4. The sign + placed before a number denotes that the number is to be added. Thus a+b denotes that the number represented by b is to be added to the number represented by a. If a represent 9 and b represent 3, then a+b represents 12. The sign + is called the plus sign, and a+b is read thus " a plus b."

5. The sign - placed before a number denotes that the number is to be subtracted. Thus a-b denotes that the number represented by b is to be subtracted from the number represented by a. If a represent 9 and b represent 3, then a-b represents 6. The sign is called the minus sign, and a-b is read thus "a minus b."

6. Similarly a+b+c denotes that we are to add b to a, and then add c to the result; a+b-c denotes that we are to add b to a, and then subtract c from the result; a-b+c denotes that we are to subtract b from a, and then add c to the result; a-b-c denotes that we are to subtract b from a, and then subtract c from the result.

=

7. The sign denotes that the numbers between which it is placed are equal. Thus ab denotes that the number represented by a is equal to the number represented by b. And a+b=c denotes that the sum of the numbers represented by a and b is equal to the number represented by c; so that if a represent 9, and b represent 3, then c must represent 12. The sign is called the sign of equality, and a=b is read thus "a equals b" or "a is equal to b."

8. The sign x denotes that the numbers between which it stands are to be multiplied together. Thus axb denotes that the number represented by a is to be multiplied by the number represented by b. If a represent 9, and b represent 3, then axb represents 27. The sign is called the sign of multiplication, and axb is read thus "a into b." Similarly axbx c denotes the product of the numbers represented by a, b, and c.

9. The sign of multiplication is however often omitted for the sake of brevity; thus ab is used instead of a xb, and has the same meaning; so also abc is used instead of axbxc, and has the same meaning.

The sign of multiplication must not be omitted when numbers are expressed in the ordinary way by figures. Thus 45 cannot be used to represent the product of 4 and 5, because a different meaning has already been, appropriated to 45, namely, forty-five. We must therefore represent the product of 4 and 5 in another way, and 4×5 is the way which is adopted. Sometimes, however, a point is used instead of the sign ; thus 4.5 is used instead of 4 x 5. To prevent any confusion between the point thus used as a sign of multiplication, and the point used in the notation for decimal fractions, it is advisable to place the point in the latter case higher up; thus 4.5 may be kept to denote 4+

5

10°

The point is sometimes placed instead of the sign × between two letters; so that a.b is used instead of a × b. But the point is here superfluous, because, as we have said, ab is used instead of axb. Nor is the point, nor the sign, necessary between a number expressed in the ordinary way by a figure and a number represented by a letter; so that, for example, 3a is used instead of 3 × a, and has the same meaning.

10. The sign÷denotes that the number which precedes it is to be divided by the number which follows it. Thus ab denotes that the number represented by a is to be divided by the number represented by b. If a represent 8, and b represent 4, then a÷b represents 2. sign is called the sign of division, and a÷b is read thus a by b."

The

a

There is also another way of denoting that one number is to be divided by another; the dividend is placed over the divisor with a line between them. Thus is b used instead of a÷b, and has the same meaning.

11. The letters of the alphabet, and the signs which we have already explained, together with those which may occur hereafter, are called algebraical symbols, because they are used to represent the numbers about which we may be reasoning, the operations performed on them, and

their relations to each other. Any collection of Algebraical symbols is called an algebraical expression, or briefly an expression.

12. We shall now give some examples as an exercise in the use of the symbols which have been explained; these examples consist in finding the numerical values of certain algebraical expressions.

Suppose a=1, b=2, c=3, d=5, e=6, ƒ=0. Then
7a+3b-2d+f=7+6¬10+0=13-10=3.
2ab+8bc-ae+df=4+48−6+0=52-6=46.

4ac 10be de 12 120 30

+

+

b cd ac 2 15 3

=6+8-10=14-10=4.

If a=1, b=2, c=3, d=4, e=5, ƒ=0, find the numerical values of the following expressions.

II. Factor. Coefficient. Power. Terms.

13. When one number consists of the product of two or more numbers, each of the latter is called a factor of the product. Thus, for example, 2×3×5-30; and each of the numbers 2, 3, and 5 is a factor of the product 30. Or we may regard 30 as the product of the two factors 2 and 15, or as the product of the two factors 6 and 5, or as the product of the two factors 3 and 10. And so, also, we may consider 4ab as the product of the two factors 4 and ab, or as the product of the two factors 4a and b, or as the product of the two factors 46 and a; or we may regard it as the product of the three factors 4 and a and b.

14. When a number consists of the product of two factors, each factor is called the coefficient of the other factor; so that coefficient is equivalent to co-factor. Thus considering 4ab as the product of 4 and ab, we call 4 the coefficient of ab, and ab the coefficient of 4; and considering 4ab as the product of 4a and b, we call 4a the coefficient of b, and b the coefficient of 4a. There will be little occasion to use the word coefficient in practice in any of these cases except the first, that is the case in which 4 is regarded as the coefficient of ab; but for the sake of distinctness we speak of 4 as the numerical coefficient of ab in 4ab, or briefly as the numerical coefficient. Thus when a product consists of one factor which is represented arithmetically, that is by a figure or figures, and of another factor which is represented algebraically, that is by a letter or letters, the former factor is called the numerical coefficient.

15. When all the factors of a product are equal, the product is called a power of that factor. Thus 7 x7 is called the second power of 7; 7×7×7 is called the third power of 7; 7×7×7×7 is called the fourth power of 7 ; and so on. In like manner a×a is called the second power of a; a×a×a is called the third power of a; a×a×a×a is called the fourth power of a; and so on. And a itself is sometimes called the first power of a.

16. A power is more briefly denoted thus; instead of expressing all the equal factors, we express the factor once, and place over it the number which indicates how often it is to be repeated. Thus a2 is used to denote a×a; a3 is used to denote a×a×a; aa is used to denote a×a×a×a; and so on. And a1 may be used to denote the first power of a, that is a itself; so that a1 has the same meaning as a.

17. A number placed over another to indicate how many times the latter occurs as a factor in a power, is called an index of the power, or an exponent of the power; or, briefly, an index, or exponent.

Thus, for example, in a3 the exponent is 3; in a” the exponent is n.

18. The student must distinguish very carefully between a coefficient and an exponent. Thus 3c means three times c; here 3 is a coefficient. But c3 means c times c times c; here 3 is an exponent. That is

3c=c+c+C,

c3=cxcxc.

19. The second power of a, that is a2, is often called the square of a, or a squared; and the third power of a, that is a3, is often called the cube of a, or a cubed. There are no such words in use for the higher powers; a4 is read thus a to the fourth power,” or briefly "a to the fourth."

66

20. If an expression contain no parts connected by the signs and, it is called a simple expression. If an expression contain parts connected by the signs + and it is called a compound expression, and the parts connected by the signs and are called terms of the expression.

Thus ax, 4bc, and 5a2c2 are simple expressions; a2 + b3 — c1 is a compound expression, and a2, b3, and c1 are its terms.

21. When an expression consists of two terms it is called a binomial expression: when it consists of three terms it is called a trinomial expression; any expression consisting of several terms may be called a multinomial expression, or a polynomial expression.