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THE

MODERN PRECEPTOR.

CHAPTER IV.

OF ALGEBRA.

ALGEBRA is an Arabic term of uncertain etymology, but generally supposed to signify the arts of restitution, comparison, resolution, and equation; meanings sufficiently denoting the nature of the art. By algebra we discover a general form of expressing the results of all questions eomprehending similar circumstances, relating to magnitude, quantity, or number; or, in other words, by algebra we perform the several operations of addition, subtraction, multiplication, and division, employing certain characters or symbols of no real intrinsic value in themselves, but qualified to represent magnitudes, quantities, and numbers of every description. For example, let us suppose any number, as 3, to be represented by the symbol or character a; 5 to be represented by b; and their sum 8 by the symbol c; then, in algebraic language, a and b added together will be equal to c, or thus, a+b=c, that is, in this example 3+5=8. But the values of the arithmetical symbols 3, 5, and 8, having by long and unvaried usage become determinate, they are not susceptable of any change; whereas the values attributable to the symbols a, b, and c, may be varied indifinitely, and operations by them still give correct results; thus a may represent 12, b 15, and c 27, then a+b=c, for 12+ 15=27.

Although

Although any letter of the alphabet may be employed to represent quantities in algebraic operations, yet it has been found convenient to use the first letters as a, b, c, d, e, &c. for quantities whose values are known or given, and the last letters, as v, x, y, z, for quantities neither given nor known; hence, as in the former example, where the values of a and b are given, and the value of their sum is required, we would say a+b=x.

Algebraic quantities are connected by means of certain signs, as(+) Plus, denoting that the quantities before and after the sign are to be added together, as 3+5 equal to 8.

The sign(-) Minus, denotes that one of the quantities is to be subtracted from the other, as 8-5 equal to 3.

The sign(x) denotes that the quantities between which it stands are to be multiplied together, as 3x5 equal to 15; or axbz. The product is also expressed by writing the symbols close together, as the letters in a word; thus axbab, and axbxc=abc.

Division is expressed by writing the dividend above a

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small line, and the divisor below it, as will signify that b is to be divided by a.

When quantities to be multiplied or divided are compound, a line, called a vinculum, is drawn over them thus; axb-c, signifying that a is to be multiplied into the difference between b and c, and axb+m+c-d+e, will signify that a is to be multiplied by the difference between the sum of b, m, and c, and the sum of d and e; but, instead of this vinculum, or tye, over the compound quantities, many algebraists inclose these quantities as within a parenthesis; thus ax (+m+c)−(d+e).

When an arithmetical figure stands before an algebraic symbol, it is called the numeral coefficient, and shows how often the algebraic quantity is to be repeated; thus 3 a will signify three times the value of a.

Equality

Equality is represented by the sign (=), as a+b=c, or the sum of a and b is equal to c.

Quantities are said to be like when they consist of the same characters; thus, 3 am and 5 am are like quantities, but 3 am and 5 am m would be unlike quantities.

Quantities having the same signs, whether or, are said to have like signs; but one having +, and another having-, have unlike signs.

Quantities having the sign+before them, are termed positive quantities; and those having before them, are negative quantities. It is true, that in the nature of things there can be no such thing as a negative quantity, that is, a quantity less than nothing; but the term is used in algebra to express such quantities whose value must be deducted from that of others with which they are connected; for example, the amount of a'person's estate may be considered as a positive quantity, and that of his debts as a negative quantity, which being deducted from the former quantity, will show how much the person's real property is.

When no sign is prefixed to an algebraic quantity, it is always considered to be + plus, or that the quantity is positive

Addition of algebraic quantities is performed in three different ways, according to the nature of the quantities.

1st. If the quantities be like, and have like signs, the rule is, to add together the co-efficients, (reckoning every character without a co-efficient for one,) annexing the common letter or letters, and prefixing the common sign.

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2d. When the quantities are like, but the signs unlike, add all the positive quantities together, and all the negative quantities together, and subtracting the one sum from the other, the remainder will be the total sum required, having the sign belonging to the greatest sum.

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To understand these two last examples, let us suppose all the positive quantities to represent the several articles of a person's effects, and the negative quantities to represent his debts; it will then be evident, that, to know the real value of his property, we must subtract the debts from the effects, and the remainder will correspond to the value of the whole positive and negative articles taken together. Hence, in the first example of this case, we have +4 mn and + 9 m = 13 m, for the effects, and 3 m and 5 m = 8m for the debts; conscquently this sum being taken away from 13 m, will leave 5 m for the value of the property remaining.

3d. When the quantities to be added are all unlike, they are to be written down in succession, with their respective signs and co-efficients, in one line, as in the fol lowing examples.

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Subtraction of algebraic quantities is performed by changing the sign of the quantity to be subtracted, and then adding the two quantities together, agreeably to the rules of addition.

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In the first example, where 3am is to be taken from 8a+ 3m, if we change the sign of 3 a, which is into -, and then add these two quantities by the second rule of addition, the result will be + 5 a; and in subtracting - m from +3 m, we change the into +; and then adding their two quantities together, the result is + 4 m. If the question had been proposed, to take away only 3 d from 8a+ 3 m, the remainder would evidently have been 5a + 3 m; but as the sum to be subtracted is less than 3 a by once the value of m, the remainder must be greater than it would have been, on the first supposition, by an additional value of m; that is, it must be 5 a + 4 m, as above shown.

Multiplication of algebraic quantities is performed according to the following rules.

When the quantities to be multiplied have like signs, the sign of the product will be +; and when they have unlike signs, it will be -.

When the quantities given are simple, find the sign of the product by the above rule; to which annex the product of the co-efficients, if any, and then all the letters, which will give the product required; thus, in the first example following, where 8 am is to be multiplied by 3x, we mul

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