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1. ALGEBRA is a branch of Mathematics in which the quantities considered are represented by letters, and the operations to be performed are indicated by signs.
2. The quantities employed in Algebra are of two kinds :
1st. Knovon Quantities, those whose values are
given; and 2d. Unknown Quantities, those whose values are
Known Quantities are generally represented by the leading letters of the alphabet; as, a, b, c, &c.
Unknown Quantities are generally represented by the. final letters of the alphabet, as, x, y, z, w, &c.
When, in the course of an operation, an unknown quantity becomes known, it is often found convenient to represent it by one of the final letters, with one or more accents, as, c', y'', '"', &c. These symbols are read, ne prime, y" second, 2'" third, &c.
3. The signs employed in Algebra are of three kinds. signs of operation, signs of relation, and signs of abbre viation.
1. The Signs of Operation are the following:
1st. The Sign of Addition, +, called plus. Wher placed between two quantities, it indicates that the second is to be added to the first. Thus, the combination, a + b, rend, a plus b, indicates that bis to be added to a.
2d. Sign of Subtraction, called minus. When placed between two quantities, it indicates that the second is to be subtracted from the first. Thus, the expression,
d, read c minus d, indicates that d is to be subtracted from C.
3d. The Sign of Multiplication, X. When placed between two quantities, it indicates that the first is to be multiplied by the second. Thus, the expression, x xy, indicates that 2 is to be multiplied by y. Instead of the sign of multiplication, a simple point is sometimes nised; or, when the factors are letters, they may be writ ten one after the other, without any intervening sign. Thus, the expressions, a? xbxe, a .b.c, and a2bc, have all the same meaning. A factor, is any one of the mulcipliers of a product.
4th. The Sign of Division, H. When written between two quantities, it indicates that the first is to be divided by the second. Thus, the expression, p:-1, indicates that p is to be divided by q. The operation may also be expressed by writing one quantity over the other, in the form of a fraction; or the sign of division may be replaced by a straight or curved line. Thus, the expres xions, p =-9,
and p, . q
5th. The Exponential Sign. The exponential sign is a number written at the right and above a quantity, and indicates the number of times which the quantity is to be taken as a factor. In the expressions, a, a, a, i, the numbers 2, 3, 5, mi, are exponents, and indicate scv erally, that the quantity a is to be taken, 2, 3, 5, m times as a factor.
The resulting products are called powers.
6th. The Radical Sign, ✓. When placed over a quantity, it indicates that a root of that quantity is to be extracted. The nature of the root is indicated by a number placed over the radical sign, called an index. Thus, the expressions, Vā, 3/a, "Ja, indicate that the square, cube, and nth roots of a, are to be extracted.
When no exponent is written over a quantity, the expo. nent 1 is always understood. When no index is written over the radical sign, the index 2 is understood.
7th. The sign +, is sometimes called the positive sign, and quantities before which it is written are called positive quantities, or additive quantities.
8th. The sign –, is sometimes called the negative sign, and quantities before which it is written are called negative quantities, or subtractive quantities.
9th. The Double Sign E, read plus and minus, is often used to indicate that the quantity before which it is placed, is first to be added to, and then to be subtracted from the preceding quantity. Thus, the expression, pvq, is equivalent to the expressions, p + Va and р
When no sign is written before a quantity, the sign. is understood.
5. The Signs of Relation are the following:
1st. The Sign of Equality, =.
When written be tween two quantities, it indicates that they are equal to each other. Thus, the expression, a = nb, indicates that a is equal to the product of n and b. 2d. The Sign of Inequality, <,>.
When written between two quantities, it indicates that they are unequal, the greater one being placed at the opening of the sign. Thus, the expressions, a>b, c< d, indicate that a is greater than 6, and that c is less than d. 3d. The Signs of Proportion, :
The single colon stands for, is to; the double colon for, as. Thus, the expression,
b :: : d,
is read, a is to b, as c is to d.
4th. The Sign of Variation, 0. When written between two quantities, it indicates that they increase or
6 diminish together. Thus, the expression, a o indicates
b that a varies as that is, that a and increase 01 decrease together.
to be opera
6. The Signs of Abbreviation are the following. 1st. The Sign ... stands for hence, or consequently. 2d. The Sign ... stands for because. 3d. The Vinculum,
the Bar, |, the Parenthe sis, (), the Brackets. " ], } }, are employed to con. nect several quantities, all of which are ted upon in the same
Thus, the expressions,
(a + b + c) x x,
to [a+b+c] x x, a+b+c] x 2, all indicate that the sum of a, b, and c, is to be multiplicit by x.
7. A Coefficient is a number written before a quantity to show how many times it is to be taken additively. Thus, the expression, 3a", is equivalent to u? + a2 + a?. The number 3 is the coefficient of a2.. The coefficient may be either numerical or literal. Thus, in the expres. yions, 3a, 3ax?, (a + b + c)X?, the quantities 3, 3a, and (a + b + c), are coefficients of x2. When no coefficient is written, the coefficient 1 is understood.
8. A Term is an expression of a single quantity. Thus,
7 al, 2a2b, ax, are terms.
9. A Monomial is a single term, unconnected with any other by the signs + or Thus, 3a2, 7a2bc, are monomials.
10. A Polynomial is a collection of terms connected
2ab by the signs + or Thus, 3a2b + c - d,
+ d, are polynomials.
11. A Binomial is a polynomial of two terms; as,
c+d, (e+fx, gx + y.
12. A Trinomial is a polynomial of three terms; as,
a+b+c, . x2 2xy + y?, - a + 363 + 2. The degree of a term is the number of its literal factors. Thus, the term 3a is of the first degree, because it cou. tains one literal factor; the term 7a? is of the second degree, because it contains two literal factors; the term
- q?bac3 is of the sixth degree, because it contains six literal factors. The degree of a term is determined by the sum of the exponents of all its letters.
T'wo terms are homogeneous, when they are of the