer to the value of the root, but we shall never be able to express the value exactly, because the cube of a mixed number can never be exactly equal to an integer, as 43 for instance. If we suppose 3 or 3 to be the cube root required, the error would be for the cube of ; is only * or 42. Thus we see that the cube root of 43 can be expressed neither by integers nor fractions. We obtain, however, a distinct notion of its magnitude, and, for the purpose of representing it, a sign &/ is placed before the number which is read cube root, to distinguish it from the square root, which is frequently merely called the root. Thus 8/43 expresses the cube root of 43, that is, the number whose cube is 43. 603. It is evident that such expressions cannot belong to rational quantities, and that, indeed, they form a particular species of irrational quantities. Between them and square roots there is nothing in common, and it is impossible to express such a cube root by a square root, as, for example, by V12, for the square of V12 being 12, its cube will be 12 viz, consequently irrational, and such cannot be equal to 43. 604. If the proposed number be a real cube the expressions become rational : &l is equal to 1 ; 3/8 is equal to 2; 8/27 is equal to 3; and, generally, o/aaa is equal to a. 605. If it be proposed to multiply one cube root by another, o/a, for example, by 8/b, the product must be 8/ab; for it has already been seen that the cube root of a product ab is found by multiplying together the cube root of its factors. Whence, also, if 3/a be divided by &b, the quotient will be 3/. And, further, 2 o'a is equal to &8a, for 2 is the same as 88; 3.3%a is equal to 3/27a, and b&a is the same as &abbb. So, reciprocally, when the number under the radical sign has a factor which is a cube, we may always get rid of it by placing its cube root before the sign. Thus, instead of 8/64a we may write 43'a, and 7 */a instead of 3/343a. Hence {/16 is equal to 28/2, because 16 is equal to 8 x 2. When a number proposed is negative, its cube root is not subject to the difficulties which we observed in speaking of square roots; for, as the cubes of negative numbers are negative, it follows that their cube roots are but negative. Thus */–8 is equal to —2, and 3/–27 to -3. So also R/–12 is the same as – 8/12, and 3/-a may be expressed by – 3/a. From which it may be deduced that the sign –, though found after the sign of the cube root, might have been as well placed before it. Hence we do not herein fall upon impossible or imaginary quantities, as we did in considering the square roots of negative numbers. of powerts in cenreitar. 606. A power is that number which is obtained by multiplying a number several times by itself. A square arises from the multiplication of a number by itself, a cube by multiplying it twice by itself, and these are powers of the number. In the former case we say the number is raised to the second degree or to the second power; and in the latter, the number is raised to the third degree or to the third power. 607. These powers are distinguished from one another by the number of times that the given number has been multiplied by itself. Thus the square is called the second power, because it has been removed to the second product by multiplication by itself; another multiplication by itself brings it to the third power or cube. When multiplied again by itself it becomes the fourth power, which is commonly called the bi-quadrate. From this will be readily comprehended what is meant by the fifth, sixth, seventh, &c. power of a number. After the fourth degree the names of the powers have only numeral distinctions. For the purpose of illustration, we may observe, that the powers of 1 must always be 1, decause how often soever we multiply 1 into itself the product must be 1. The following table shows the powers of 2 and 3. |