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ELEMENTS

OF

ALGEBRA.

Κ

ALGEBRA.

ALGEBRA treats of magnitude in general. It is a

kind of universal Arithmetic. The characters therein employed are the letters of the alphabet, which, having no determined value, are thereby very proper to represent all kinds of magnitude or quantities. We perform upon algebraical quantities, the same operations as on numerical ones; we add, subtract, multiply, and divide them, but as these operations can only be indicated, in quantities purely algebraical, we have had recourse to certain signs, the use of which is an invention which particularly characterizes Algebra.

The sign of Addition is +, and is pronounced more. The sign of Subtraction is and is pronounced

·less.

The sign of Multiplication is X, and signifies multiplied by; we often put only a dot between the factors of the multiplication, and we sometimes also write the letters beside each other, without separating them by any sign; so that a × b, or ab, or ab, express the same thing.

The sign of Division, is, as in arithmetic, a horizontal bar between the dividend and divisor; we some

times also put two dots between the two quantities ;

a

so that -, or ab express the same thing.

b

We call a monomial, every quantity, the constituent parts of which are not separated by the sign +, or the sign; and we call a polynomial every quantity composed of several monomials, separated by the sign + or the sign A binomial is a polynomial, composed of two monomials, or of two terms;* a trinomial of three, &c.

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Algebra, in its generality, comprehends indifferently positive and negative quantities. It is easy to conceive what is meant by a positive and a negative quantity.

Let us consider the condition of a man who has property and debts; it is evident, that if I count his property as a positive quantity, I ought on the contrary to look upon his debts as negative quantities, since the latter tend to diminish the former.

P

If then I call a his property, and b his debt, his situation will be expressed by a-b. Let us also consider a body P in motion on the line M N, and suppose we wish to compute its distance, with relation to the

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MA CB D

point A; if the body move towards N, and be arrived at the point D, its distance from the point A will be

* We also give the naine of term, to a quantity, fuch a+b

as, though it is not a monomial: In general, every a+b-c+&c.

quantity, fuch as

m+n+ &c.

joined by the fign of division,

is called a term in the fame manner as this, (a+b+&c.) X'd.

expressed by AB+BD; but if, on the contrary, the body be in motion towards M, and be arrived at the point C, its distance from the point A will then be AB-CB; we see, therefore, that if in the first supposition the course BD performed by the body P, be taken as a positive quantity, in the second supposition the course BC, performed by the same body, must be taken as a negative one.

This distinction being once established, it was essential to know what sign ought to effect the result of the combinations, which we could make of positive quantities with negative ones; this is what we are going to explain, by giving the rules of Addition, Subtraction, Multiplication and Division.

ADDITION.

Addition of algebraical quantities, consists in writing the quantities one after another, and in preserving their signs such as they are. Thus to add +6 and +a, we must write a+b, (every quantity which is not preceded by any sign, is always supposed to have the sign+); and to add - b & a, we must write a-b. The reason of this rule, results from what we have said, that preceded by the sign, being considered as a debt, it is evident, that to add a debt to an estate, is to diminish the estate, so that a being the estate and b the debt, to add them together we must write a—b. If we had to add a and a we would then write a+a, but we abridge this expression by putting 2a; and in like manner, if we had 2a+5a, we would write 7a. The figure written before a letter or a quantity, is called the coefficient of that letter or of that quantity.

We shall remark, that the unit is always supposed to be the coefficient of every quantity which has none, for a or 1 Xa, is the same thing.

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