Definitions. If ac=b, the process of finding c when a and b are given is called division. b is the dividend, a the divisor, b c the quotient, and we write b÷ a = c, or c. For the case, α = 0, see §§ 24, 25. α If a • c = a, a ÷ 0, then the number c is called unity, and is written 1. That is, α α = 1. Multiplying both sides of the equality ac, which by hypothesis equals b. b aac, α number and then multiplying by the same number gives as a result the original number operated upon. As a Axiom IX is called the uniqueness axiom of division. direct consequence of this axiom we have: If equal numbers are divided by equal numbers, the quotients are equal numbers. 12. Axioms I, IV (in case the subtrahend is not greater than the minuend), V, and IX underlie respectively the processes of addition, subtraction, multiplication, and division, from the very beginning in elementary arithmetic. Axioms II, III, VI, VII, and VIII are also fundamental in arithmetic, where they are usually assumed without formal statement. E.g. Axiom VIII is used in long multiplication such as 125 × 235, where we multiply 125 by 5, by 30, and by 200, and then add the products. 13. Negative Numbers. Axiom IV, in case the subtrahend is greater than the minuend, does not hold in arithmetic because of the absence of the negative number. This axiom therefore brings the negative number into algebra. We now proceed to study the laws of operation upon this enlarged number system. In the Elementary Course concrete applications were used to show that certain rules of signs hold in operations upon positive and negative numbers. We shall now see that the same rules follow from the axioms just stated. 14. Definitions. If a + b = of a and a the negative of b. 0, then b is said to be the negative If a is a positive number, that is an ordinary number of arithmetic, then b is called a negative We denote the negative of a by number. a + ( − a) = 0. a and a. Hence, a have the same absolute value. b is positive, then a is said to be greater than b. to be less than b. This is written ab. If a b = 0, then b, and if a = b then a α = b=0. See § 6. THEOREMS ON ADDITION AND SUBTRACTION Definition. A theorem is a statement to be proved. A corollary is a theorem which follows directly from some other theorem. 15. Theorem 1. Adding a negative number is equivalent to subtracting a positive number having the same absolute value. That is, a+(-b) = ab See § 48, E. C. Proof. Let a+(-b) = x. (1) Such a number x exists by Axiom I. (2) a + (b) + b = a + [( − b) + b] = a + 0 = a. (3) From (2) and (3) by § 3, x + b = a. (4) From (4), by the definition of subtraction, § 6, a − b = x. (5) From (1) and (5) by § 3, a + (-b) = α b. It follows from theorem 1 that either of the symbols, +(— b) orb, may replace the other in any algebraic expression." 16. Corollary. A parenthesis preceded by the plus sign may be removed without changing the sign of any term within it. See § 28, E. C. For, since by the theorem b α = = b + ( − a), each subtraction is reducible to an addition, so that the associative law, § 5, applies. Thus a+ (b −c + d) = a + [b + (−c) + d] = a + b + (−c)+d=a+b-c+d. Hence an expression may be inclosed in a parenthesis preceded by the plus sign without changing the sign of any of its terms. 17. Theorem 2. Subtracting a negative number is equivalent to adding a positive number having the same absolute value. That is, From (1) by § 2, a-(-b)+(-b) = x + (-b). From (2) by §§ 6 and 15, See § 60, E. C. (1) (2) (3) (4) (5) a=x+(-b)=x-b. a+b=x. Hence by the definition of subtraction, From (1) and (4) by § 3, a-(-b) = a+b. It follows from theorem 2 that either of the symbols -(-b) orb may replace the other in any algebraic expression. 18. Theorem 3. A parenthesis preceded by the minus sign may be removed by changing the sign of each term within it. That is, a− (b = c + d) = a-b+c-d. See § 28, E. C. Adding (-b), c, and (d) to each member and using § 4, a+(-b)+c+(- d) = x + b From (4), by §§ 14, 15, From (1) and (5) by § 3, It follows from equation (6), read from right to left, that an expression may be inclosed in a parenthesis preceded by a minus sign, if the sign of each term within is changed. 19. Corollary 1. a-b-(b − a). -a+(-b) = - [a + b]. For by §§ 15 and 4, Hence by § 18, a-b= a + ( − b) = − b + a. (1) (2) 20. Corollary 2. -a+(-b) = − (a+b). See § 48, E. C. For by § 18, -a+(-b)= [a - (-b)]. (1) Hence by § 17, (2) it follows that addition and subtraction of positive and negative numbers are reducible to these operations as found in arithmetic, where all numbers added and subtracted are positive, and where the subtrahend is never greater than the minuend. THEOREMS ON MULTIPLICATION AND DIVISION 22. Theorem 1. The product of any number and zero is That is, a0=0. zero. Proof. By definition of zero, § 6, a ⋅ 0 = a (b − b). By the distributive law of multiplication, § 10, Notice that by the commutative law of multiplication, § 8, a.0=0.a. It follows from this theorem and § 9, that a product is zero if any one of its factors is zero; and conversely, by § 11, if a product is zero, then at least one of its factors must be zero. 23. Corollary 1. 0 a =0, provided a is not zero. Since by the theorem 0= a. 0, the corollary is an immediate consequence of the definition of division (§ 11). 24. Corollary 2. represents any number whatever. That is, 0=k, for all values of k Since 00k, this is an immediate consequence of the definition of division. 25. Corollary 3. There is no number k such that &=k, provided a is not zero. This follows at once from k·0 = 0 for all values of k. From §§ 24, 25, it follows that division by zero is to be ruled out in all cases unless special interpretation is given to the results thus obtained. By §§ 2 and 26, (− a)(-b)+(-a)b = x - ab. By §§ 10, 14, 22, (-a)(b+b)=0x (2) ab. (3) |