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 (square root), which is used to express the square roots of all numbers that are not perfect squares. Thus 12 signifies or represents the square root of 12, or that number which, multiplied by itself, produces 12. So √2 represents the square root of 2, that of 3, and, generally, a 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 √. 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 √2 × 2 will produce 2, and that √ √ makes ; and so of any other number, and, generally, that produces c.

ax da

587. When, however, it is required to multiply a by b, the product is ab, 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 ab, by multiplying the square root of a, or a, by the square root of b, or b. And from this it is evident that if b were equal to a, aa would be the product of ✅a by b. Now, there can be no doubt that √aa must be a, for aa is the square of a.

a

588. In division, if it be required to divide a by b, the quotient must be ✅ in which it may be, that the irrational number may vanish in the quotient. Thus, in the case of dividing 18 by 8, the quotient is g, which is reduced to ✅, and, consequently, to being the square of

589. When the number to which the radical sign is prefixed happens to be a square, the expression of the root follows the ordinary course. Thus, 4 is equivalent to 2; 9 is the same as 3; 81 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 5 produces 2/5, and 3 multiplied by 2 produces 3./2. In the last instance, however, as 3 is equal to √9, the expression is also 3 times 2 by 9 multiplied by 2 or by 18. In the same way of considering this matter, 2a is the same as 4a, and 3a is equivalent to 9a. Generally, ba is equivalent to the square root of bba or ✅✔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 ba instead of √bba. From this it will be easy to comprehend the following expressions:

On the foregoing principles the operations of division are based, for a divided by ✔✅b must

591. It is unnecessary to follow this out in division and subtraction, because the numbers are merely connected by the signs + and -. For example, √2 added to √3 is expressed √2+ √3; and 6 subtracted from 10 is written √10/6.

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 OR IMAGINARY QUANTITIES.

-a

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 produces + aa, 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 O, and that all negative numbers are less than 0; hence whatever exceeds O is a positive number, and that which is less than 0 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 O multiplied by O is 0, and does not, therefore, produce a negative number. But as all conceivable numbers are greater or less than 0, or are O 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 -1, −2, √ −3, √-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 ✔ -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 -3, for instance, or the product of ✔-3 by √ -3 will be -3; that the product of -1 by -1 is -1; and, in general, that by multiplying a by√ Now -a we obtain —a. -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 ora is equal to a multiplied by -1. Buta is a possible or real number, consequently the whole impossibility of an imaginary quantity may be always reduced to 1. Thus -4 is equal to 4 multiplied by -1, and equal also to 2√−1, for the √4 is equal to 2; and so also -9 is reduced to 9x-1 or 3-1, and similarly ✔-16 is equal 4-1. Thus, also, as a multiplied by b produces ab, we have b for the value of -2 multiplied by -3; and 4 or 2 for the value of the product of -1 by -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,✓−3 by √ +5 gives √−15. 596. The same species of results prevail in division; for, as a divided by b makes it is clear that -4 divided by ✔-1 will make ✔+4 or 2, that +3 divided by √−3 gives ✓−1; and that 1 divided by ✔-1 gives ✓ or -1, because 1 is equal It has been already stated that the square root of a number has universally two values, one positive and the other negative; that 4, for example, is both +2 and -2; and that, generally, -va as well as +va exhibit equally the square root of a. It is

Va

to +1.

+1

the same in the case of imaginary numbers, for the square root of a is both +✔―a and --a, but the signs and - before the radical sign ✔ 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+√−4 and 6-√-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. 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 37. 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, which is 125 or 18, and the cube of 1 is that of 3, or, or 33.

As aaa is the cube of a, that of ab will be aaubbb; 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 2a 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+aaa; but the cube of a negative will be negative, for a by -a gives +aa, and that again multiplied by -a gives - ааа. So that it is not the same as with squares, for these are always positive.

CUBE ROOTS AND THE IRRATIONAL NUMBERS THAT RESULT FROM THEM.

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 the cube root is, and that of is. And last, in the case of a mixed number, as 2,9, the cube root will be or 1, because 219 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 to be the cube root required,

8

the error would be, for the cube of is only 343 or 427. 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 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 12, for the square of 12 being 12, its cube will be 12/12, consequently irrational, and such cannot be equal to 43.

604. If the proposed number be a real cube the expressions become rational: 1 is equal to 1; 8 is equal to 2; 27 is equal to 3; and, generally, Vaaa is equal to a.

605. If it be proposed to multiply one cube root by another, a, for example, by 3b, the product must be 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 a be divided by b, the quotient will be . And, further, 2 Va is equal to 8a, for 2 is the same as 8; 3a is equal to 27a, and ba 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 64a we may write 4 'a, and 7 Va instead of 343a. Hence 16 is equal to 2/2, 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 −27 to 3. So also -12 is the same as - 3/12, and 3-a may be expressed by Va. 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 POWERS IN GENERAL.

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.

R

608. Of powers, those of the number 10 are the most remarkable, as being the foundation of our system of arithmetic. We will range in order a few of them, as under: —

To consider which more generally, we may take the powers of any number a, as placed in the following order: —

But in this mode of writing powers there is much inconvenience, because of the trouble of counting the figures and letters; for the purpose of ascertaining the powers intended to be represented, as, for example, the inconvenience of representing the hundredth power would be so great as to incumber almost to impossibility the expression of it. To avoid this inconvenience, an expedient has been devised which is sufficiently convenient, and which we have now to explain. To express, for example, the hundredth power of a, we write just above it to the right the power in question; thus, a100 means, conventionally, a raised to the hundredth power. The number thus written above that whose power or degree it represents is called an exponent, from its expounding the power or degree to which the number is to be raised, which, in the example we have adduced, is 100. Thus, then, a2 represents the square or second power of a, which, as we have seen, may be also represented by aa, either of these expressions being understood with equal facility. To express the cube or third power of a or aaa, a3 is written, by which mode less room is occupied. So a1, as, aa, &c. respectively represent the fourth, fifth, and sixth powers of a. We may in this manner represent a by al, which, in fact, shows nothing more than that this letter is to be written only once. Such a series of powers as we here have noticed is called also a geometrical progression, because each term is once greater than the preceding.

a

609. As in this series of powers each term increases by multiplying the preceding term by a, thereby increasing the exponent by 1, so where any term is given the preceding one may be found if we divide by a, because it diminishes the exponent by 1: thus showing that the first term a1 must necessarily be or 1; hence, if we proceed according to the exponents, we immediately perceive that the term which precedes the first must be ao, from which follows this remarkable property, that ao is always equal to 1, however great or small the value of the number a may be, even if a be nothing.

a

610. The series of powers may be continued in a retrograde order, and in two different ways: first, by dividing continually by a; and, secondly, by diminishing the exponent by unity. In either mode the terms will be equal, The decreasing series, exhibited in both forms, is shown in the subjoined table, which is to be read from right to left.

Thus we come to the knowledge of powers whose exponents are negative, and are able to assign the precise value of those powers.

apparent that

a-l

a-2

And hence, from what has been said, it will be

-18

H:

is equal to

аа

or
a2

a-3

a3

&c.

This gives us the facility of finding the powers of a product ab; for they must be evidently ab, or a1b1, a2b2, a3b3, aaba, a5b5, &c.; and the powers of fractions are found in the same