Example 3.

Let the numbers be 10000 and 31416, or the ratio 10000

10000) 31416 (3

30000

1416) 10000 (7

9912

88 ) 1416 (16

31416

CHAPTRE II.

ALGEBRA.

SECTION I.-Definitions and Notation.

Algebra is the science of the computation of magnitudes in general, as arithmetic is the particular science of the computation of numbers.

Every figure or arithmetical character has a determinate and individual value; the figure 5, for example, represents always one and the same number, namely, the collection of 5 units, of an order depending upon the position and use of the figure itself. Algebraical characters, on the contrary, must be, in general, independent of all particular signification, and proper to represent all sorts of numbers or quantities, according to the nature of the questions to which we apply them. They should, moreover, be simple and easy to trace, so as to fatigue neither the attention nor the memory. These advantages are obtained by employing the letters of the alphabet, a, b, c, &c. to represent any kinds of magnitudes which becomes the subjects of mathematical research. The consequence is that when we have resolved by a single algebraical computation all the problems of the same kind proposed in the utmost generality of which they are susceptible; the application of the investigation to all particular cases requires no more than arithmetical operations.

It is usual, though by no means absolutely necessary, to represent quantities that are known by the commencing letters of the alphabet, as a, b, c, d, &c. and those that are unknown by the concluding letters, w, x, y, z. But it is often convenient, especially as it assists the memory, to represent any quantity, whether known or unknown, which enters an investigation, by its initial letter, as sum by s, product by p, density by d, velocity by v, time by t, and so of others.

Now, if s denotes the sum of four numbers represented by a, b, c, and d, then, adopting the other symbols explained at the beginning of arithmetic, we should express this algebraically by writing s=a+b+c+d.

If the four quantities be all equal, or s = a + a + a + a, this evidently reduces to s = 4 x a, or simply s = 4 a, dropping the sign of multiplication, which is here understood. The figure 4 is named the coefficient. In the quantities 3 a, 5 a, 7 a, n a; 3, 5, 7, and n, are respectively the coefficients.

The continual product of three or more quantities is expressed either by interposing the sign of multiplication, as a xbx c x d; or by interposing dots, which have the same signification, as a .b c.d; or, lastly, by placing the letters in juxtaposition,

as a b c d.

When the quantities are equal, their continued multiplication produces powers, as a a, a a a, a a a a, &c. which are usually represented, instead of repeating the letters, by placing a figure a little above the single letter, to expound or tell how many equal factors are multiplied together; this figure is called the exponent. Thus, instead of a a, a a a, a a a a, we put a2, a3, a', the figures 2, 3, 4, being the exponents.

Since roots are the reverse of powers, they are expressed by exponents, which are the reciprocals of those that express the corresponding powers. Thus the square root of a is represented either by ✔a, or by a; the cube root of a + b, either by ✔a + b, or by (a + b); the fourth root of a + b — c, either by (a + b—c), or by (a + b — c)3.

We give the name term to any quantity separated from another by the sign + or - A monomial has one term; a binomial has two terms, as a + b, a c — 4 a b; when the second term of a binomial is it is frequently called a residual. A trinomial has three terms, as a + b + c, a d 4 ab +5 b c. A quadrinomial has four, as a+b+c―d. A multinomial, or polynomial, has many terms.

The signs and, which in arithmetic simply indicate the operations of addition and subtraction, are employed more extensively in algebra, to denote, besides addition and subtraction, any two operations or any two states which are as opposed in their nature as addition and subtraction are. And if, in an algebraical process, the sign is prefixed to a quantity to mark that it exists in a certain state, position, direction, &c., then, whenever the sign occurs in connexion with such quantity, it must indicate precisely the contrary state, position, &c. and no intermediate one. This is a matter of pure convention, and not of metaphysical reasoning.

Other characters might have been contrived to denote this opposition; but they would be superfluous, because the cha

racters and, though originally restricted to denote addition and subtraction, may safely be extended to other

And two such

And so on in every kind of contrariety. quantities connected together in any case destroy each other's effect, or are equal to nothing, as + a a = 0. Thus, if a man has but 107. and at the same time owes 107. he is worth nothing. And, if a vessel which would, otherwise, sail six miles an hour, be carried back six miles an hour by a current, it makes no advance.

Like signs are either all positive (+), or all negative (—). And unlike are when some are positive and others negative. If there be no sign before a quantity, the sign + is understood..

An equation is when two sets of quantities which make an equal aggregate are placed with the sign of equality ( = ) between them :

As 12+5= 20

3, or x + y

a+b = c d.

The quantities placed on both sides the sign of equality are called respectively the members of the equation.

SECTION II.-Addition and Subtraction.

1. Properly speaking, there is not in algebra either addition or subtraction, but a reduction, namely, the algebraic operation, by which several terms are, when it is possible, combined into one term. This, however, can only be effected upon quantities that differ in their coefficients and their signs, while they are formed of the same letters and the same exponents.

Thus, 3 a,+4a, + 7 a,

}

5a3b,-3 ab, + 8 ab, are evidently reducible.