4. Divide a+b=1 by c+d√

The multiplier for rationalising the denominator is evidently c—d√ — 1; hence

a+b√√1 (a+b√—1) (c—d√

=

c+d√1

(c+d-1) (c-d√—1)

acbd-(ad-be)√1

c2 + d2

EXERCISES.

1. Divide 8-18 by 2√2

2. Divide 156 by 3√3

3. Divide 4-2-1 by 2-3√1
4. Divide 3-5√3 by 2—3 √.

5. Divide ab by c√ — d

2

OF EQUATIONS.

(222.) 'An equation is an expression stating the equality of two quantities, and generally containing at least one unknown quantity."

Thus, x -3=4 is an equation which states the equality between a -3 and 4; in which is the unknown quantity. By the equation, it appears that if 3 be subtracted from æ, the remainder is 4; and hence ≈ must be 7. The equation 5 cannot be said to contain an unknown quantity, as the value of x is evident; and so of the equations = 5—3, x = a, or x=b+c; in which a, b, and c, are known quantities.

=

(223.) A quantity is known when its value in numbers is given; and when this value is not given, it is called an unknown quantity."

(224.) The last letters of the alphabet x, y, z, are used to represent unknown quantities, and the known quantities are either numbers, or they are denoted by the first letters of the alphabet, a, b, c,

...

(225.) The two parts of an equation on the opposite sides of the sign of equality, are called members or sides of the equation; that on the left is called the first member or side, and the other the second member or side. Members are composed of one or more terms.

(226.) The method of solution of an equation is the process for finding the value of the unknown quantity; and an equation is said to be solved when the unknown quantity stands alone on one side, and its value in known terms on the other side. #

(227.) An equation consisting of literal quantities, when their numerical values are known, is called simply an equality."

Thus, if the numerical values of a, b, c, and d, be known, and be such that

a+b=c―d,

the expression is an equality. If, for example, a=2, b=3, c=8, d= 3, then the expression becomes

2+3=8-3 or 5=5

(228.) A self-evident equality is called an identity or a verified equality.

As 12. 8 4 or 3 a=

26 -a2a-5b+ 3b (229.) If in an equation containing only one unknown quantity, a number be substituted for the unknown quantity, such that the equation is reduced to an identity, this number is called a value of the unknown quantity, or a root of the equation."

Thus, in the equation

X 5=16-3+2,

if 20 be substituted for a, it becomes

2015-16−3+2

This result is an identity; and hence 20 is the value of x (230.) An equation expresses some relation between the unknown and the known quantities. The value of the unknown quantity is said to satisfy or fulfil the equation." In the equation

X= 8-5

the relation between a and known quantities is just that x shall be equal to the difference between 8 and 5, that is, 3; and this value of a being put in place of X, would reduce the equation to the identity 3=8—5, or it would satisfy or fulfil the given equation.

(231.) To verify a value of the unknown quantity is to substitute this value for it in the given equation; if the result be an identity, the value is then verified, or proved to be correct.

(232.) If only one quantity in any proposed question is unknown, only one relation or condition must be given." For example, if a number is required, such that being increased by 3, the sum shall be 12, then if x be that number, the equation

x+3=12

expresses this relation or condition, and a is evidently 9. for 9+3=12. But it is unnecessary to give another relation such as that the number required being increased by 6, the sum is 15, as nothing would be deduced from this further than that the number required is just 9. If, however, a second condition be added, so as to give a new value

to the unknown quantity different from that given by the first equation, the question becomes impossible, as the conditions would be inconsistent were x to be = 9, and, at the same time, equal to some other number, while by the question only one number is sought. If, therefore, when only one unknown number is sought, a second condition be added, though it be consistent with the first, it is unneces sary, the first being sufficient for determining the required number; and if the second condition be inconsistent with the first, the question becomes impossible.

(233.) Equations are divided into numerical and literal; the former contain only numbers combined with the unknown quantity or quantities, and the latter contain also literal quantities, whose numerical values are known, or are supposed to be so.

Equations are also divided into determinate and indeter minate; the former term being applied when there are as many equations as unknown quantities, and the latter when there are more unknown quantities than equations.

(234.) Any power of a quantity, or any product of two or more quantities, or of their powers, is called a dimension. The order of the dimension depends on the number of times that the simple power of the quantity or quantities enters into the dimension, or on the sum of the exponents in the product. When the simple power is repeated twice, it is called the second dimension; when three times, the third dimension, and so on; and, generally, if it be repeated n times, that is, if the sum of the exponents be equal to î,

it is called the nth dimension."

Thus, if x, y, z, be unknown quantities, a3, y, xyz, are each of the third dimension; a, ay, ayz, xz, are of the fifth dimension, and so on.

(235.) Equations containing only one unknown quantity are divided into different classes, according to the highest power of the unknown quantity contained in them. An equation which contains only the first power of the unknown quantity, is called a simple equation; one in which the highest power is the second, is called a quadratic; when the highest power is the third, the equation is called a cubic, or an equation of the third degree; when the highest

power is the fourth, an equation of the fourth degree; when the fifth, of the fifth degree; and, in general, when the highest power of the unknown quantity has the exponent n, it is called an equation of the nth degree, and sometimes it is called an equation of n dimensions.

(236.) Equations containing two or more unknown quantities are similarly classified. If the highest dimension of the unknown quantities be the first, second, or third, it is called a simple, quadratic, or cubic equation; if it contains the fourth dimension, it is called an equation of the fourth degree or dimension; and similarly for other dimensions. (237.) The dimension of an equation is the same as that of any term in it that contains the highest dimension of the unknown quantity or quantities in the equation (40). It is understood, however, that the equation contains no fractional or negative exponents of the unknown quantity, or that it is freed of radicals and of denominators containing unknown quantities."

OF SIMPLE EQUATIONS.

I. EQUATIONS CONTAINING ONLY ONE UNKNOWN QUANTITY. Equations must undergo some alterations in their form, in order to prepare them for solution. These preparatory transformations depend on the following axioms :—

(238.) If equal quantities be added to the two members of an equation, or subtracted from them, or if the two members be multiplied by the same quantity, or divided by the same quantity, the members will still be equal, or the equation will still subsist."

From these axioms, the following rules for the process of solution may be derived :

(239.) I. Any term may be transposed from one side of an equation to the other, by changing its sign."

EXAMPLES.

Transpose the known quantities to the second member, and the unknown to the first, in the following equations:

L