ALGEBRA.

CHAPTER I.

DEFINITIONS, NOTATION, AND ALGEBRAIC OPERATIONS.

DEFINITIONS AND NOTATION.

(1.) ALGEBRA is that branch of mathematics, in which the letters of the alphabet, and other symbols are used, for the purpose of abridging and generalizing the reasoning required in the resolution of questions relating to numbers.

In the solution of algebraic problems, there are always certain quantities whose values are known, from which known quantities, it is proposed to determine others which are unknown; and it is usual to express known quantities by the first letters of the alphabet, as a, b, c, d, &c., and unknown quantities by the last, as x, y, z, t, u, &c.

The signs which denote algebraic operations and conditions, are the same as in arithmetic, with the addition of the following:

The sign of inequality, >, which is used to denote that one quantity is greater than another. The opening of the sign is always put towards the greater quantity, thus a>b signifies that a is greater than b, and a <b shows that a is less than b.

When quantities multiplied together, are represented by letters, the sign of multiplication is omitted. Thus, axbxc is written a, b, c: but when numbers are used, the sign is retained, for 5×6 is a very different thing from 56, on account of the local value of the figures.

For division, the sign

α

b'

or writing the numbers in a frac

tional form with the divisor beneath, is most commonly used.

The coefficient is a number written before a letter, to denote how many times this letter is taken in the expression, or how many times+ONE, the quantity it expresses is added to itself. Instead of writing a+a+a+a, we write 4 a, to denote that a is taken 4 times; or which is the same thing added to itself 3 times: ab + ab + ab, is the same as 3ab; abc +abc is the same as 2abc, and va + va is the same as 2vā. When no coefficient is written, 1 is always understood.

The exponent is a number written to the right of a letter, and a little above, to denote the number of times it is taken as a factor; or how many times + ONE it is multiplied by itself. Instead of writing a×a×a×a×a, or aaaaa, we write a3, which is read a, fifth power, and signifies that a is a factor 5 times in the quantity; or that it is multiplied by itself 4 times. When no exponent is written, 1 is always understood.

(2.) Every quantity written with algebraic signs, is called an algebraic quantity. Thus, 3a, 3a-b, 4b2c2 — a3, and 5b2c+a are algebraic quantities.

When an algebraic quantity is not connected with any other by the sign of addition or subtraction, it is called a monomial: 3a, 262, and 3c are monomials.

When it consists of two terms, it is a binomial, as 3a-b, 4a3 + b2.

When it consists of three or more terms, it is called a polynomial.

The numerical value of an algebraic expression, is the number which would be obtained by giving to each letter that enters into it, its particular numerical value. Thus, the numerical value of the expression 3a+25, if we suppose a = = 4, and b=7, is 3x4+2x7, or 12+14=26. On the same supposition, 4a2b2+2ab is equal to

4x4272+2x4x7-64-495671.

The numerical value of a polynomial is not altered by changing the places of its terms. Thus,

4a2+b-3a2c-b2+17 abc, is the same as

17 abc-3a2c+b+4a2-b2.

In any polynomial, the terms which have the sign + before them are called additive terms, and those which have the sign, subtractive terms; when no sign is written, + is always understood. Additive terms are sometimes called positive, and subtractive terms, negative quantities,

The number of literal factors which enter into any term, is the degree of that term; ab is of the second degree, a2b of the third, ab2c2 of the fifth. In general, the degree of an algebraic term is found by taking the sum of the exponents of the letters which enter into that term. An algebraic quantity which has all its terms of the same degree, is said to be homogeneous: 4a5+2ab3-3abc3+b'c, is therefore homogeneous, and of the 5th degree.

Similar terms are those which contain the same letters in the same powers, 2a2b and 5a2b are similar. But, 3ab2 and 3ab are not similar terms, for the letters, although the same, are not in the same power.

If an algebraic quantity contain similar terms, it may be simplified or reduced. Thus, the quantity Za3b2-4a3b2 is evidently equal to 3a3b2; 2ab+6ab is equal to 8ab; and 16a2b8b3+8b3 +8a2b-21a2b+3c2 is equal to 3a2b+3c2; since 16a2b + 8a3b = 24a2b, and 24a2b -21a2b3a2b.

A parenthesis (), indicates that whatever arithmetical operations are performed upon one of the letters contained within it, are to be performed upon the whole: thus, (a - b)×c shows that both a and b are to be multiplied by c; a-(b+c) shows that b and c are both to be taken from a; (a + b + c) (d+e) signifies that a+b+c is to be multiplied by d+e;

and (a+b)2, (abc)3, and ( —o)

abc, and

C

a

b α

4

indicates that a + b,

are to be raised to their 2d, 3d and 4th powers, respectively. A vinculum or bar drawn over any number of letters, indicates the same as a parenthesis; thus, va +x indicates the square root of the sum of a and x, the X, square root of the difference of a and x.

Va

(3.) The following practical rules and exercises in algebraic notation will be of great use to the pupil before proceeding farther.

To write a quantity algebraically, express by means of algebraic symbols all the arithmetical operations and con ditions belonging to that quantity.

EXAMPLES.

1. From the difference of the products of three times a multiplied by b, and 4 times a multiplied by the third power of c, subtract the quotient of b divided by c.

b

3ab-4ac3 is the algebraic expression.

C

2. Write the following in algebraic language:-4 times the square root of the difference of a and c, added to the sum of twice a of the second power and b of the third power, is equal to the cube root of the fifth power of a, added to the third power of c.

Ans. 4 va―c+2a2 + b3= }√a3 + c3.

3. Express in algebraic language, the quotient of 8 times the product of a and a subtracted from 9 times a of the third power multiplied by b of the second power, divided by the dif

ference between a2 and a.

4. Write the following:

6 a diminished by x, the difference increased by the square root of 2 a1, diminished by the cube root of 16, is equal to twice the product of 3 a3 by b3, diminished by the quotient of 21, divided by the square root of 6.

Ans. 6a-x+ √2a1

3

21

5. Write the difference between the square root of the sum of the squares of a and b, and the square root of the difference of the squares of a and b.

Ans. Va2+b2. Va2 — b2.

To find the value of an algebraic quantity.

(4.) Substitute the numbers which the letters represent, and perform upon them all the arithmetical operations indicated.

EXAMPLES.

1. Let a=6, b=5, c=4, and d=1, then will a3-2ab+c-3d-63--2×6×5+4-3×1

=216—60+4—3—157, the numerical value.

To reduce the similar terms of an algebraic expression to a single term.

(5.) Add together the coefficients of all the additive quantities, and also the coefficients of the subtractive quantities, and take the difference between the sums: affect this difference with the sign which belonged to the greatest sum: this difference, written before the letters common to all the similar terms, will be the single term required.

EXAMPLES.

1. Reduce 12a3b2 · 3ab3 — 2a3b2 — 6a3b2 — 4ab3 +2ab3 + a3b2. In this example we have 4 terms containing