HIGH SCHOOL ALGEBRA ADVANCED COURSE CHAPTER I FUNDAMENTAL LAWS 1. We have seen in the Elementary Course that algebra, like arithmetic, deals with numbers and with operations upon numbers. We now proceed to study in greater detail the laws that underlie these operations. THE AXIOMS OF ADDITION AND SUBTRACTION In adding numbers we assume at the outset certain axioms. 2. Axiom I. Any two numbers have one and only one sum. Since two numbers are equal when and only when they are the same number, it follows from this axiom that if a=b and cd then a +c=b+d. For if a is the same number as b, and c is the same number as d, then adding b and d is the same as adding a and c, and by Axiom I the sums are the same and hence equal. Therefore from Axiom I follows the axiom usually given : If equal numbers be added to equal numbers, the sums are equal numbers. Since Axiom I asserts that the sum of two numbers is unique, it is often called the uniqueness axiom of addition. 3. If ac and b =c then ab, since the given equations assert that a is the same number as b. Hence the usual statement: If each of two numbers is equal to the same number, they are equal to each other. 4. The sum of two numbers, as 6 and 8, may be found by adding 6 to 8 or 8 to 6, in either case obtaining 14 as the result. This is a particular case of a general law for all numbers of algebra, which we enunciate as Axiom II. The sum of two numbers is the same in whatever order they are added. This is expressed in symbols by the identity: a+b=b+a. [See § 37, E. C.*] Axiom II states what is called the commutative law of addition, since it asserts that numbers to be added may be commuted or interchanged in order. Definition. Numbers which are to be added are called addends. 5. In adding three numbers such as 5, 6, and 7, we first add two of them and then add the third to this sum. It is immaterial whether we first add 5 and 6 and then add 7 to the sum, or first add 6 and 7 and then add 5 to the sum. This is a particular case of a general law for all numbers of algebra, which we enunciate as Axiom III. The sum of three numbers is the same in whatever manner they are grouped. In symbols we have a+b+c=a+ (b+c). When no symbols of grouping are used, we understand a + b + c to mean that a and b are to be added first and then c is to be added to the sum. Axiom III states what is called the associative law of addition, since it asserts that addends may be associated or grouped in any desired manner. It is to be noted that an equality may be read in either direction. Thus a+b+c=a+(b+c) and a + (b + c) = a + b + c are equivalent statements. 6. If any two numbers, such as 19 and 25, are given, then in arithmetic we can always find a number which added to *E. C. means the Elementary Course. the smaller gives the larger as a sum. tract the smaller number from the larger. That is, we can sub In Algebra, where negative numbers are used, any number may be subtracted from any other number. That is: Axiom IV. For any pair of numbers a and b there is one and only one number c such that a + c = b. The process of finding the number c when a and b are given is called subtraction. b is the minuend, a the subtrahend, and c the remainder. This operation is also indicated thus, bac. If a + c = a, then the number c is called zero, and is written 0. That is, a + 0 = a, or a а 0. Adding a to each member of the equality b α = c, we have b-a+a=c+a, which by hypothesis is equal to b. Hence subtracting a number and then adding the same number gives as a result the original number operated upon. Α Axiom IV is called the uniqueness axiom of subtraction. direct consequence is the following: If equal numbers are subtracted from equal numbers, the remainders are equal numbers. THE AXIOMS OF MULTIPLICATION AND DIVISION 7. Axioms similar to those just given for addition and subtraction hold for multiplication and division. Axiom V. Two numbers have one and only one product. This is called the uniqueness axiom of multiplication. It is a direct consequence of this axiom that: If equal numbers are multiplied by equal numbers, the products are equal numbers. 8. The product of 5 and 6 may be obtained by taking 5 six times, or by taking 6 five times. That is, 5.6 6.5. This is a special case of a general law for all numbers of algebra, which we enunciate as Axiom VI. The product of two numbers is the same in whatever order they are multiplied. This axiom states what is called the commutative law of factors in multiplication. 9. The product of three numbers, such as 5, 6, and 7, may be obtained by multiplying 5 and 6, and this product by 7, or 6 and 7, and this product by 5. This is a special case of a general law for all numbers of algebra, which we enunciate as Axiom VII. The product of three numbers is the same in whatever manner they are grouped. The expression abc without symbols of grouping is understood to mean that the product of a and b is to be multiplied by c. This axiom states what is called the associative law of factors in multiplication. Principles III and XV of E. C. follow from Axioms VI and VII. 10. Another law for all numbers of algebra is enunciated as Axiom VIII. The product of the sum or difference of two numbers and a given number is equal to the result obtained by multiplying each number separately by the given number and then adding or subtracting the products. In symbols we have a(b + c) = ab + ac and a(b − c) = ab — ac. Axiom VIII states what is called the distributive law of multiplication. When these identities are read from left to right, they are equivalent to Principle IV, E. C., and when read from right to left (see § 5) they are equivalent to Principles I and II, E. C. In the form a(b±c) = ab± ac this axiom is directly applicable to the multiplication of a polynomial by a monomial, and in the form ab ± ac = a(b±c), to the addition and subtraction of monomials having a common factor. 11. Axiom IX. For any two numbers, a and b, provided a is not equal to zero,* there is one and only one number c such that a. c = b. *The symbol for the expression a is not equal to zero is a 0. |