that the quantity represented by b is to be added to the quan tity represented by a. 6. The sign is called minus, and when placed between two quantities, indicates that the one on the right is to be subtracted from the one on the left. Thus, c d is read c minus d, and indicates that the quantity represented by d is to be subtracted from the quantity represented by c. The sign, is sometimes called the positive sign, and the quantity before which it is placed is said to be positive. The sign, is called the negative sign, and quantities affected by it are said to be negative. 7. The sign X, is called the sign of multiplication, and when placed between two quantities, indicates that the one on the left is to be multiplied by the one on the right. Thus, a × b, indi cates that a is to be multiplied by b. The multiplication of quantities may also be indicated by placing a simple point between them, as a.b, which is read a multiplied by b. The multiplication of quantities, which are represented by letters, is generally indicated by simply writing the letters one after another, without interposing any sign. Thus, ab is the same as a × b, or a.b; and abc, the same as a xbx c, or a.b.c. It is plain that the notation last explained cannot be employed when the quantities are represented by figures. For, if it were required to indicate that 5 was to be multiplied by 6, we could not write 5 6, without confounding the product with the number 56. The result of a multiplication is called the product, and each of the quantities employed, is called a factor. In the product of several letters, each single letter is called a literal factor. Thus, in the product ab there are two literal factors a and b; in the product bed there are three, b, c and d. 8. The sign, is called the sign of division, and when placed between two quantities, indicates that the one on the left is to be divided by the one on the right. Thus, ab indicates that a is to be divided by b. The same operation may be indicated by writing a ¿ under a, and drawing a line between them, as; or by writing b b on the right of a, and drawing a line between them, as a|b. 9. The sign, is called the sign of equality, and indicates that the two quantities between which it is placed are equal to each other. Thus, a − b = c +d, indicates that a diminished by bis equal to c increased by d. 10. The sign >, is called the sign of inequality, and is used to indicate that one quantity is greater or less than another. Thus, a >b is read, a greater than b; and a <b is read, a less than b; that is, the opening of the sign is turned toward the greater quantity. 11. The sign ~ is sometimes employed to indicate the difference if two quantities when it is not known which is the greater. Thus, ab, indicates the difference between a and b, without showing which is to be subtracted from the other. 12. The sign ∞, is used to indicate that one quantity varies as 1 to another. Thus a ∞ 1 b' indicates that a varies as 13. The signs and, are called the signs of proportion; the first is read, is to, and the second is read, as. Thus, a: b :: cd, is read, a is to b, as c is to d. The sign .., is read hence, or consequently. 14. If a quantity is taken several times, as a+a+a+a+a, it is generally written but once, and a number is then placed before it, to show how many times it is taken. Thus, a+a+a+a+a may be written 5a. The number 5 is called the co-efficient of a, and denotes that a is taken 5 times. Hence, a co-efficient is a number prefixed to a quantity, denoting the number of times which the quantity is taken. When no co-efficient is written, the co-efficient 1 is always understood; thus, a is the same as la. 15. If a quantity is taken several times as a factor, the product may be expressed by writing the quantity once, and placing a number to the right and above it, to show how many times it is taken as a factor. Thus, αχαχαχαχα may be written as. The number 5 is called an exponent, and indicates that a is taken 5 times as a factor. Hence, an exponent is a number written to the right and above a quantity, to show how many times it is taken as a factor. If no exponent is written, the exponent 1 is understood. Thus, a is the same as a1. 16. If a quantity be taken any number of times as a factor, the resulting product is called a power of that quantity: the exponent denotes the degree of the power. For example, ala is the first a2 = axa is the second power, or square of a, a3 = a × a × a is the third power, or cube of a, a5 = axaxaxaxa is the fifth power of a, in which the exponents of the powers are, 1, 2, 3, 4 and 5; and the powers themselves, are the results of the multiplications. It should be observed that the exponent of a power is always greater by one than the number of multiplications. The exponent of a power of a quantity is sometimes, for the sake of brevity, called the exponent of the quantity. 17. As an example of the use of the exponent in algebra, let it be required to express that a number a is to be multiplied three times by itself; that this product is then to be multiplied three times by b, and this new product twice by c ; we should write axaxaxa x b x b x b xc x c = aab3c2. If it were further required to take this result a certain number of times, say seven, we should simply write 7a4b3 18. A root of a quantity, is a quantity which being taken a certain number of times, as a factor, will produce the given quantity. The sign is called the radical sign, and when placed over a quantity, indicates that its root is to be extracted. Thus, or simply a denotes the square root of a. 3 a denotes the cube root of a. a denotes the fourth root of a. √a The number placed over the radical sign is called the index of the root. Thus, 2 is the index of the square root, 3 of the cube root, 4 of the fourth root, &c. 19. The reciprocal of a quantity, is 1 divided by that quantity. Thus, 20. Every quantity written in algebraic language, that is, by the aid of letters and signs, is called an algebraic quantity, or the algebraic expression of a quantity. Thus, 2a2 За 5a2 7a3b2 3a - 56 3ab + 462 { } { is the algebraic expression of three times the quantity denoted by a; is the algebraic expression of five times the is the algebraic expression of seven times the 21. A single algebraic expression, not connected with any other by the sign of addition or subtraction, is called a monomial, or simply, a term. Thus, 3a, 5a2, 7a3b2, are monomials, or single terms. An algebraic expression composed of two or more terms con nected by the signor, is called a polynomial. For example, 3a - 56 and 2a2-3cb + 462, are polynomials. A polynomial of two terms, is called a binomial; and one of three terms, a trinomial. 22. The numerical value of an algebraic expression, is the num ber obtained by giving a particular value to each letter which enters it, and performing the operations indicated. This numerical value will depend on the particular values attributed to the letters, and will generally vary with them. For example, the numerical value of 2a3, will be 54 if we make a = 3; for, 33 = 3 × 3 × 3 = 27, and 2 × 27 = 54. = The numerical value of the same expression is 250 when we make a = 5; for, 535 × 5 × 5 125, and 2 x 125 = 250. We say that the numerical value of an algebraic expression generally varies with the values of the letters which enter it; it does not, however, always do so. Thus, in the expression ab, so long as a and b are increased or diminished by the same number, the value of the expression will not be changed. For example, make a 7 and b = 4: there results a-b 3. Now, make a = 7+ 5 = 12, and b=4+5=9, and there results, as before, a b = 12 9 = 3. = The 23. Of the different terms which compose a polynomial, some are preceded by the sign +, and others by the sign former are called additive terms, the latter, subtractive terms. When the first term of a polynomial is plus, the sign is generally omitted; and when no sign is written before a term, it is always understood to have the sign +. 24. The numerical value of a polynomial is not affected by changing the order of its terms, provided the signs of all the terms remain unchanged. For example, the polynomial 4a3 - 3a2b + 5ac2 = 5ac2 — 3a2b + 4a3 = 3a2b+5ac2 + 4a3. 25. Each literal factor which enters a term, is called a dimen sion of the term; and the degree of a term is indicated by the number of these factors or dimensions, Thus, |