CHAPTER XXVI. DISCUSSION OF EQUATIONS Character of Roots of Quadratics. Relations of Roots CHAPTER I INTRODUCTION AND EVALUATION 1. Algebra is merely an extension or continuation of Arithmetic. The rules, principles and methods of Arithmetic are also used in Algebra. The one great difference between them is, that while in Arithmetic, numbers are expressed by the figures or digits, 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9, in Algebra we use figures and letters to represent these numbers. By employing these figures and letters,(we are able to solve problems with less work and in many instances, solve problems that could not be solved by arithmetic alone. Any letters of the alphabet may be used, but it is usual to represent known numbers by those in the begining of the alphabet as a, b, c, and to represent unknown numbers, that is, numbers whose values are to be found, by the letters at the end of the alphabet, as, x, y, z. Quantity. A number of certain units is called a quantity. Thus 5 cents, 6 pounds, 3 feet are called quantities. In arithmetic, these quantities were called concrete numbers, in distinction from numbers without names, as 5, 12, or 7, which were called abstract numbers. 2. Signs of Operation. The principal signs of operation are the same in Algebra as in Arithmetic. The Sign of Addition, +, which is read plus, meaning more, shows that the numbers connected by this sign are to be added. Thus, 3+ 4, shows that the numbers 3 and 4 are to be added. In the same way, a + b, means that the numbers which a and b represent are to be added. In the case of the numbers 3 and 4, this addition can actually be performed by combining them, giving us the sum 7, but when we are asked to add the numbers a and b, we cannot combine them, but only indicate the addition and write as our sum, a + b. の 7 The Sign of Subtraction, -, read minus, shows that the second number is to be subtracted from the first. Thus, 4, which is read, seven minus four, means that 4 is to be subtracted from 7, and the difference is 3. In the same way, a b, read a minus b, means that b is to be subtracted from a. Here, as in the similar case in addition, the numbers a and b cannot be combined and we indicate the subtraction, calling the difference a- b. The Sign of Multiplication, X, read multiplied by or times, shows that the first number is to be multiplied by the second. Thus, 4 X 3, is read four times three, or four multiplied by three, and a X b, is read a multiplied by b or a times b. We can actually multiply 4 by 3 and obtain the product 12, but in multiplying a by b, we have the same difficulty as before, and can only indicate the product, a b. Where letters are used to represent numbers, that is, in the case of literal numbers, we frequently use a dot,, placed between the numbers and above the line, to distinguish it from a decimal point, to indicate the multiplication as, ab, which means a X b, or the sign may be omitted altogether and the letters written beside each other, as ab, which also means, a × b. The Sign of Division,÷, read divided by, shows that the first number is to be divided by the second number. Thus, 82, read eight divided by two, means that 8 is to be divided by 2, and, a÷b, read, a divided by b, means that the number a is to be divided by the number b. The division of a by b can only be indicated and the quotient is written ab. As every fraction is an indicated division, the numerator being the dividend and the denominator, the divisor, ab may also be written as a b or a/b. OTHER SIGNS USED IN ALGEBRA 3. The Sign of Equality, =, is read is equal to or equals and means that the two numbers, between which it is placed, are equal. Thus 4+ 7 = 11, is read, four plus seven is equal to eleven and means that the sum of the numbers 4 and 7 is equal to the number 11. Also, a + b = 6, is read a plus b equals 6 and means that the sum of the numbers a and b is equal to the number 6; x = a + b, means that the number x is equal to the number, a + b. 4. The Signs of Aggregation.-Since, in the case of literal numbers, the arithmetical operations of addition, subtraction, multiplication and division cannot be performed by actually combining them, it is frequently found necessary to treat such indicated operations as single numbers. Thus, the sum, a+b+c, is to be multiplied by 3. To show that a + b + c is to be treated as a single number, we place a parenthesis around it, thus (a + b + c) and to indicate the multiplication by 3, we simply connect the two numbers, 3 and (a+b+c) by the usual sign of multiplication, 3 x (a+b+c) or 3(a+b+c), no sign between the numbers, indicating a multiplication. In addition to the parenthesis, (), we have the Bracket [], the Brace {}, the Vinculum and the Bar. All these signs perform the same duty of aggregating, that is, taking any number of figures and letters together, to be treated as a single number. Thus x [b c], means that the number x is to be divided by the number [b c]; {ab + x} ÷ y, means that the number {ab + x} is to be divided by the number y; C- x+y, means that the number x + y is to be subtracted from the number c. The value of these signs can readily be appreciated. Suppose we omitted the vinculum in the last expression and wrote it, cx + y; this would mean that the number x is to be subtracted from the number c and the number y added to the resulting number. 5. Factors. The parts of a number, which, when multiplied, produce the number, are called its Factors. Thus, 3 and 4 are factors of 12, because 3 × 4 will produce 12; 6xy will have as its factors, 3, 2, x and y, because 3 × 2 × Xy equals the number 6xy; similarly, the factors of c(a + b) are c and (a + b), since their product will produce |
