33×25, or 36.25, is read, 36 multiplied by 25, or the product of 36 by 25. 7. The multiplication of quantities, which are represented by`letters, is indicated by simply writing them one after the other, without interposing any sign. Thus,ab signifies the same thing as axb, or as a.b; and abc the same as axb×c, or as a.b.c. It is plain that the notation ab, or abc, which is more simple than a×b, or axb×c, cannot be employed when the quantities are represented by figures. For example, if it were required to express the product of 5 by 6, and we were to write 5 6, the notation would confound the product with the number 56. 8. In the product of several letters, as abc, the single letters, a, b and c, are called factors of the product. Thus, in the product ab, there are two factors, a and b; in the product acd, there are three, a, c and d 9. There are three signs used to denote division. Thus, a÷b denotes that a is to be divided by b, a denotes that a is to be divided by b, alb denotes that a is to be divided by b. 10. The sign, is called the sign of equality, and is read, is equal to. When placed between two quantities, it denotes that they are equal to each other. Thus, 9-5-4: that is, 9 minus 5 is equal to 4: Also, a+b=c, denotes that the sum of the quantities a and b is equal to c. 11. The sign >, is called the sign of inequality, and is used to express that one quantity is greater or less than another. Thus, ab is read, a greater than b; and a<b is read, a less than b; that is, the opening of the sign is turned towards the greater quantity. 12. If a quantity is added to itself several times, as a +a+a+a +a, we generally write it but once, and then place a number before it to express how many times it is taken. Thus, a+a+a+a+a=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; and it also indicates the number of times plus one, that the quantity is added to itself. When no co-efficient is written, the co-efficient 1 is always understood. 13. If a quantity be multiplied continually by itself, as a×a×a Xaxa, we generally express the product by writing the letter once, and placing a number to the right of, and a little above it: thus, αχαχαχαχαπά The number 5 is called the exponent of a, and denotes the number of times which a enters into the product as a factor. Hence, the exponent of a quantity shows how many times the quantity is a factor; and it also indicates the number of times, plus one, that the quantity is to be multiplied by itself. When no exponent is written, the exponent 1 is always understood. 14. The product resulting from the multiplication of a quantity by itself any number of times, is called the power of that quantity: and the exponent, which always exceeds by one the number of mul. tiplications to be made, denotes the degree of the power. Thus, a3 is the fifth power of a. The exponent 5 denotes the degree of the power; and the power itself is formed by multiplying a four times by itself. 15. In order to show the importance of the exponent in algebra, suppose that we wish to express that a number a is to be multiplied three times by itself, that this product is to be multiplied three times by b, and that this new product is to be multiplied twice by c, we would write simply a1 b3 c3. If, then, we wish to expess that this last result is to be added to itself six times, or is to be multiplied by 7, we would write, 7aab3c2. This gives an idea of the brevity of algebraic language. 16. The root of a quantity, is a quantity which being multiplied by itself a certain number of times will produce the given quantity. The sign, is called the radical sign, and when prefixed to a quantity, indicates that its root is to be extracted. Thus, Va or simply va denotes the square root of a. Va, denotes the fourth root of 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. 17. Every quantity written in algebraic language; that is, with the aid of letters and signs, is called an algebraic quantity, or the algebraic expression of a quantity. Thus, За 5a2 7a3b2 3a-5b 2a-3ab+462 { { { { { is the algebraic expression of three times the number a; is the algebraic expression of five times the is the algebraic expression of seven times the is the algebraic expression of twice the square by b, augmented by four times the square of b. 18. When an algebraic quantity is not connected with any other by the sign of addition or subtraction, it is called a monomial, or a quantity composed of a single term, or simply, a term. Thus, 3a, 5a2, 7a3b2, are monomials, or single terms. 19. An algebraic expression composed of two or more parts, separated by the sign + or —, is called a polynomial, or quantity involving two or more terms. For example, 3a-56 and 2a-3cb+46 are polynomials. 20. A polynomial composed of two terms, is called a binomial; and a polynomial of three terms is called a trinomial. 21. The numerical value of an algebraic expression, is the number which would be obtained by giving particular values to the letters which enter it, and performing the arithmetical operations indicated. This numerical value evidently depends upon the particular values attributed to the letters, and will generally vary with them. For example, the numerical value of 2a3-54 when we make a=3; for, the cube of 3=27, and 2×27=54. The numerical value of the same expression is 250 when we make a=5; for, 53=125, and 2×125=250. 22. We have said, 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 a-b, so long as a and b increase 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 a-b-12-9=3, as before. 23. The numerical value of a polynomial is not affected by changing the order of its terms, provided the signs of all the terms be preserved. For example, the polynomial 4a3-3a2b+5ac2= 5ac2-3a2b+4a3-3a2b+5ac2+4a3. This is evident, from the nature of arithmetical addition and subtraction. The 24. Of the different terms which compose a polynomial, some are preceded by the sign+, and the others by the sign first are called additive terms, the others, subtractive terms. The first term of a polynomial is commonly not preceded by any sign, but then, it is understood to be affected with the sign +. 25. Each of the literal factors which compose a term is called a dimension of this term; and the degree of a term is the number of 3a is a term of one dimension, or of the first degree. 7a3bc2=7aaabcc is of six dimensions, or of the sixth degree. In general, the degree, or the number of dimensions of a term, is estimated by taking the sum of the exponents of the letters which enter this term. For example, the term 8a2bcd3 is of the seventh degree, since the sum of the exponents, 2+1+1+3=7. 26. A polynomial is said to be homogeneous, when all its terms are of the same degree. The polynomial 3a-2b+c -4ab+b2 is of the first degree and homogeneous. is of the second degree and homogeneous. 5a2c-4c+2c'd is of the third degree and homogeneous. 8a3-4ab+c 27. A vinculum or bar or a parenthesis (), is used to express that all the terms of a polynomial are to be considered together. Thus, a+b+cxb, or (a+b×c)×b denotes that the trinomial a+b+c is to be multiplied by b; also a+b+cxc+d+f or (a+b+c)x(c+d+f) denotes that the trinomial a+b+c is to be multiplied by the trinomial c+d+f. When the parenthesis is used, the sign of multiplication is usually omitted. Thus (a+b+c)xb is the same as (a+b+c) b. The bar is also sometimes placed vertically. Thus, +ax +b +c is the same as (a+b+c) x or a+b+cxx 28. The terms of a polynomial which are composed of the same letters, the same letters in each being affected with like exponents, are called similar terms. Thus, in the polynomial 7ab+3ab-4ab2+5a3, the terms Tab and 3ab, are similar; and so also are the terms-4a32 and 5a3b3, the letters and exponents in each being the same. But in the bino |