Algebra - Fundamental Principles Assumed

ALGEBRA - FUNDAMENTAL PRINCIPLES ASSUMED There are certain axioms which are often set forth in textbooks, as the statement "If ab=p, then a=p/b," but others, equally important from the standpoint of logic are properly assumed tacitly. Such for example, are the statements that a+b=b+a, the commutative law of addition; that ab=ba, the commutative law of multiplication; that a+b+c= (a+b)+c=a+(b+c), the associative law of addition; that abc=(ab)c=a(bc), the associa tive law of multiplication; and that a(b+c) =ab+-ac, the dis tributive law. Such axioms and laws seem so evidently true to the mind of the child that any exhaustive discussion of them in the early years is profitless. They may or may not be universally true ; in fact, some of them are not ; but they are true in the immediate and limited field of the beginner's study.

Laws of Equations.

Asstated above, the axiom "tf ab=p, then a=p/b" is often set forth in elementary textbooks, but it ceases to have any meaning for the pupil if b=o. Similarly the statement "If a>b, then na>nb" ceases to be true if n is o or negative; and the statement "If a=b, then V/ a= `/ b" also fails unless it is understood which of the nth roots in each case is to be taken, requiring a limitation upon the meaning of the symbol V. These laws are usually stated as axioms, with a statement of the limitations to be placed upon them in the elementary field.

(See MATHEMATICS, FOUNDATIONS OF.) Validity of Definitions.—Justas laws of operation which appear axiomatic to the beginner are seen to be wanting in uni versal validity as mathematics broadens, and as the laws of equa tions which are true in the early stages need close scrutiny when we leave the domain of the positive integers -0, +2, ..., -}-n (finite), so the definitions of the early years of instruction soon cease to have meaning, even before the pupil leaves the elementary stage. For example, ab does not mean that b is taken a times unless a is a positive integer; "minus a times" has no meaning unless and until the primitive use of "times" is changed, and similarly when a is equal to 3, to V2, to log 7 or to V —3 ; and an does not mean that a is taken n times as a factor when n ceases to be a positive integer. For this reason rigid definitions are not desirable.

Growth Imposes Consideration

of Limitations.—The teacher of elementary algebra should early recognize that, as the science grows or as the pupil's knowledge increases, the early laws, axioms, definitions, postulates and significance of symbols constantly require further consideration of their limitations. For example, the statement that must have the limita tion imposed that m and n are positive integers and that m > n, unless negative and fractional exponents are admitted.

It is desirable that even the beginners in the study of algebra should see the necessity for maintaining the fundamental laws as the meaning of terms and symbols is broadened. For example, it would be unfortunate if a fractional or a negative exponent were to be so defined that aman would cease to be equal to am+n, or that multiplication were to be given such a meaning that ab should not be equal to ba. As the pupil progresses in his study of mathematics he will meet with branches in which some of these laws cease to be valid, but his maturity will then permit of the necessary modifications.