Thus division by zero is entirely excluded from algebraic processes. Before a division can safely be performed one must be assured that the divisor cannot vanish. In the equation 4.0 2.0, if we should allow division of both sides of our equation by zero, we should be led to the absurd result 4 = 2. 8. Fundamental operations. The operations of addition, subtraction, multiplication, and division we call the four fundamental operations. Any numbers that can be derived from the number 1 by means of the four fundamental operations we call rational numbers. Such numbers comprise all positive and negative integers and fractions. Positive or negative integers are called integral numbers. 9. Practical demand for negative and fractional numbers. In the preceding discussion negative numbers and fractions have been introduced on account of the mathematical necessity for them. They were needed to make the four fundamental operations always possible. That this mathematical necessity corresponds to a practical necessity appears as soon as we attempt to apply our four operations to practical affairs. Thus if on a certain day the temperature is + 20° and the next day the mercury falls 25°, in order to express the second temperature we must subtract 25 from 20. If we had not introduced negative numbers, this would be impossible and our mathematics would be inapplicable to this and countless other everyday occurrences. 10. Laws of operation. All the numbers which we use in algebra are subject to the following laws. Commutative law of addition. This law asserts that the value of the sum of two numbers does not depend on the order of summation. Symbolically expressed, a+b=b+a, where a and b represent any numbers such as we have presented or shall hereafter introduce. Associative law of addition. This law asserts that the sum of three numbers does not depend on the way in which the numbers are grouped in performing the process of addition. Symbolically expressed, a + (b + c) = (a + b) + c = a + b + c. Commutative law of multiplication. This law asserts that the value of the product of two numbers does not depend on the order of multiplication. Symbolically expressed, a. b = b. a. Associative law of multiplication. This law asserts that the value of the product of three numbers does not depend on the way in which the numbers are grouped in the process of multiplication. Symbolically expressed, (ab) c = = a. (b. c): = a.b. c. Distributive law. This law asserts that the product of a single number and the sum of two numbers is identical with the sum of the products of the first number and the other two numbers taken singly. Symbolically expressed, a · (b + c) = a⋅ b + a. c. All the above laws are readily seen to hold when more than three numbers are involved. 11. Integral and rational expressions. A polynomial is integral when it may be expressed by a succession of literal terms, no one of which contains any letter in the denominator. Thus 4 x5 x32x2x + 1 is integral. The quotient of two integral expressions is called rational. 12. Operations on polynomials. We assume that the same formal laws for the four fundamental operations enunciated in §§ 2-6 and the laws given in § 10 hold whether the letters in the symbolic statements represent numbers or polynomials. In fact the literal expressions which we use are in essence nothing else than numerical expressions, since the letters are merely symbols for numbers. When the letters are replaced by numbers, the literal expressions reduce to numerical expressions for which the previous laws have been explicitly given. 13. Addition of polynomials. For performing this operation we have the following RULE. Write the terms with the same literal part in a column. Find the algebraic sum of the terms in each column, and write the results in succession with their proper signs. When the polynomials reduce to monomials the same rule is to be observed. 4. 9(a+b)6(b+c) + 7 (a + c); 4 (b+c) - 7 (a + b) — 8 (a + c); (a+c) (a + b) + (b + c). 14. Subtraction of polynomials. For performing this operation we have the following RULE. Write the subtrahend under the minuend so that terms with the same literal part are in the same column. To each term of the minuend add the corresponding term of the subtrahend, the sign of the latter having been changed. It is generally preferable to imagine the signs of the subtrahend changed rather than actually to write it with the changed signs. EXERCISES 1. From a2b2 – 3 a2b + 8 ab + 6b subtract 9 a2b2 – 6 ab + 4 a2b + a. Solution: a2b2 - 3 a2b+ 8 ab + 6b 9a2b2+4a2b - 6 ab -8a2b27a2b + 14 ab + 6 b 4 abx. cm + dn + (b − a) q and (a — b) q − (a + d) n − cm. 15. Parentheses. When it is desirable to consider as a single symbol an expression involving several numbers or symbols for numbers, the expression is inclosed in a parenthesis. This parenthesis may then be used in operations as if it were a single number or symbol, as in fact it is, excepting that the operations inside the parenthesis may not yet have been carried out. RULE. When a single parenthesis is preceded by a + sign the parenthesis may be removed, the various terms retaining the same sign. When a single parenthesis is preceded by a sign the parenthesis may be removed, providing we change the signs of all the terms inside the parenthesis. When several parentheses occur in an expression we have the following RULE. Remove the innermost parenthesis, changing the signs of the terms inside if the sign preceding it is minus. Simplify, if possible, the expression inside the new innermost parenthesis. Repeat the process until all the parentheses are removed. It is in general unwise to shorten the process by carrying out some of the steps in one's head. The liability to error in such attempts more than offsets the gain in time. 5. x2 y2 [4x+3(y-9x (y-x)) - 9y (x − y)]}. {5b [a — (3 c − 3 b) + 2c - 3 (a 2b — c)]} 16. Multiplication. It n terms is customary to write a a = a2; We have then by the associative law of multiplication, § 10, a2 · a3 = (a · a) (a · a · a) = a · a · a · a · a = a3, Equation (I) asserts that the exponent in the product of two powers of any expression is the sum of the exponents of the factors. Hence we may multiply monomials as follows: |