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difference between + 3 and 12 is 20, or between + b and
cis b + c. Hence it follows, that to subtract a quantity which has the signa we must give it the opposite sign, that is, it must be added.
X. The learner, by this time, must have some idea of the use of letters, or general symbols, in algebraic reasoning. It has been already observed that, strictly speaking, we cannot actually perform the four fundamental operations on these quantities, as we do in arithmetic; yet in expressing these operations, it is frequently necessary to perform operations so analogous to them, that they may with propriety be called by the same -names. Most of these have already been explained ; bat in order to impress them more firmly on the mind of the learner, they will be briefly recapitulated, and some others explained which could not be introduced before,
Note. Algebraic quantities, which consist of only one term, are called simple quantities, as + 2 a, -- 3 ab, &c.; quantities which consist of two terms are called binomials, as a +-b, a- - b, 3b + 2 c, &c.; those which consist of three terms are called trinomials ; and in general those which consist of many terms are called polynomials.
The addition of simple quantities is performed by writing them after each other with the sign + between them. To express that a is added to b, we write a +b. To
a, b, c, d, and e are added together, we write a +b+c+dte. It is evidently unimportant which term is written first, for 3 + 5 + 8 is the same as 5 + 3 + 8, or as 8 + 5 + 3. So a+b + c has the same value as b + a + c.
It has been remarked (Art. I.) that x + x + x may be written 3 x. This is multiplication, and it arises, as was observed in Arithmetic, Art. III., from the successive addition of the same quantity. 3 x, it appears, signifies 3 times the quantity X, that is, x multiplied by 3. So b+b+b+b + b may be written 56. In the same manner, if x is to be repeated, any number of times, for instance as many times as there are units in a, we write a x, which signifies a times w, or x multiplied by a.
N. B. The learner should constantly bear in mind that the letters, a, b, c, &c. may be used to represent any known number; or they may be used indefinitely, and any number may afterwards be substituted in their place.
Again, ab + ab + ab may be written 3 ab, that is, 3 times the product ab; also c times the product a b may be written cab.
It may be remarked that a times b is the same as b times a; for a times 1 is a, and a times b must be b times as much, that is, b times a. Hence the product of a and b may be written either ab or ba. In the same manner it may be shown that the product cab is the same as a b c. Suppose a = 3,6=5, and c= 2, then abc= = 3 X 5 X 2, and cab= 2 X 3 X 5. In fact it has been shown, in Arith. Art. IV., that when a product is to consist of several factors, it is not important in what order those factors are multiplied together. The product of a, b, c, d, e, and f, is written abcdef. They may be written in any other order, as a cdbef, or f bedca, but it is generally more convenient to write them in the order they stand in the alphabet.
Let it be required to multiply 3 ab by 2 cd. The product is 6 abcd; for d times 3 ab is 3 ab d, but cd times 3 ab is c times as much, or 3 abcd, and 2 c d times 3 a b must be twice as much as the latter, that is, 6 abcd.
Hence, the product of any two or more simple quantities must consist of all the letters of each quantity, and the product of the coefficients of the quantities.
N. B. Though the product of literal quantities is expressed by writing them together without the sign of multiplication, the same cannot be done with figures, because their value depends upon the place in which they stand. 3 a b multiplied by 2cd, for instance, cannot be written 32 abcd. If it is required to express the multiplication of the figures as well as of the letters, they must be written 3 ab 2dc, or 3 x 2 abcd, or 3.2 a b cd. That is, the figures must either be separated by the letters or by the sign of multiplication.
Examples in Multiplication. 1. Multiply 3 ab by 4cdf. Ans. 12 abcdf. 2. 5bcd by
Ans. 5 abbccd. 3. 9 egh
8. 4. 13 ac
7 aacd. 5. 35 abc by
a b c.
42 ay y
12 x x y.
6. Multiply 138 by 5acd.
11 abr. 8.
by It frequently happens, as in some of the above examples, that a quantity is multiplied several times by itself, or enters several times as a factor into a product; as 3 a a abb, into which a enters three times and b twice as a factor. In cases like this the expression may be very much abridged by writing it thus, 3 a* b. That is, by placing a figure a little above the letter, and a little to the right of it, to show how many times that letter is a factor in the product. The figure 3 over the a shows, that a enters three times as a factor; and the 2 over the b, that b enters twice as a factor, and the expression is to be understood the same as 3 a a abb. The figure written over the letter in this manner is called the index or exponent of that letter. The exponent affects no letter except the one over which it is written.
Care must be taken not to confound exponents with coefficients. The quantities 3 a and a have very different values. Suppose a = -4, then 3 a 5
12; whereas 3 = 4 X 4 X 4 = 64. In the product 3a*b* suppose a = 4 and b = :5, then
3 ao 12 = 3 X 4 X 4 X 4 X 5 X 5 = 4800. The expression a' is called the second power of a, a' is called the third power, as the fourth power, &c. To preserve a uniformity, a, without an exponent, is considered the same as a', which is called the first power of a.
Figures as well as letters may have exponents. The first power of 3 is written
3 the second power
32 3 X 3 = 9 the third power
33 = 3 X 3 X 3 = 27 the fourth power
34 = 3 X 3 X 3 X 3 = 81 the fifth power
3 = 3 X 3 X 3 X 3 X 3 = 243. The multiplication of quantities in which some of the factors are above the first power, is performed in the same manner as in other cases, by writing the letters of both quantities together,
* In most treatises on algebra a2 is called the square of a, and a3 the cube of a. The terms square and cube were borrowed from geometry, but as they are not only inappropriate, but convey ideas very foreign to the present subject, it has been thought best to discard them entirely.
a? 12 c.
taking care to give them their proper exponents. 2 am X 3c" di is the same as 2 a mm x 3ccdd, which gives
6 ammccdd=6 a m* c* d'. a' multiplied by a' gives a' a' ; but a = a a a and a' =
=αα ; hence a d = a a a aa = a. In all cases the product consists of all the factors of the multiplicand and multiplier. In the last example a is three times a factor in the one quantity, and twice in the other ; hence it will be five times a factor in the product. The exponents show how many times a letter is a factor in any quantity; hence if any letter is contained as a factor one or more times in both multiplier and multiplicand, the exponents being added together will give the exponent of that letter in the product. a Xa=a' x a'=a'+ = a'. a' x a' = a + = a'.
a' x a = m+ = a', &c. 9. Multiply ao lo by a 6%.
Ans. a6. 10.
by abc. 11.
6 a3 c d by abc. 12.
a ca 13. 7 at ry
5 a' b cx" y, 14. 17 18 di e by
23 ao x8 by 2 a a b x x. 16, 18 a ayy by
ба” уух. It has already been remarked that the addition of two or more quantities is performed by writing the quantities after each other with the sign + between them. The sum of 3 ab, 2 acd, 5 a' b, 4 u b, and 3 db, is 3 ab + 2 acd + 5 a*b+ 4 ab + 3 a*b. But a reduction may be made in this expression, for 3 ab + 4 ab is the same as 7 ab; and 5 a*b + 3 a' b is the same as 8 a' b; hence the expression becomes
7 ab + 2 acd + 8 a' b. Reductions of this kind may always be made when two or more of the terms are similar. When two or more terms are composed of the same letters, the letters being severally of the same powers, they are said to be similar. The numerical co efficients are not regarded. The quantities 4 a b and 3 a b are similar, and so are 5 a' b and 3 a' b; but 4 ab and 5 a* b are not similar quantities, and cannot be united.
The subtraction of algebraic quantities is performed by writing those, which are to be subtracted, after those from which they are to be taken, with the sign – between them.
If b is to be subtracted from a it is written a
5 a 62 to be subtracted from 8 a b>, is written 8 a b? 5 a b?. This last expression may be reduced to 3 ab. In all cases when the quantities are similar, the subtraction may be performed immediately upon the coefficients.
XI. The addition and subtraction of simple quantities, pro duce quantities consisting of two or more terms which are called compound quantities. 2a +cd - 36 is a compound quantity.
Addition of Compound Quantities.
The addition of two or more compound quantities, when all the terms are affected with the sign + will evidently be the same, as if it were required to add together all the simple quantities of which they are composed ; that is, they must be written one after the other with the sign + before all the terms except the first. The sum of the quantities 3a + 2 c and 6 + 2 d. is 3 a + 2c+b+ 2 d.
If the quantities 3 ab + 5 d and b c be added, in which some of the terms have the sign the sum will be 3 ab +5d +b-; for b c is less than h, therefore, if b be added the sum will be too large by the quantity c. Hence c must be subtracted from the result.
This may be illustrated by figures. Add together 17 + 10 and 20 6. Now 20- - 6 is 14 and 17 + 10 + 20 - 6 is equal to 17 + 10 + 14.
From the above observations we derive the following rule for the addition of compound quantities.
Write the quantities after each other without changing their signs, observing that terms which have no sign before them are understood to have the sign +.
A sign affects no term except the one immediately before which it is placed ; hence it is unimportant in what order the terms are written, for 14. - 5 + 2 has the same value as 14 + 2-5 or as 5+2 + 14. Those which have the sign + are to be added together, and those which have the sign to be subtracted from their sum. If the first term has the sign