THE

.ACCOMPLISHED TUTOR.

CHAP. VII.

OF ALGEBRA.

SECT. I.

OF NOTATION.

I.

Definitions.

ALGEBRA is the art of refolving difficult queftions more readily than by the rules of common arithmetic."

In algebra, the value of quantities is expreffed. by .fome letters of the alphabet, which have fometimes figures, and *certain characters added to them, whereby their value is increafed, or diminished; and each letter may represent any quantity at pleafure. But, generally, the first letters in the alphabet, a, b, c, d, &c. are used to fignify quantities, the value whereof is known; and the latter letters, as w, x, y, %, &c. are used for quantities which are unknown: the letters are then managed according to the rules of art.

2. The fign+ fignifies addition, aud in algebra it is called plus; it denotes that the characters or letters placed on each fide of it are to be added together, thus, a+b fignifies that

VOL. II.

the

the quantity expressed by a is to be added to that representçd by b. Thus, if a stand for 3, and b for 6; then a+b will be equal to 9.

3. The fign -fignifies fubtraction; and fhows that the quantity following it is to be fubtracted from the quantity. preceding it: thus, a-b fignifies that the quantity repre fented by b is to be fubtracted from that reprefented by a; as, if a was 8, and b was 3, then a-b would be equal to 5: this fign is called minus.

4. The fign + representing addition, is called a pofitive, or an affirmative fign. The fign, fignifying subtractiop, is called a negative fign.

5. Like figus are when feveral quantities have all the fign +or; and, ulike figns are quantities where fome have the fign+,and others the fign —.'

=

6. The fign denotes equality, and is placed between two quantities, to fhow they are equal: thus, a=b fignifies that a and b are equal to each other.

7. The fignftands for multiplication, and fignifies that the quantities, placed on each fide are to be multiplied together: thus, fignifies the quantity a is to be multiplied by the quantity 6; as, if a be equal to 5, and 6 equal to 6, they will, with the product, ftand thus axb30, which fignifies that a multiplied by is equal to 30. But the product of two or more fimple quantities is generally gnified by merely joining the letters. Thus, the product of the above quantity is expressed a b, and if there be three or more quantities to be mutiplied together, as a xc, they will be expreffed thus, à·b c.*..

8. The fign expresses division thus, a fignifies that a is to be divided by b; but this fign is not much used, for divifion is generally expreffed in the manner of a fraction: a-b

thus,

and b

c+d

fignifies that a is to be divided by 6, and

divided by c+d.

9. The figno fignifies the difference between two quan

tities

ties: thus, ab ftands for the difference between a and b. Thus if a stand for 9, and 6 for 4, ab represents 5.

10. The fignor 7 are figns of majority, and flow that the quantity placed before the fign is greater than that which follows it: thus, ab, or a7b, fhows that a is greater than b.

11. The fign or x fignifies minority, and fhows that the quantity placed before the fign is lefs than that which follows it; thus, ab, or ab, fignifies that a is less than 6.

12. The fign is the fign of the Square root. It also expreffes the cube root, biquadrate root, &c. by placing 3 or 4, &c. over it; thus, √a, ora, or a, denote the square root, cube root, and biquadrate root of a refpectively.

13. Involution is the raising of a quantity to any power, according as it is joined to the figures 2, 3, 4, &c. refpectively.

14. The fign lu fignifies evolution, and denotes that the quantity to which it is joined is the fquare or cube root, &c. as it is joined to the numbers 2, 3, &c. refpectively.

The power of a quantity is often expressed in algebra by placing a figure over the quantity; thus, a, a3, and a*, denote the square, cube, and biquadrate, of a respectively; or the fecond, third, and fourth power; and the figures 2, 3, and 4, placed over a, are called the indices or exponents of a. 15. Like quantities are those that confift of the fame letters, as a, 4a+2a, or b—2b+3bb, &c.

16. Unlike quantities confift of different letters; as a, 26,

3c; or 2a, cd-d.

17. Simple quantities confift of one term only; as 46, or 3a, or 12d, &c.

18. Compound quantities confift of several terms; as a+c, 2b-d, &c.

19. A vinculum is a line drawn over feveral quantities, and shows that they are to be taken as a compound quantity; as a+bc.

20. The coefficient of a quantity is the number prefixed

to it; as 6d; here 6 is the coefficient, and fignifies that the quantity dis multiplied thereby.

21. A binomial quantity confifts of two terms; as b+c. A trinomial quantity, of three terms; as a+b+c. A qua drinomial quantity, of four terms; as a+b+c+d.

22. A refidual quantity is a binomial, where one of the terms is a negative one; as a-b.

23. A rational quantity has no radical fign.

24. A furd quantity is that which has not a proper root; as the fquare root of b (b), the biquadrate root of b (Abb).

25. The fign:::: fignifies proportion; as 5: 10 :: 40: 80, that is, as 5 to 10, fo is 40 to 80.

26. An equation is the comparison of two quantities which are equal to one another, having the fign of equality between them, as 5, 9=6, 8, which fignifies that 5 and 9 are equal to 6 and 8, or 14. Of equations there are feveral forts:

1. A dependant equation is that which may be deduced from fome others.-2. An independent equation, that which cannot be deducible from another.-3. A pure equation, that which contains but one power of the unknown quantity.-4. An affected equation, that which has several powers of the unknown quantity.

Axioms.

1. If equal quantities be added to equal quantities, the fums will be equal. And if equal quantities be taken from equal quantities, the remainders will be equal.

2. If equal quantities be multiplied by equal quantities, the products will be equal. And if equal quantities be divided by equal quantities, the quotients will be equal.

3. If equal quantities be raised to equal powers, the products will be equal.

4. Quantities equal to any other quantity are equal to one another.

5. The whole is equal to all its parts taken together.

SECT.

SECT. II.

OF THE FOUR SINGLE RULES OF ALGEBRA.

OF ADDITION.

RULE I. When like quantities having like figns are to be added together, add together the coefficients (if there be any), and to the fum prefix the fign, and fubjoin the common quantity.

2. When the quantities are like, but have unlike signs, take the difference between the fum of the affirmative coefficients, and the sum of the negative ones; to which difference prefix the fign of the greater fum, and annex the common quantity.

3. But when the quantities are all unlike, they cannot be brought into one fum, but must be written down one after another, prefixing to each its proper fign; as in the following

Where there is no coefficient prefixed to a quantity, the coefficient is 1. And when there is no fign prefixed, the quantity is affirmative; as in the quantities of the first and fecond of the foregoing examples.

Examples