we conclude that the change ofBigm does not alter

since -15 divided by .5 is 3, and 15 dividedby 5 is also 3, we conclude that the change ofBigm does not alter the final answer. The samereasoning may be applled in the followingample, in which the aign of the fraction itself tsnegatlve:

When the fraction itself has a negative sigIn this example, the fraction may be enclosedIn parentheses temporarily, for the purpose ofworking with the numerator and denominatoronly, Then the sign of the fraction is appliedseparately to the result, as follows

All of this can be done mentally.

Hf a fraction has a negative sign in one of thethree sign positions, this sign may be moved toanother position. Such an adjustment is an ad-vantage in some types of complicated expressions involving fractions. Examples of thisype of sign change follow

In the first expression of the foregoing exle, the algn of the numerator is positivenderstood) and the sign of the fraction isor the umrat or and the d igm f hedetor, Observe that the slgm changes in each casevolve a patr of signs. Thig leads to the lawof aigns for fractioe thresigma d a fraction may be changed without altering the value of the fractionAXIOMS AND LAwSAn axiom is a gelf-evident truth. It isnot requre proo ersally accepted that it doesat "a straighthe shortest distancen axom fromretry. One tends to accept the traxiom without proof, because anything which isaes and relationshlps and then developing logl-cal conclusions from the definitionsAXIOMS OF EQUALITYThe four axioms of equalitywhicstated as followe1. I the same quantity is added to each dd出体四时情antities, the resultngequals are added to equals, the results areety () to both mmpi: bf hd to tw ng me auanwe obtain two sums which are equal:

2. I the same quantity is subtracted fromeach of two equal quantities, the resulting quan-tities are equal. This is sometimes stated afollows: If equals are subtracted from equals,the results are equal. For example, by subtracting 2 from both sides of the following equa-tion we obtain results which are equal

3. If two equal quantities are multiplied bythe same quantity, the resulting products areequal. This is sometimes stated as follows: Hequals are multiplied by equals, the productsare equal. For example, both sides of the following equation are multiplied by -3 and equalresults are obtained:

4. If two equal quantities are divided by theamequantity, the resulting quotients are equal.This is sometimes stated as follows: If equalsare divided by equals, the results are equalFor example, both sides of the following equation are divided by 3, and the resulting quotientsare equal

These axioms are especially usefletters are used to represent numbers. lf weknow that 5x a-30, for instance, then dividingboth 5x30 by 5 leads to the conclusionthat xuLAWS FOR COMBINING NUMBERSNumbers are combined in accordance withthe following basic laws:1. The associative laws of addition and mul-implication2. The commutative laws of addition andmultiplication.3. The distributive law

Associative Law of AdditionThe word"assocIative"suggests associatioor grouping. This law states that the sum ofthree or more addends is the same regardlessin which they are grouped. F(example, 6+3+1 is the same as 6+(3+ 1)or(6+3)+1.This law can be applied to subtraction bysigns are treated as number signs rather thanerational signsof thedends can be negative numbers. For exam4·2 can be rewritey the associative law, this is the sa6+【(-4)+(-2)]or[6+(-4)]+(-2)However, 6-4-2 is not the same as 6-(4-2);applying the associative law of additAssociative Law of MultiplicationThis law states that the product of three ormore factors is the same regardless of themanner in which they are grouped. For e3.2 is the same as (6.3).2 orFo(-2) is the same6·(-4)(-2)or6·[(4)·(-2)lCommutative Law of AdditionThe word"commute" means to change, ststates that thtwo or more addends is the same regardless of4+3+2 is thd to subtraction byanging signs sns of oper3-2 is chang5+(-3)+(-2), which is the s5+(2)+(-3)or(3)Commutative Law of MultiplicationThis law states that the product of two ormore factors is the same regardless of therder in whexample,3·4·5 is the same as5:3:4or

4·3·5. Negative signs require no apecialtreatment in the application of this Law. Forexample,2·(-4)·(-3)is(-4)(-3)·2or(-3)·2·(-4)Distributive LawThis law combines the operations o additionand multiplication, The word"distributive"refers to the distribution of a common multiplieramong the terms of an additive expressioFor example,

2(3+4+5)·22·4+2:510To verify the distributive law, we note that2(3+4+5)is the same as 2(12)or 24. Also6+8+10 Lg 24. For application of the dis.tributive law where negative signs appear, thefollowing procedure is recommended2)

The emphasis in prevous chapters of thiscourse has been on integers (whole numbers).In this chapter, we turn our attention to num-bers which are not integers. The simplestof number other than an integer is a coMMINFRACTION, Common fractions and integerstogether comprise a set of numbers called theRATIONAL NUMBERS: this set is a subset ofthe set of real numbersThe number line may be used to show therelationshlp between integers and fractionsFor example, if the interval between 0 and 1 ismarked off to form three equal spaces( thirds)then each space so formed is one.thi rd of thetotal interval. If we move along the number linefrom o toward 1, we will have covered two ofthe three "thirds" when we reach the secondmark. Thus the position of the second markrepresents the number 2/3.(See flg. 4-1.)