ADVANCED ALGEBRA ALGEBRA TO QUADRATICS CHAPTER I FUNDAMENTAL OPERATIONS 1. It is assumed that the elementary operations and the meaning of the usual symbols of algebra are familiar and do not demand detailed treatment. In the following brief exposition of the formal laws of algebra most of the proofs are omitted. 2. Addition. The process of adding two positive integers a and b consists in finding a number x such that a+b = x. For any two given positive integers a single sum x exists which is itself a positive integer. 3. Subtraction. The process of subtracting the positive numberb from the positive number a consists in finding a number x such that b + x = a. (1) This number x is called the difference between a and b and is denoted as follows: a b = x, a being called the minuend and b the subtrahend. If ab and both are positive integers, then a single positive integer x exists which satisfies the condition expressed by equation (1) If a <b, then x is not a positive integer. In order that the process of subtraction may be possible in this case also, we introduce negative numbers which we symbolize by (-a), (b), etc. When in the difference a-b, a is less than b, we define a − b − (− (b − a)). The processes of addition and subtraction for the negative numbers are defined as follows: = 4. Zero. If in equation (1), ab, there is no positive or negative number which satisfies the equation. In order that in this case also the equation may have a number satisfying it, we introduce the number zero which is symbolized by 0 and defined by the equation or a + 0 The processes of addition and subtraction for this new number zero are defined as follows, where a stands for either a positive or a negative number 5. Multiplication. The process of multiplying a by b consists in finding a number x which satisfies the equation *The symbol for a positive integer might be written (+ a), (+ b), etc., consistently with the notation for negative numbers. Since, however, no ambiguity results, we omit the sign. Since the laws of combining the + and - signs given in this and the following paragraphs remove the necessity for the parentheses in the notation for the negative number, we shall omit them where no ambiguity results. When a and b are positive integers x is a positive integer which may be found by adding a to itself b times. When the numbers to be multiplied are negative we have the following laws, (− a) · (—b) = a. b, (a)⋅ b = a. (b) = — (a · b), 0.α = α.00, where a is a positive or negative number or zero. (1) These symbolical statements include the statement of the ́ following PRINCIPLE. A product of numbers is zero when and only when one or more of the factors are zero. This most important fact, which we shall use continually, assures us that when we have a product of several numbers as first, if e equals zero, it is certain that one or more of the numbers a, b, c, or d are zero; second, if one or more of the numbers a, b, c, or d are zero, then e is also zero. 6. Division. The process of dividing a by ẞ consists in finding a number x which satisfies the equation where a and ẞ are positive or negative integers, or a is 0. When a occurs in the sequence of numbers ...3 ß, 2ß, - ß, 0, ß, 2ß, 3ß, ..., (1) x is a definite integer or 0, that is, it is a number such as we have previously considered. If a is not found in this series, but is between two numbers of the series, then in order that in this case the process may also be possible we introduce the fraction which we symbolize by a + B or%2 and which is defined by the equation α = α. The operations for addition, subtraction, multiplication, and division of fractions are defined as follows: The last two equations are expressed verbally as follows: Both numerator and denominator of a fraction may be multiplied by any number without changing the value of the fraction. Changing the sign of either numerator or denominator of a fraction is equivalent to changing the sign of the fraction. The laws of signs in multiplication given on p. 3 may now be assumed to hold when the letters represent fractions as well as integers. ас Thus for example ( bd' The positive or negative number a may be written in the fractional form α 1 7. Division by zero. If in equation (1), § 6, ß single number x which satisfies the equation, since by (1), § 5, whatever value x might have, its product with zero would be zero. |