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In these four examples, as well as in Ex. 22, 23, 2 25, we have used the sign =, to signify that the ser on one side of it, is merely the result of the operati which, on the other side, is indicated and not performe we shall examine in the next chapter, under what c cumstances an arithmetical equality can be considered existing between them.

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If we should make k = (a b'a'b), the process of division, as well as the form of the resulting series, would be considerably simplified.

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(35) Divide a +B+yx2 by a + bx.

a + bx) a + ẞx + y2

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+ya2, where k=aß-ab

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x − a) x3 — px2 + q x − r (x2+(a−p)x+a2 —pa+q+~+ &c.

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The third remainder a3-pa2 + qa-r or k, is a quantity identical with the dividend, putting a in the place of x; and it is obvious that the quotient will not be finite, unless k=0.

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In this case, the division does not terminate unless the fourth remainder, or a-pa3 + qa2-ras, be equal

to zero.

CHAP. III.

OBSERVATIONS UPON THE FIRST PRINCIPLES AND FUNDAMENTAL

OPERATIONS OF ALGEBRA.

operations

Algebra

metic.

47. THE operations of Algebra have been named The from those operations in Arithmetic, with which they are and terms analogous, or partly identical, and most of the terms used in made use of in this science have had a similar origin: derived the consequence has been, that the peculiar and in some from Arithrespects limited meaning which those operations and terms possessed in Arithmetic, has been attached to them when used in a much more general signification: and hence also has arisen the custom of considering Algebra as merely such a generalization of the processes of Arithmetic as was derived from the use of symbolical language.

of the

and

48. In order to ascertain to what extent, or in what The extent and nature sense, Arithmetic may be considered as the basis of the science of Algebra, it will be expedient to examine the symbols nature of the symbols employed in Arithmetic, the extent operations of their representation and the meaning and limits of the of Arithoperations to which they are subject.

metic.

49. The symbols of Arithmetic are the nine digits Symbols. and zero, and no others.

The quantities which they represent are numbers, Their reprewhether abstract or concrete. As far, however, as the sentation. operations of Arithmetic are concerned, we may consider them as abstract only, inasmuch as the relative magnitude of quantities denoted by numbers is alone considered, without any reference to the affections or specific properties of the quantities themselves.

Funda

mental operations.

Addition.

Subtrac tion.

Inverse operations.

Multiplication.

Division.

Inverse

The position of the symbols determines their numerical value: there is nothing arbitrary, either in the symbols themselves, or in the notation by means of them.

50. There are four fundamental operations in Arithmetic; Addition, Subtraction, Multiplication, and Division.

51.

Addition is the union of two or more numbers into one sum.

Subtraction is equivalent to the determination of a number (the remainder) which added to another (the subtrahend) will produce a given number for their sum: the operation is of course limited to those values of the subtrahend, which are less than the number from which they are to be subtracted.

Addition and Subtraction are inverse operations in the following sense: if to one number another be first added and then subtracted, or conversely, its value will remain unaltered.

52. Multiplication is equivalent to the perpetual addition of one number, which is called the multiplicand, to itself, as often as unity is contained in another number, which is called the multiplier. The multiplicand and the multiplier are likewise convertible quantities; that is, it is indifferent whether we make the multiplier the multiplicand, and the multiplicand the multiplier, or conversely, the product in both cases being the same.

Division may be considered as equivalent to finding a number (the quotient) which multiplied into another (the divisor) will produce a given number (the dividend).

Division is the inverse of multiplication in the followoperations. ing sense: if a number be first multiplied and then divided by the same number, or conversely, its value is not altered.

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