is called a numerical factor. Thus, in the expression 5ab, 5 is a numerical factor, while a and b are literal factors. 14. The sign of division is a short horizontal line with a point above and one below, ÷. When placed between two quantities, it indicates that the first is to be divided by the second. Thus, 24÷6 indicates that 24 is to be divided by 6, making 4. So, also, a÷b indicates that a is to be divided by b. Generally, however, the division of two numbers is indicated by writing the divisor under the dividend, and drawing a line between them. Thus, 24÷6 and a÷b are usually writ24 and 2 ten а 15. The products formed by the successive multiplication of the same number by itself, are called the powers of that number. Thus, 2x2=4, the second power or square of 2. 2×2×2=8, the third power or cube of 2. So, also, 3×3=9, the second power of 3. Also, 3×3×3=27, the third power of 3, etc. axaxa=aaa, the third power of a, etc. In general, any power of a quantity is designated by the number of equal factors which form the product. 16. The sign of involution is a number written above a quantity, at the right hand, to indicate how many times the quantity is to be taken as a factor. A root of a quantity is a factor which, multiplied by itself a certain number of times, will produce the given quantity. The figure which indicates how many times the root or factor is taken, is called the exponent of the power. Thus, instead of aa, we write a2, where 2 is the exponent of the power; instead of aaa, we write a3, where 3 is the expo nent of the power; instead of aaaaa, we write a", where 5 is the exponent of the power, etc. When no exponent is written over a quantity, the exponent 1 is always understood. Thus, a1 and a signify the same thing. Exponents may be attached to figures as well as letters. Thus the product of 3 by 3 may be written 33, which equals 9. 3×3×3 3×3×3×3×3 แ 33, 35, 27. 243. 17. The sign of evolution, or the radical sign, is the character√. When placed over a quantity, it indicates that a root of that quantity is to be extracted. The name or index of the required root is the number written above the radical sign. Thus, V9, or simply V9, denotes the square root of 9, which is 3. 64 denotes the cube root of 64, which is 4. V16 denotes the fourth root of 16, which is 2. So, also, Va, or simply Va, denotes the square root of a. Va denotes the fourth root of a. а Va denotes the nth root of a, where n may represent any number whatever. When no index is written over the sign, the index 2 is understood. Thus, instead of Vab, we usually write Vab. Symbols which indicate Relation. 18. The sign of equality consists of two short horizontal lines, When written between two quantities, it indicates that they are equal to each other. Thus, 7+6=13 denotes that the sum of 7 and 6 is equal to 13. In like manner, a=b+c denotes that a is equal to the sum of b and c; and a+b=c―d denotes that the sum of the numbers designated by a and b, is equal to the difference of the numbers designated by c and d. 19. The sign of inequality is the angle > or <. When placed between two quantities, it indicates that they are unequal, the opening of the angle being turned toward the greater number. When the opening is toward the left, it is read greater than; when the opening is toward the right, it is read less than. Thus, 5>3 denotes that 5 is greater than 3; and 6<11 denotes that 6 is less than 11. So, also, a>b denotes that a is greater than b; and x<y+z denotes that x is less than the sum of y and z. is employed to 20. A parenthesis, ( ), or a vinculum, connect several quantities, all of which are to be subjected to the same operation. Thus the expression (a+b+c)×x, or a+b+c×x, indicates that the sum of a, b, and e is to be multiplied by x. But a+b+c×x denotes that c only is to be multiplied by x. When the parenthesis is used, the sign of multiplication is generally omitted. Thus, (a+b+c)× x is the same as (a+b+c)x. 21. The sign of ratio consists of two points like the colon : placed between the quantities compared. Thus the ratio of a to b is written a:b. 22. The sign of proportion consists of a combination of the sign of ratio and the sign of equality, thus, := ; or a combination of eight points, thus, : :: Thus, if a, b, c, d, are four quantities which are proportional to each other, we say a is to b as c is to d; and this is expressed by writing them thus: 23. The sign of variation is the character. When written between two quantities, it denotes that both increase or diminish together, and in the same ratio. Thus the expression so tv denotes that s varies in the same ratio as the product of t and v. 24. Three dots are sometimes employed to denote therefore, or consequently. A few other symbols are employed in Algebra, in addition to those here enumerated, which will be explained as they occur. Combination of Algebraic Quantities. 25. Every number written in algebraic language—that is, by aid of algebraic symbols—is called an algebraic quantity, or an algebraic expression. Thus, 3a2 is the algebraic expression for three times the square of the number a. 7a3b4 is the algebraic expression for seven times the third power of a, multiplied by the fourth power of b. 26. An algebraic quantity, not composed of parts which are separated from each other by the sign of addition or subtraction, is called a monomial, or a quantity of one term, or simply a term. Thus, 3a, 5bc, and 7xy2, are monomials. Positive terms are those which are preceded by the sign plus, and negative terms are those which are preceded by the sign minus. When the first term of an algebraic quantity is posi tive, the sign is generally omitted. Thus a+b-c is the same as +a+b+c. The sign of a negative term should never be omitted. 27. The coefficient of a quantity is the number or letter prefixed to it, showing how often the quantity is to be taken. Thus, instead of writing a+a+a+a+a, which represents 5 a's added together, we write 5a, where 5 is the coefficient of In 6(x+y), 6. is the coefficient of x+y. When no coefficient is expressed, 1 is always to be understood. Thus, la and a denote the same thing. α. The coefficient may be a letter as well as a figure. In the expression nx, n may be considered as the coefficient of x, because x is to be taken as many times as there are units in n. If n stands for 5, then nx is 5 times x. When the coefficient is a number, it may be called a numerical coefficient; and when it is a letter, a literal coefficient. In 7ax, 7 may be regarded as the coefficient of ax, or 7a may be regarded as the coefficient of x. 28. The coefficient of a positive term shows how many times the quantity is taken positively, and the coefficient of a negative term shows how many times the quantity is taken negatively. Thus, +4x=+x+x+x+x; 29. Similar terms are terms composed of the same letters, affected with the same exponents. The signs and coefficients may differ, and the terms still be similar. Thus, 3ab and 7ab are similar terms. Also, 5a2c and -3a2c are similar terms. 30. Dissimilar terms are those which have different letters or exponents. Thus, axy and axz are dissimilar terms. Also, 3ab2 and 4a3b are dissimilar terms. 31. A polynomial is an algebraic expression consisting of more than one term; as, a+b; or a+2b-5c+x. A polynomial consisting of two terms only is usually called a binomial; and one consisting of three terms only is called a trinomial. Thus, 3a +56 is a binomial; and 5a-3bc+xy is a trinomial. 32. The degree of a term is the number of its literal factors. Thus, 3a is a term of the first degree. In general, the degree of a term is found by taking the sum of the exponents of all the letters contained in the term. Thus the degree of the term 5ab2cd3 is 1+2+1+3, or 7; that is, this term is of the seventh degree. B |