« ΠροηγούμενηΣυνέχεια »
Connect the whole to be added by the sign + ; and, if two or more of these quantities are to be found, of the same kind, they must be united into one simple quantity, or term, as above.
Example 1.-Add a, b, c, d, together; then a +b+c+d=S.
Join all the quantities into one expression for the sum; observing to prefix the same sign to each quantity that it had before the whole were united.
Example.-Add a, – bx, cd, - 2bx, together; then, a + cd – 3bx = S. 100. To add Compound Quantities together :
Connect all the quantities, in the several parts, to be added into one expression, giving each quantity the same sign that it had before, in each separate part, and observing to unite such terms as may be found of the same kind.
Example.-Add 5 bx + 4, 5 ab 5, 2bx – 3 ab +3ex, together. The answer will be 7bx + 2 ab +
+3ex = S.
SUBTRACTION OF ALGEBRA.
101. To subtract one simple quantity from another:
Join the quantity to be subtracted to that from which the subtraction is to be made, with a different sign to the original one. Let the difference be put equal to D.
Example 1.-Subtract n from m; then m- n= D.
102. To subtract a compound quantity either from a simple or compound quantity.
Subjoin the terms of the quantity to be taken away, with their signs changed, to that from which they are to be taken ; observing that, when two terms are of the same kind, they must be united.
Ex.–From xy +46-3c, subtract bx—56+4c; then xy+96–7c-bx = D.
It is evident that changing the signs of the terms of the quantity to be taken away cannot affect its aggregate or value, considered independently of
its signs. For, if they are of different kinds, the difference must be the same after the change as before it took place. Thus, let 5 – 2+7–3, be a quantity to be subtracted.
Then 5-2 +7-3=7
and – 5+2-7+3= -7 Hence we see the reason for changing the signs of the quantity to be subtracted.
MULTIPLICATION OF ALGEBRA.
103. To find the algebraic product of two compound factors, or of one simple and the other compound.
If one of the factors be a simple quantity, let that factor be made the multiplier; then join the multiplier to every term of the multiplicand, and prefix the sign + to each product when its factors have like signs; but prefix the sign – when its factors have unlike signs; then the sum of all the products is the total product.
If the multiplier consist of more than one term, proceed with every term of the multiplier in the same manner as if it had but one term; then the sum of all the simple products is the whole product ; observing that, all such simple products, as have a common factor, may be united together.
Example 1.-Multiply a + b by a + b.
Operation ... a + b
a + b
a® + ab
+ ab + 72
a + 2ab + b2 = (a + b)2
104. So that the square of any binomial consists of the square of each part and twice their product.
Hence, if we see such a quantity as a2 + 2ax + a', we shall immediately know that it is the square of the binomial x + a.
Example 2.-Multiply a-b by a-b, or find the square of a-b.
a’ - ab
a’ - 2ab+b+=(a - b)2
105. Hence the square of any binomial, which has one of its parts negative, is the same which ever of the parts be negative; for the square of each of the parts is always affirmative, and twice the product of the two parts is negative; so that the square of a-b is the very same as the square of b-a.
106. Hence we shall always know, by bare inspection only, that the difference of two squares is the product of the sum and difference of the roots : hence x? — y'=(x+y)(x - y), and, reciprocally, that the product of the sum and difference of any two quantities is equal to the difference of their squares.
Thus (a+x) (a —x)=a? — ?.
ALGEBRAIC DIVISION AND FRACTIONS.
107. Division is the converse of Multiplication ; therefore, if the signs be alike in the divisor and dividend, the quotient will be affirmative; but if unlike, the quotient will be negative. The general rule is to place the dividend in the form of a numerator, and the divisor in that of a denominator; expunge like quantities from both, and divide the co-efficients by the greatest common
108. Powers of the same root are divided by subtracting their exponents.
Example 1.--Divide by by b?. ..
109. A Fraction is multiplied by any quantity by joining it to the numerator; thus, the product of r and is represented by " , which is the product of rm divided by n, or the number of times that the product rm contains n.
Since the operation of division is opposite to that of multiplication, if a fraction be multiplied by a quantity equal to its denominator, both the denominator and the multiplier may be taken away from the result ; thus, if, be multiplied by b, the result is ab, which is equivalent to the quantity a only.
110. The Terms of a fraction are its numerator and denominator.
If the terms of a fraction be equally multiplied, that is, multiplied by the same quantity, the value of that fraction will be the same as before; thus
: and, if the terms of a fraction be equally divided by the same quantity, the result will be equal to the original quantity.
111. The resolution of an equation is the mode of finding the value of the unknown quantity, in terms of those which are given.
The principles of this resolution depend on the following Axioms, which are similar to those already given at the beginning of the Geometry, in page 15.
1. If to each side of an equation the same quantity be added, the sums will be an equation still.
2. If from each side of an equation the same quantity be subtracted, the remainders will still be an equation.
3. If each side of an equation be multiplied by the same quantity, the products will be an equation.
4. If each side of an equation be divided by the same quantity, the quotients will still be an equation.
112. DEFINITION.-Four quantities are proportionals when the first contains some part of the second, as often as the third contains the like part of the fourth.
113. If four quantities, a, b, c, d, are proportionals, the product of the two extremes will be equal to the product of the two means.
Let the first, a, contain the nth part of the second b, m times ; then, by the definition, the third, c, will contain the nth part of d also m times; now the nth part of b is, and the nth part of d is ; therefore =m; and a = m; wherefore = a.
Multiply each side of this equation by bd, and =; therefore, dividing the terms of the fraction on the first side by b, and the terms of the fraction on the second side by d, which are common, ad=bc.
114. If any equation consist of the product of two quantities on each side, and if the four factors be placed in a row, so that the two factors on either side may occupy the middle place of the four, and the other two each one of the extreme places; the four factors, thus taken in order, will be proportionals.
Let ad=bc: Now here are four different ways of taking out the quantities; but in which ever of these ways they are taken, we shall always have ad=bc, by multiplying the two extreme terms together, and the two middle terms together.
Let, therefore, a, b, c, d, be one of the four; then, since ad=bc, divide both sides by bd and ad = b; that is, g = a; now let h be the common measure of a