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The only matter remaining, then, is the consideration of the powers of negative numbers.
CALCULATION or POWERs.
611. The addition and subtraction of powers is effected by means of the signs + and when the powers are different; for example, a3+ a” is the sum of the third and second powers of a; and a” —a is the remainder when the fourth power of a is subtracted from the fifth ; neither of which results can be abridged. If the powers are of the same kind or degree it is not necessary to connect them by signs, thus as + as makes 2a3, &c.
612. But in the multiplication of powers, we must observe, first, that any power of a multiplied by a, gives the succeeding power, that is to say, the power whose exponent is one unit greater. Thus a” multiplied by a produces as ; and as multiplied by a produces a+. Similarly, if it be required to multiply by a, the powers of that number having negative exponents, 1 must be added to the exponent. Thus, a "multiplied by a produces ao or 1;
and this becomes most clearly seen by considering that a" is equal to: and that the product of # being #, it is consequently equal to 1. So a-* multiplied by a produces
Or #, and a "multiplied by a produces a ', and so on.
613. If it be required to multiply a power of a by aa or the second power, the exponent then becomes greater by 2. Thus the product of a” by a2 is at ; that of as by as is as; that of a by a' is a'; and, generally, a” multiplied by as makes a”. In the case of ne
gative exponents, a' or a is the product of a-' by a”. For a-l being the same as 1, it is d
just the same as if we had divided aa by a ; hence the product required is : or a. In the
same way, a-* multiplied by a produces a” or 1, and a "multiplied by a” produces a-'.
It is equally clear that to multiply any power of a by as, its exponent must be increased by three units, consequently the product of a” by a” is a”. And as often as it is required to multiply two powers of a, the product must be a power of a whose exponent is equal to the sum of those of the two given powers. For instance, a multiplied by as will make a”, and a12 multiplied by a produces a 19, &c.
614. On the principles here exhibited, it is easy to determine the highest powers. Thus, to find the twenty-fourth power of 2, multiply the twelfth power by the twelfth power; because 224 is equal to 219 x 212. But we have already seen that 212 is equal to 4096 ; hence the number 16777216, being the product of 4096 by 4096, is 224, or the required power of 2.
615. In division we must observe that to divide a power of a by a the exponent must be diminished by unity. Thus a divided by a gives a”; a” or 1 divided by a is equal to a-' or : ; a "divided by a gives a". So, if we have to divide a given power of a by a”, the exponent must be diminished by 2, and if by as, three units must be subtracted from the exponent of the power proposed; and, generally, if it be required to divide any power of a by any other power of a, the rule is to subtract the exponent of the second from the exponent of the first of those powers. Thus a” divided by a” gives a ; as divided by as will give a ". So a" divided by a will give a ".
616. It is not difficult, then, from what has been said, to find the powers of powers, for it is effected by multiplication. Thus, if we have to seek the square or second power of a”, we find as, and for the cube or third power of a we have a”. To obtain the square of a power it is only necessary to double the exponent; for its cube, to triple the exponent, and so on. Thus a” is the square of a”, a” is the cube of a”, and the seventh power of a" is a”. The square of a', or square of the square of a, being a', is hence called biquadrate. The square of as is as; hence the sixth power has received the name of the square-cubed. To conclude, the cube of a being a”, the ninth power has received the name of the cubo-cube.
Root's RELATIVELY TO POWERS IN GENERAL.
617. The square root of a given number is a number whose square is equal to that
number; the cube root, that whose cube is equal to the given number: hence, whatever
number be given, such roots of it will exist that their fourth, their fifth, or any other
power, will be equal to the given number. For distinction sake, we shall call the square
root the second root, the cube root the third root, the bi-quadrate the fourth root, and so on. As the square or second root is marked by the sign V, and the cubic or third root by the sign & ; so the fourth and fifth roots are respectively marked by the signs & and y, and so on. It is evident, according to this method of expression, the sign of the square root should be v; but by common consent the figure is always left out; and we are to recollect that when a radical sign has no number prefixed to it, the square root is always meant. To give a still better explanation, we here subjoin some different roots of the numbera, with their respective values: –
618. Whether a be a small or a great number, we know what value to affix to all these roots of different degrees. If unity be substituted for a the roots remain constantly l; for all powers of 1 have unity for their value. But if the number a be greater than 1, the roots will also all exceed unity; and further, if a represent a less number than 1, all the roots will be less than unity.
619. When the number a is positive, from what has been before said of square and cube roots, we know that all the other roots may be determined, and that they will be real and possible numbers. But if the number a is negative, its second, fourth, sixth, and all even roots become impossible, or imaginary numbers; because all the powers of an even order, whether of positive or of negative numbers, are affected by the sign + ; whereas the third, fifth, seventh, and all odd roots become negative, but rational, because the odd powers of negative numbers are also negative. Hence an inexhaustible source of new kinds of surd or irrational quantities; for, whenever the number a is not a power represented by some one of the foregoing indices, it is impossible to express the root either in whole numbers or fractions, and it must therefore be ranked among the numbers called irrational.
THE REPRESENTATION OF POWERS BY FRACTIONAL ExPoNENTS,
620. In the preceding subsections we have seen that the square of any power is found by doubling its exponent, and that in general the square or second power of a” is a”. Hence the converse, that the square root of the power a” is found by dividing the exponent of
that power by 2. Thus the square root of a” is al.; that of as is as; and as this is general, the square root of as is necessarily al, and that of a is . Thus we have a! for the square root of a', and hence a' is equal to va; a new method of expressing the square root, which requires our particular attention. 621. To find the cube of a power, as a”, we have already shown that its exponent must be multiplied by 3, hence its cube becomes a”; and, conversely, to find the third or cube root of the power a”, we have only to divide the exponent by 3; hence the root is a ". Thus, also, al or a is the cube root of as, as that of as, at that of ale, and so on. The same reasoning is applicable to those cases in which the exponent is not divisible by 3; for it is evident that the cube root of a” is as, as the cube root of a' is ałor als. Hence the
third or cube root of a or a will be a!, which is the same as R/a. 622. The application is the same with roots of a higher degree: thus the fourth root of
a will be ał, which expression is of the same value as :/a. The fifth root of a will be
a', which is equivalent to :/a, and so on in roots of higher degree. It would be possible, therefore, to dispense altogether with the radical signs, and to substitute fractional exponents for them; but as custom has sanctioned the signs, and as they are met with in all works on algebra, it would be wrong to banish them altogether from calculation. There is, however, sufficient reason to employ, as is frequently done, the other method of calculation; because it clearly corresponds with the nature of the thing. Thus, in fact, it is manifest that a' is the square root of a, because we know that its square is equal to at or a.
623. What has been said will be sufficient to show how we are to understand fractional
exponents; thus, if d should occur, it means that we are first to take the fourth power of a and then extract its cube or third root, and hence ad is the same as 3'a". Again, to find
the value of at the cube or third power of a or a must first be taken, and the fourth root
of that power extracted, so that a' is the same as *a*. So a is the same as £/a4, &c. But when the fraction which represents the exponent is greater than unity, the value of the given quantity may be otherwise expressed. Let it, for instance, be a', now this quantity is equivalent to a” which is the product of a by a!. Now a' is equal to va, wherefore aí is equal to a 2 x/a. So a', or a”, is equal to a 3:/a8; and a', that is, ask expresses 'va'. From these examples the use of fractional exponents may be properly appreciated. This, however, extends also to fractional numbers, as follows.
w. is given, we know that it is equal to '' now we have already seen
that a fraction of the form : may be expressed by a "; and instead of .# we may use
the expression a ". Also, 3. is equal to a". So let the quantity : be proposed,
it is transformable into #. which is the product of a” by a", and this is equivalent to ai
a;, or to a”, or, lastly, to Va". These reductions will be facilitated by a little practice.
625. Each root may be variously represented, for Va is the same as ał, and 1 being equivalent to the fractions 3, #, #, #, #, &c., it is clear that Va is equal to :/aş, to 5'as, to wa', and so on. Similarly, 3/a is equal to a”, and to 8/as, to 8/as, and to 3'a". It is, moreover, manifest, that the number a, or a might be represented by the following radical expressions:
3/a2, 3/a3, .3/a4, 3/as, &c.
a property of great use in multiplication and division; for, suppose we have to multiply 3/a by 3/a, we write 3/a3 for 3/a, and 3/a2 - instead of 3/a, thus obtaining the same radical sign for both, and the multiplication being now performed, gives the product vas. A similar result arises from a+**, the product of a multiplied by a', for | +} is #, and, consequently, the product required is ai, or %as. If it were required to divide va or a by
METHODS OF CALCULATION AND THEIR MUTUAL CONNECTION.
626. In the foregoing passages have been explained the different methods of calculation in addition, subtraction, multiplication, and division, the involution of powers, and the extraction of roots. We here propose to review the origin of the different methods, and to explain the connection subsisting among them, in order that we may ascertain if it be possible or not for other operations of the same kind to exist; an inquiry which will illustrate the subjects that have been considered. We shall, for this purpose, here introduce a new sign =, which means that equality exists between the quantities it is used to join, and is read equal to. Thus, if I write a=b, it means that a is equal to b; and so 3 x 8 =24. 627. Addition, the process by which we add two numbers together and find their sum, is the first mode of calculation that presents itself to the mind. Thus if a and b be two given numbers whose sum is expressed by c, we shall have a + b =c. So that, knowing the two numbers a and b, we are taught by addition how to find the number c. Recollecting this comparison a +b=c, the question may be reversed by asking in what way b can be found when we know the numbers a and c. Let us, then, ascertain what number must be added to a that the sum may be c. Now, suppose, for instance, a =3, and c=8, it is evident we must have 3+b=8, and b will be found by subtracting 3 from 8. So, generally, to find b, we must subtract a from c, whence arises b=c-a; for, by adding a to both sides again, we have b+a=c-a+a, that is, as was supposed, =c. And this is the origin of subtraction, being, indeed, nothing more than an inversion of the question from which *ddition arises. Now it is possible that it may be required to subtract a greater from a *er number; as, for example, 9 from 5. In this case we are furnished with the idea of * new kind of numbers, which are called negative numbers, because 5–9 = -4. 628. If several equal numbers are to be added together, their sum is found by multiplication, and is called a product. Thus ab expresses the product of the multiplication of a by b, or from a being added to itself b times. If this product be represented by c, we have ab=c, and we may, by the use of multiplication, determine the number c where the num** a and b are known. Suppose, for example, a =3, and c=15, so that 3b=15, we ave to ascertain what number b represents, merely to find by what number b is to be multiplied, in order that the product may be 15, for to that is the question reduced: and * is division; for the number sought is found by dividing 15 by 3; hence, in general,
the number b is found by dividing c by a, whence results the equation b=#.
But, frequently, the number c cannot be actually divided by the number a, the letter b having a determinate value; hence a new kind of numbers, called fractions, arises. For, suppose a=4, c=3, so that 4b =3, in this case b cannot be an integer, but must be a fraction, and we shall find that b can be no more than #. Multiplication, then, as we have seen, arises from the addition of equal quantities; so, from the multiplication of several equal quantities together, powers are derived, and they are represented in a general manner by the expression a', which signifies that the number a must be multiplied by itself as often as is pointed out by the number b, which is called the exponent, whilst a is called the root, and a the power. If this power be represented by the letter c, we have a*=c, an equation in which are found the letters a, b, c. In treating of powers, it has been shown how to find the power itself, that is, the letter c, when the root a and its exponent b are given. Suppose, for instance, a =4, and b = 3, we shall have c=43, or the third power of 4, which is 64, whence c=64. If we wish to reverse this question, we shall find that there are two modes of doing it. Let, for instance, two of the three numbers a, b, and c be given. If it be required to find the third, it is clear that the question admits of three different suppositions, and hence, also, of three solutions. The case has been considered in which a and b were the numbers given; we may therefore suppose, further, that c and a or c and b are known, and that it is required to determine the third letter. Now, it must be observed, that between involution and the two operations which lead to it there is a great difference. For when, in addition, we reverse the question, there was only one way of doing it, and it was of no consequence whether we took c and a or c and b for the given numbers, for it is quite indifferent to the result whether we write a + b or b +a. And it is quite the same with multiplication; the letters a and b might be placed in each other's places at pleasure, the equation ab=c being exactly the same as ba–c. But in the calculation of powers, we cannot change the places of the letters; for instance, we could on no account write b for a'. This we will illustrate by one example. Thus, let a =4, and b-3, we have a' = 4s =64. But b =34 =81, two very different results. 629. We may propose two more questions; one to find the root a by means of the given power c, and the exponent b, the other to find the exponent b, the power c and the root a being known. The former of these questions has been answered in the subsection which treats of the extractions of roots : since, if b = 2, and a2+ c, we know that a is a number whose square is equal to c, and consequently a =Vc. So, if b = 3 and a' =c, we know that the cube of a is equal to the given number c, and hence that a = &c. We conclude, generally, from this, how the letter a may be determined by means of the letters c and b; for a must necessarily be =&c. 630. We have already seen that if the given number is not a real power (a contingency of frequent occurrence), the required root a can be expressed neither by integers nor fractions; nevertheless, as it must have a determinate value, the same consideration led us to the numbers called surd or irrational numbers, which, on account of the great variety of roots, are divisible into an infinite number of kinds. We were also, by the same enquiry, led to the knowledge of imaginary numbers. 631. Upon the second question, that of determining the exponent by means of the power c and the root a, is founded the very important theory of logarithms; an invention so important that without them scarcely any long calculation could be effected.
632. Resuming, then, the equation ab–c, we in the doctrine of logarithms assume for the root a number taken at pleasure, but supposed to preserve its assumed value without variation. This being the case, the exponent b is taken, such that the power a becomes equal to a given number c, and this exponent b is said to be the logarithm of the number c. To express this, we shall use the letter L or the initial letters log. Thus, by b-Lie or : logic, we mean that b is equal to the logarithm of the number c, or that the logarithm of c is b. 633. If the value of the root a is once established, the logarithm of any number c is but the exponent of that power of a which is equal to c. So that c being = a, b is the logarithm of the power of a . If we suppose b = 1, we have 1 for the logarithm of a ; hence L-a-1. Suppose b=2, we have 2 for the logarithm of a”; that is Lae=2. Similarly, La"=3, L. a' =4, L.a5–5, and so on. 634. If b be made =0, o must be the logarithm of ao; but ao =1; consequently, L.1 =0, whatever the value of the root a. If b=-1, then-1 will be the logarithm of a ". Now a-'-'. ; therefore, L.' =-1. So, also, L. := –2 ; L.:- –3; L. := -4; &c. 635. Thus, then, may be represented the logarithms of all the powers of a, and even those of fractions wherein unity is the numerator, and the denominator a power of a. We see, also, that, in all those cases, the logarithms are integers: but if b were a fraction it would be the logarithm of an irrational number. For suppose b=\, then is the logarithm of a ', or of va; consequently we have L. va–4; and in the same way, L. :/a+\, L. &a=1 &c. 636. If it be required to find the logarithm of another number c, it will be readily seen that it can neither be an integer nor a fraction. However, there must be such an exponent b, that the power a may become equal to the number proposed; we have, therefore, b=Lc, and, generally, a”*=c.
637. If we consider another number d, whose logarithm is represented in a similar manner by L.d.; then a”=d; and multiplying this expression by the preceding one a*=c, we have a “**=cd. The exponent being always the logarithm of the power L. c + L.d =L.cd. If instead of multiplying, we divide the former expression by the latter, we obtain al-La-; ; hence L.c -L-d=L.'.
638. From this we are led to the two principal properties of logarithms which are contained in the equations Lc + L.d=Led, and L.c-L.d=L. #: by the former whereof we learn
that the logarithm of a product, as cd, is found by adding together the logarithms of the factors; by the last, that the logarithm of a fraction is determined by the subtraction of the logarithm of the denominator from that of the numerator. Whence it follows that to multiply or divide two numbers by one another, we have only to add or subtract their logarithms. This constitutes the immense advantage of logarithms in calculation; for when a question is incumbered with large quantities, it is, of course, much easier to add or subtract than to multiply and divide. In the involution of powers and the extraction of roots, logarithms are yet more useful. Thus, if d=c, we have by the first property L. c + Le=L-ce; consequently, L.cc=2L.c. Similarly, we have L.c" = 3L.c, L.c4 =4L.c, and, generally, L.c" = n L.c. Substituting fractional numbers for n, we shall have, for example, L.c', that is L've -#L.e. Lastly, if n represents negative numbers, we have L.c-" or L. #= –L.c; L.c-”
also from L.1 =0, as we have before shown. In tables of logarithms which are calculated for all numbers, great assistance is rendered in performing the most prolix calculations. Suppose, for instance, the square root of the number c is sought, having found the logarithm of c, which is L.c, we have only to divide it by 2, that is, take the of it, and we have the logarithm of the square root required; and the number in the table answering to that logarithm is the number required. We have seen that the numbers 1, 2, 3, 4, 5, 6, &c., that is, all positive numbers, are logarithms of the root a, and of its positive powers, and consequently logarithms of numbers greater than unity; and, on the other hand, that negative numbers, -1, -2, &c., are
logarithms of the fractions # . , &c., which are less than unity, but, nevertheless, greater
than nothing; from whence it follows, that if the logarithm be positive, the number is always greater than unity, but, if negative, the number, though less than one, is yet greater than 0. Thus we cannot express the logarithms of negative numbers, and must conclude that they are impossible, and belong to the class of imaginary quantities. That this may be better understood, let us fix on a determinate number for the root a, such, for instance, as the number 10, on which the common logarithmic tables are formed, and which is, moreover, the basis of our arithmetic. Any other number, however, provided it be greater than unity, would answer the same purpose. The reason why the a=1 would not suit is, that all the powers would be but equal to unity.
LoGARITHMIC TABLES NOW USED.
639. We set out with the supposition that the root a = 10. Then the logarithm of any number c is the exponent to which the number 10 must be raised, so that the power resulting from it may be equal to the number c.; or if we denote the logarithm of c by L.c, we shall always have 10L.c=c. I' reader will recollect that the logarithm of 1 is always 0, and we have 100 = 1. CIlce
I-k=-1, L.1, -2, L.T.: = -8 L.T.'"--4, L.T.'"--5, L. "--6, &c.
The logarithms of the principal numbers are therefore readily determined; but those between them, as inserted in the tables, are not so easy to find. Our object here, however, is only a general view of the subject, with which we shall proceed. And, first, since I-1=0 and L. 10=1, it is manifest that the logarithms of all numbers between 1 and 10 lie between o and 1, that is greater than 0 and less than 1. Let us, then, consider the number 2, whose logarithm is certainly greater than o, and yet less than unity. Now, if we represent this logarithm by the letter r, so that L.2=x, the value of x must be such as