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37 CEN 1 VIIMAMS.

ʼn the Certe's Office of the Southern Jistrer of New York, and in the Clerk's Office of the District Court of Massachusetts.

The following Works prepared by Joan D. WILLIAMS.

Key to Daboll's Arithmetic, 18mo. 50 cts.

A New Syarm of Arithmetic, and Key to the same, 12mo. Each 50 cts. Arithmetical and Algebraical Amusements, or a Key to Mathematics; Beo. $125.

Key to Hutton's Mathematics, containing all the Questions and Solutions that is left unsolved in all the editions, I believe. Bound in sheep. $2,00. A New System of Algebra. In two parts.

An Elementary Treatise on Algebra, with the Application of Algebra to Geometry, containing 620 pages 8vo., with more than 2,000 Questions and Solutions. $1,50.

Hutton's Algebra. 18mo. 300 pages. 75 cts.

THE

ELEMENTS OF ALGEBRA.

DEFINITIONS AND NOTATION.

1. ALGEBRA is a general method of computation, in which abstract quantities are represented by letters, and their connexion pointed out by means of certain indeterminate characters or symbols, which have been invented for this purpose.

2. The leading rules in Algebra are the same as those in Arithmetic, viz: Notation, Addition, Subtraction, Multiplication and Division.

3. Known quantities are usually expressed by the first letters of the alphabet, a, b, c, &c., and unknown quantities by the last, x, y, z, &c., and this must be always understood, unless the contrary be expressed.

4. A simple equation, or an equation of the first degree, is that which contains the unknown quantity simply; that is, without any of its powers except the first. A quadratic equation, or an equation of the second degree, is that which contains the square, but no higher power, of the unknown quantity.

5. The sign is called greater than, and the sign is called less than; and the expression in which either of these signs occurs is called an inequality; thus, the inequality a> b denotes that a is greater than b, and the inequality a b denotes that a is less than b; the greatest quantity being always placed at the opening of the sign.

6. The sign Plus, or +, denotes addition, and means that the quantities between which it is placed are to be added together; thus, ab means that the quantity represented by b is to be added to the quantity represented by a, and is read, a plus b. If a represents 9, and b 5, then a +b represents 14.

7. This character (or minus) is the sign of subtraction, and means that the quantity to which it is prefixed is to be taken from the former. If a represents 9, and b 5, then a b represents 4. 8. The sign is called the sign of multiplication, and placed between two quantities, denotes that they are to be multiplied together. A point is often used instead of this sign, or, when the quantities to be multiplied together are represented by letters in the form of a word, the sign may be altogether omitted; thus,

2 X 4 X 5, or 2.4.5, is the continued product of 2, 4 and 5. Likewise, 7X a X b, or 7.a.b., or 7ab, is the continued product of 7, a, and b.

9. divided by. This character is the sign of division, and signifies that the former of the quantities between which it is placed is to be divided by the latter; thus, ab means that the quantity a is to be divided by b. The division of one quantity by another is frequently represented by placing the dividend over the divisor, with a line between them, in which case the expression is Dividend, a

called a fraction; thus, Divisor,, signifies a divided by b; then

a is the numerator and b the denominator of the fraction.

10. A quantity in the denominator of a fraction is also expressed by placing it in the numerator, and prefixing the negative sign to its 1 1 1 1 index; thus, a ̄1, a, a, a", signify, respectively; these are called the negative powers of a.

11. Points are generally made use of to denote proportion; thus, a: b::c: d, signifies that a bears the same proportion to b that c bears to d.

-by

=

equal to. This sign means that the quantities between which it is placed are equal to each other; ax cd+ad, signifies that the quantity ax — by is equal to the quantity cd+ad. 12. The sign between two quantities means or signifies their difference; thus, a x, is a x or xa, according as a or x is the greater; a 4 x signifies the sum or difference of a and x. 13. A vinculum is a line drawn over several quantities, and signifies that the terms under it are to be taken as one whole, and to be affected with the same operation. The modern method of expressing the same thing is by the parenthesis () or bracket, [ ]. Thus (a+b) × x, a +b × x, or [a + b] × x, means that the quantity represented by a +b is to be multiplied by the quantity represented by x. Let a 3, b= 4, then (a + b) × x=7x, and if x = 1, then 7x= = 7.

=

14. The powers of algebraic quantities are denoted by placing a small figure, called the index or (indices) exponent of the power, to the right hand of the letter, as a2, a3, a1, &c.; so when the index is a fraction the numerator shows the power to which the quantity is first to be raised, and the denominator expresses the root to be extracted; thus a denotes the cube root of a square; a is the square root of a3; and a* denotes the square root of a, the same as a; as, the cube root of a, the same as ✔a; the fifth root of

the cube of a, or a3; and a is the ninth root of a fifth power of a, or ✔a3; and (a + b) denotes the nth root of the sum of a and b.

15. A surd, or irrational quantity, is that of which the value cannot be accurately expressed in numbers; thus 2, 3, 5, and 7, the root of quantities, are denoted by the radical, with the proper index in it. The signs, or ↓, J, ✔, &c., are used to express the square, cube, biquadrate, &c., roots of the quantities before which they are placed; thus, 2, a, a3 = a, a1 = a.

16. The reciprocal of any quantity is that quantity inverted, or unity divided by it ; so the reciprocal of a, or f, is, and the recip rocal of is 2.

៖

b

17. A rational quantity is that which has no radical sign or index annexed to it, as a, a, or Sa.

18. The words therefore, consequently, or that is, are usually expressed by the symbol..; thus the sentence therefore a+b is equal to cd, is expressed by.. a+b=c+d.

19. Positive or affirmative quantities are those which are to be added, as a or+a, or 3ax, or + 3ax. See ex. 1, p. 5. For when a quantity is found without a sign it is understood to be positive, or to have the sign + prefixed, that is, always when it is a leading quantity; and a quantity without any coefficient is supposed to have one or unity before it; thus, a once a, (la.)

2by2,

20. Negative quantities are those which are to be subtracted, as Ex. 11, viz: 6by', — by, - Sby3, — by?. 21. When two quantities are multiplied together, each, considered separately, is termed the coefficient of the other; but when one of them is a known quantity, it alone is termed the coefficient. Thus in the quantity ab, a is the coefficient of b, and b of a; but in 4x, the numeral 4 is the coefficient of x, and x is never termed the coefficient of 4. Sometimes the coefficient is a compound quantity, as (a + b)x, or ax + bx; here the coefficient is a+b. 22. Like signs are such as are all positive, (+), plus, or all negative, (—), minus.

23. Unlike signs are when some are positive, plus, (+), and others minus, (—).

24. Like quantities are such as do not differ except in their coefficients; as 2ax2, and -3ax2. Unlike quantities are those which differ in letters or power, as 2, a and b, or 6ax, and 4a3y2.

25. A monomial, or simple quantity, is a quantity consisting of one term only, as a; 3bx; 2xy; +3xy.

is some

26. A binomial quantity consists of two terms, as a + x, or a-b, the latter of which, being connected by the signtimes called a residual.

27. A trinomial consists of three, as a + bx-3y.

28. A quadrinomial is a quantity consisting of four terms, as ax2 + 3xy — 3y + bxy2, or a + b −x + y.

29. A multinomial, or polynomial, is a quantity consisting of many terms, or of an indefinite number of terms, as a +b c+d- x + y, &c.

30. When two equal quantities are compounded with the sign =between them, the comparison is called an equation; thus 3+4 7, or 2x+4 8, is an equation; the quantity on the left hand side of the sign is called the first member of an equation, and the quantity on the right hand of it the second. In the equation 2x+o a; 2x+b is the first member, and a is the second. When the quantities which compose a number are separated by the signs or, each quantity, so separated, is called a term; thus, the first member of the equation 2x+b: a, contains two terms, viz. 2x and b; also a and b are the terms of ab or and a, b, x, y are the terms of the proportion a:b::x : y.

b

a

31. A quantity is said to be a multiple of another when it contains it a certain number of times exactly;, thus 16a is a multiple of 4a, as it contains it exactly four times.

32. An equation of the third, fourth, &c., degree, is one in which the highest power of the unknown quantity is the third, fourth, &c., powers; and in general, an equation, in which the unknown quantity is called an equation of the mth degree, and each of the two members of an equation is called a side.

33. If equal quantities be either increased or diminished by the same quantity, the results will be equal, or, in other words, if each side of an equation be either increased or diminished by the same quantity, the result will be an equation.

34. If each side of an equation be either multiplied or divided by the same quantity, the result will be an equation.

35. If each side of an equation be either involved to the same power, or evolved to the same root, the result will be an equation.

36. Hence, generally, whatever operations be performed on one side of an equation, if the same operations be performed on the other side, the result will be an equation.

37. Notation in Algebra is the method of representing any proposed quantity by means of certain symbols.

1. What is the numeral value of a + b b=5, and c=8?

c, supposing a 7,