Notes on Algbra G Chrystal

8/11/2019 Notes on Algbra G Chrystal

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~x + x~y = ~x (1 + ~y) + x~y = ~x + ~x~y + x~y = ~x + ~y (x + ~x) = ~x + ~y.

NOTES ON UNDERSTANDING THIS LAST COMPLICATED RESULTS AND IMPLICATION ABOUT EXPANDING BEFORE SIMPLIFYING:1.1 Thus the very first step shows how a single variable may need to be expandedinto several. Thus in the first step ~x is expanded to become ~x(1+ ~y). Lateron this allows us to multiply out to get ~x + ~x~y + x~y, after which we then group the last two terms to get ~x + ~y (x + ~x), and finally ~x + ~y.1.2 Thus the reason we expanded ~x to become ~x(1+ ~y), is so that we can regroup it with another term later on. In fact any single variable x at all can be expanded to become x(1 + y), because of combining the absorption property, with the elemento absorbant 2.16. 1.2.1 ON WHY THIS IS SO POWERFUL EXPLAINED DEF 13!:The power of this lies in the fact that since any variable x can be written asx*1, and taken together with the fact that 1 can be written as (1 + y), where yis any varaible, we can strategically choose our variable "y", such that we planto re-group it with other terms in the expression and use another simplifying property we may know (ie the distributive property)1.3 DEF 13! ON IMPLICATION IN ALGEBRA AS WRITING ANY VARIABLE X AS X*1 AND USINGANOTHER PROPERTY AND REGROUPING AS A STRATEGY IN ALGEBRA: ....Thus another "operation" or stragegy in algebra is writing any variable x as x*1 instead, and ifwe have an expression equal to 1, we can subtitute that into x*1 and then expand as we may need. 1.3.1 Thus parentheses can also be seen as an "operation" inst

ead of just grouping symbols. 1.3.2 This finding equivalent expression to 1 andsubstituting it into x*1 is likely to be used mostly with the allowing of regrouping and then the distributive property (rather than some other property) to help simplify. Identities such as trig identities are likely to be used with this method as well. 1.3.3 One could also use the additive identity x + 0 and find an expression equivalent to 0 and do the same thing.

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In the first place, I have kept the fundamental principles

of the subject well to the front from the very beginning; I may instance the treatment of the derivation ofequations in Chapter VI., a subject usually dealt with as ifit were a separate science.

NOTE: Thus says deals with the fundametnal principles in the very beginning.

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A word regarding the first steps in teaching Algebra.I hold, in common, I believe, with most teachers of Mathematicswho have deeply considered their business, thatthe teaching of Algebrathat is, of the science of arithmeticaloperationsshould commence with the teaching ofArithmetic itself.

NOTE IMPORTANT: Thus says that algebra is the science of arithmetical operations.


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For example, the beginner should notbe allowed simply to learn that 3 and 5 together make up8, and to write mechanically the scheme35Ans, 8and the like ; but ought as well to be made to write3-1-5 = 8, or even -f- 3 -f 5 = -f 8. It should also bepointed out to him that 3 + 5 = 8 = 5 + 3; that 3 + (3 - 2)= 3 + 3 - 2 ; and so on. The laws herein involved neednot be named to him at first by their long forbiddingnames ; but they should be illustrated by means of concreteinstances, and especially by geometric figures. Aftera course of this kind, extended over the earlier years of hisarithmetical training, the learner should be made to statethe solutions of the little problems which he works asconcatenations of numerical operands and operating symbols.The next stage is to learn to generalise a problemby substituting letters or hypothetical operands for theactual numbers of earlier essays. Then, and not till then.

NOTE IMPORTANT: Thus implies that the first step in learning algebra should be

experimentin withconcrete arithmetic problems and those equalities >before< even thinknig about substituting in letters.Also certain results should be observed and illustrated without necessarily giving them names. (Comenius things before names).

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tion to be stated presently ; and the former of these two,

owincj to the readiness with which it can be either written orprinted, is now coming much into use. The fractional notation- has certain advantages, but it is difficult to print and takes upbmore room than the others. We shall have occasion to remarklater on that the fractional notation is not in all cases exactlyequivalent to -^ or to / ; it has in some cases the effect of abracket, e.g. is not the same thing as a-^h + c or ajb + c,h + cbut is really equivalent to a-^{b + c) or aj{b + c).

NOTE ON FRACTIONAL AND DIVISION NOTAITON 12!: Thus says fractional and divisionnotation is >almost< equivalent, but not in all cases.

Such as a/b+c , in this case it is not the same thing as a divided by b+ c but rather isthe same thing as a divided by (b+c), and thus parentheses must be included.

The fact that when the dividend and divisor are integralnumbers, as in |^, the fractional and divisional notation are notdistinguishable is of no consequence, because in Algebra, wheneverwe regard ^ not as a whole, but with respect to 3 and 4separately, we regard it as a quotient, and, on the other hand,


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when ^ is regarded as a whole, i.e. as an operand, it has thesame abstract properties whether we consider it as " threefourths" or as three divided by four : e.g. from either point ofview I X 4 = 3 ; and this may be regarded as the fundamentalor defining property of ^.

NOTE 12!: Says whether we regard it as a quotient with 3 and 4 separate, or ifwe regard it as a whole as an operand, it has thesame abstract properties both ways whether we consider it as "three-fourths" oras three divided by four.Says from either point of view 3/4 * 4 = 3.

4. Whatever concrete or other meaning the learner mayhave hitherto attached to addition and subtraction, he willsee that the two operations are mutually Inverse in the sensethat, if we first add any quantity and then subtract the same,or first subtract any quantity and then add the same, the resultis the same as if we had not operated at all ; that is to sayrt-f-6 5 = a, a b + b = a.Multiplication and division are inverse to each other inexactly the same sense, viz. we have

axb-^b = a, a-7-b xb = a.Rightly considered, the above remark leads us to see thatwhen addition and multiplication are fully defined by concreteinterpretation or otherwise, the nature and laws of their inverses,subtraction and division, are determined (see A. Ch. I.).*

NOTE IMPLCTN ON ADD/SUB MULT/DIV INVERSE OPERATIONS 11.75!: Says adidtion and subtraction are inverse in the sense that they undo each other's operation.Thus if you subtract and add by the same quantity it's as if you had not operated at all.The same thing goes with multiplication and division.

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CHAPTER IIFUNDAMENTAL LAWS OF ALGEBRA LAWS OF COMMUTATIONAND ASSOCIATION FOR ADDITION AND SUBTRACTION 11. It will be advisable for our present purposes that the

learner should attach some convenient concrete meaniugs toaddition and subtraction. In the first instance, we shall usethe notion of credit and debit ; later we shall employ a moreimportant but perhaps less familiar illustration.We shall suppose that + a means a pounds to be paid bysome debtor to a merchant A, and that - h means 6 j)o^^^^s tobe paid by A to some creditor of his. It will facilitate mattersif we suppose that A collects his debts and pays his creditorsthrough an agent B, who may be supposed to have a certainamount of spare cash of his own, in case it may happen on his


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rounds that he may either have more to pay out than to collect,or that he may, in the first instance, have to pay out somemoney before the money he has to collect for A has come in tocover his outlay.

The chain of additions and subtractions +a-\-h - c-\- d c -fwill then represent a collected, b collected, c paid out, dcollected, etc., in a particular order on a certain round.+ h + a-ird - c - e -f will evidently, so far as A is concerned,represent the same final result ; it might indeed representsimply a different way of arranging B's round of businesscalls. There is, in fact, from our present point of view, noreason why any or all of the creditors should not be visitedfirst, B in the meantime paying out of his own cash, andthen we should have fur the symbolic representation of B'sround c e-f+a-\-h-\-d.

NOTE IMPORTANT ON THINKING OF ADDITION AND SUBTRACTION AS CREDIT AND DEBIT 12!:Thus not only thinking of negative numbers as debit, but thinkingof the >operations< of addition and subtraction as credti and debit.1.1 Says there is no reason why any or all of the creditors should not be visited first. Thus this reasoning is the law of

commutation.

"We are thus led to see that in a chain of additions and

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subtractions the order of the oi^erations is indifferent^ ijrovided eachajperand carry ivith it the ojjerating symbol + or originallyattached to it.

It will be immediately perceived that, although in whatprecedes we have not gone beyond the limits of common sense,we have already transcended the boundaries of Arithmetic asordinarily understood ; for, although +1 + 4-3 is at everystep a perfectly intelligible arithmetical sequence, +1-3 + 4,which from the ]3oint of view above explained is the samething, directs that 3 shall be subtracted from 1, and 3 + 1+4,according to the ordinary arithmetical notions, has no meaningat all. To this point we shall return hereafter ; all thatwe need note at present is that the notion of debit andcredit has led us to a generalisation of the operation of

subtraction.

NOTE: Thus says it's the notion of debit and credit that has led us to a generalization of the operation of subtraction.

12. Since two separate debts of cl each, both supposed


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good, are from the merchant's point of view the same thing asa single debt of


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NOTE: Thus says this is the law of asociation stated fully for either a positive in front of brackets or for a negative in front of brackets.

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16. If we consider any two quantities which differ by cc, saya and a + X, we have-\- (a + x) a= +x,+ a {a + x)= X.If now we make x smaller and smaller, a + x becomes moreand more nearly a ; and therefore if, as is usual in aritlimetic,we denote a quantity which is smaller than any assignable quaiitityby 0, we have+ a-a= +0 ;+ a - a= - 0.These two equations may be taken as the definition of asan operand in Algebra (so far as it is admissible in that capacity).We see at once that has the special property possessed by noother operand, that+ = - 0.This agrees perfectly w4th arithmetical notions ; for we have

h + = b = b-0 ;and this again is consistent with our algebraical notion of themutually inverse character of addition and subtraction ; for if4- stand for + a a, we have b + = b + a a = b (see 4).

NOTE: Thus in algebra, 0 has this special property emphasized, the additive identity property.

In the first place, they lead us to the idea of the cumulationof operative symbols with the law that the concurrence of two likesymbols, i.e. + ( + a) or ( a), gives the direct symbol, viz. in

each case +a, wliile the concurrence of two unlike symbols, i.e.- ( + a) or + ( - a), gives the inverse symbol, viz. in each case- a.

NOTE: Thus two like symbols give addition and two unlike symbols give subtraction.

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two parts, viz. the removal of the bracket and the determinationof the sign. Thus we may first write- ( - a + & 4- c) = - ( - ft) - ( + 6) - ( + c);then apply the " law of signs," which gives- ( - ft) = + ft, -( + &)=-&, - ( + c) = - c.The cmuiilation of the signs + and - thus suggested may becarried to any extent by repeated appUcations of the four fundamentalcases, +{ + a)= +a, +{-a)= -a, -{ + a)= -a, -(-)=+.


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we have+ =+( + )= +( + ( + a))=+( + (- (-ft)));-a=+{-a)=+i-{ + a))=+{-{-{-a)));and so on ; the rule for reduction to a single operation being obviouslythat the reduced sign is + , if there be no - or an even number of -signs in the sequence ; and - , if there be an odd number of - signsin the sequence.

NOTE IMPLCTN FR FASTER DETERMINATION IF RESULT IS POSITIVE OR NEGATIVE 12.25!:Thus says the cumulation of signs can be carried to any extent by repeated applicationof hte four fundamental cases.1.1 Says that it's reduced to plus if there is no "-" or if there is an even number of "-". It is reducedto "-" if there is an odd number of "-" signs in the sequence.

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CHAPTER III

fundamental laws of algebraThe Laws of Commutation and Association for MultiplicationAND DivisionLaws of Indices 22. The reader who has been rationally taught the fundamentalprinciples of Arithmetic is already aware that in a chainof multiplications and divisions, unbroken by additions or subtractions,the order in which these operations are performed isindifferent.Thus, for examplex3x2x6= X3x6x2=x6x2x3, etc.xl6-r2x8= xl6x8-^2= x8-^2xl6= -=-2 X 8 X 16, etc.

'v^J2-'v^l4-^l-l-l vJL-^-JLvl-i-l fife 3 "5'~2 3'^4-"5~ ''^'and, in general= X c X a-^d-^b,= -i-h X ax c-^d, etc.It will be observed that we have here a law formally identicalwith the Law of Commutation already stated for an algebraicsum : it is called the Law of Commutation for Multiplicationand Division, and may be verbally stated as follows :In any chain of tnultvplications and divisions the order of theconstituents is indifferent, provided the proffer sign he attached toeach operand and move with it.

Just as in an algebraic sum, when the first operation is thedirect one, in this case multiplication, the sign is usuallyomitted : thus w^e write axh-=rc instead of xa xb-=rc.

NOTE IMPLCTN FR UNDRSTNDNG ALGBRA ND CLCLTNG 12.5!: Thus just like with addition and subtraction the "+" and "-" signsfollow wherever they go and we have the metaphor of credit and debit, so with multiplication and division we can rearrange the order to whatever we likeas long as the "*" and "/" signs follow their original numbers, though the credi


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The equivalences (1), (2), (3), (4) are examples of the Law ofAssociation for multiplication and division, which may be statedthus :A chain of multi'plications and divisions associated into a singleoperand hy means of a bracket may he dissociated into the constituentoperations by removi7ig the bracket, leaving all the signsunchanged if the bracket is preceded by x , reversing each sign if thebracket is preceded by -^.Or, in symbolsX {x a^b-^cxd)= X a-^b-^cx d . (5);-^{xa-^b-^cxd)=-^axbxc-^d . (6).

NOTE IMPLCTN FR UNDERSTNDNG ALGEBRA AND CALCULATION ON LAW OF ASSOCIATION FOR MULTIPLICATION AND DIVISION 12.25!: Thus a chain of mulitiplications and divisions associated into a single operandby means of a bracket, can be dissociated into the constituent operations by removing the bracket, and just like in the addition case (in which we inverse the signs inside if the parentheses is to be subtracted), if theexpression in the parentheses is to be divided (preceded by a division symbol),

we reverse each sign.1.1 This can be seen as making everything inside the parentheses a reciprocal aswell.1.2

24. As in the case of the Law of Association for additionand subtraction, we may resolve the process of dissociation formultiplication and division into two partsthe removal ofthe bracket and the determination of the signs. Thus (5) and(6) could be writtenx(xa-^6-^cxf?)=x(xa)x(-^&)x(^c)x(xf?);-^ ( X a -^ 6 -^ c X d) = ^ ( X a) -f ( -f 6) -^ ( -^ c) -f- ( X cQ :

with the following Law of Signs :X ( X rt) = X a, X ( -^ ft) = -^ a,-7- { X a) = -7- a, ~{-^a)= X a:

NOTE: Thus as with addition and subtraction, two like signs make a multiplcation and two different signs make a division.

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that is to say, the concurrence of like signs gives the directsign ( X ), the concurrence of unlike signs the inverse sign ( -i- ).This dissection of the Law of Association is of less importancein the present case ; because there is in ordinary algebra, at leastas yet, no important development of multiplicative and divisivequantity as there is of additive and subtractive quantity, whichgave us the notion of so-called algebraic quantity.


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From the definition of 1 just given, and the laws ofassociation and commutation for multiplication and division,and the mutual relation between multiplication and division,we have6x l=hx{xa-â) = bxa-â = b.b-^l=b-i-(xa-ra)==b-âxa=:b.Alsoxl=x(x-ra) = xa-â = -r-axa =-^(xc^-â);that is to say, x 1 = 4-1, which is analogous to + = - 0.

NOTE IMPORTANT IMPLCTN FR ALGEBRAIC OPERATIONAL DEFINITION OF "1" ANALOGOUS TO ALGEBRAIC OPERATIONAL DEFINITION OF "0" 12.5!: Thus when we said +a -a = 0. Wewere >operationally defining< 0 by this property. Similarly with multiplicationand division we >operationally define< 1 to be *a/a, except the only differenceis that we restrict the a in a/a to be >different from< 0.1.1 Thus using the associative property any (*a/a) can be treated as a 1.1.2 In addition we also have the result that *1 = /1, just like we had +0 = -0.

27. It is interesting at this stage to notice that the lawsfor the transformation of fractions by multiplying or dividingnumerator and denominator by the same quantity, and formultiplying and dividing fractions, are instances of the two

laws which we have just been discussing. Thus3x 5_34 X 5 ~ 4may be established as follows. We have|^ = (3X5)-(4X6),since from the algebraical point of view the two sides ofthis equation are only different notations for the same function.Next, by the law of association (3x5)4-(4x5) = 3x54-44-5;and, by the law of commutation 3x54-44-5 = 3-^4x54-5.

NOTE ON WHY MULTIPLYING THE TOP AND BOTTOM OF FRACTIONS BY THE SAME THING LEAVESTHE FRACTION EQUIVALENT: Thus this section shows through the law of association and commutation for multiplication and division >why< when we multiply the topand the bottom of fractions by the same thing, we get the same thing.

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27 OPERATIONS WITH FRACTIONS 27Finally, from the definition of multiplication and division asmutually inverse operations 3-^4x5^5 = 3-^4,whicli establishes our result, since 3 -^ 4 and | are the samething.


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any algebraic operands whatsoever, simple or complex. Thus,for example a X (1) a . a a6x(-l)^6' '^'^' ~^^1){x-l){x+2) _x-\^^ (a; + 3)(x+2)~x + 3are cases of (1) ; andx-l^x+2_x-\ a;-2_(a;- l)(-2)a; + 3~cc-2~a; + 3 a + 2 ~ (x + 3)(x + 2)is a case of (4).EXERCISES IV.Simplify each of the following as much as you can :

NOTE IMPORTANT ON WHY TWO NEGATIVES IN A FRACTION ARE A POSITIVE AND BASIC CANCELING 12!: Thus shows that a negative in both the numerator and denominator is apositive becausethey are divided by each other and by the law of signs seem to "cancel" and have

a positive.

Monomial Integral FunctionsLaws of Indices forIntegral Exponents 28. Technical Use of the Word Term.The word termis often used in Algebra in a technical sense, which it will beconvenient here to define. A function, or 'part of a function ofany operands which involves only multiplication and division, andnot addition and subtraction, is called a term. Thus 3x4,axh-^c, arffi Sa^ are called terms. On the other hand, a^ + b'^,

ab c'^ja are not in themselves terms ; but + a^ and + 6^ are theterms of a^ _]_ 52 . g^j^^i _j_ ^^j ^j^^i _ ^2^^^ ^]^g terms of ab c-ja. Afunction which consists of a single term is called a Monomial ;a function which is the algebraic sum of two terms a Binomial;and so on.

NOTE IMPORTANT IMPLCTN ON "TERM" MEANS OPERANDS INVOLVING >BOTH< MULTIPLICATIONOR DIVISION ONLY 12.5!: Thus says a term in algebra means any operands which involves only multiplication and division and notaddition and subtraction. Thus in ab - c^2/a, ab is a term, and c^2/a >taken as

a whole including the division< is one term as well.

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29. In dealing with rational terms (rational monomialfunctions), such as


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(3(1^) X (2a)3 ^ {ahcf x (a2)3 x ( _j(1),it is found convenient to arrange so that all the multiplicationsor divisions by merely numerical ojDerands shall be brought togetherand, usually, condensed into a single number ; and, inlike manner, all the multiplications and divisions by the sameletter brought together and replaced by a multiplication ordivision ])y a single power of that letter.Thus we shall presently show that the monomial (1) can bereduced to 24(1^"^ -^h'' -^c^ or 2-ia^^/h~c\

NOTE IMPLCTN ON EVEN IF A STRING OF MULTIPLICATIOSN AND DIVISIONS LOOK LIKE SEPARATE TERMS THEY ARE STILL ONE TERM AND CONDENSED TO BE THAT 12.5!: Thus in thisexample the entire thing is one term, but to see this it may be necessary to simplify all of the multiplications and divisions. To do this it's convenient toarrange by the law of commutation so that all the multiplications or divisionsby merely numerical operands shall be brought together usually condensed into asingle number. Likewise all the multiplications and divisions by the same letter are brought together and replaced by a multiplication or division by a singlepower of that letter. Thus don't view it as multiplying the operands out, thinkof it as condensing all of the multiplications by the same letter into a singleletter.

1.1 Thus the result is 24a^19/b^7c^3 which is indeed a single term all condensed to it's simplest form.

This reduction is greatly facilitated by the establishment ofrules1. For expressing the product of any powers of one andthe same base, or the quotient of two powers of one and thesame base, by means of a single power of that base.2. For expressing any power of a power of one base as asingle power of that base.

3. For expressing a power of the product of any bases, or apower of the quotient of two bases, as a product or quotient ofsingle powers of those bases.These rules, commonly spoken of as the Laws of Indices,are as follows :I. (a) a"^ X rt^^ X rt2' X . . .=a"^+^^+i'+ * '{/3) a"' -^ a'' = '-, if m > 7i;= 1 4- a^* ~ "^, if m < n.11. (a*)" = a'III. (a) (a X 6 X c X . . .)' = iO"- xh'*^xc'^x. . .;

or, in wordsI. (a) TJie jirodud of any iiowers of one and the same base is a ijowerof that base v:hose irule.c is the sum of the indices of the given j^owers.(f3) The quotient of two different 2)owers of the same base is a'power of the base ivhich is tlie absolute difference of the two powers^or unity divided by the same^ according as the index of the dividendis greater or less than the index of the divisor.


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NOTE IMPLCTN ON THE USE OF AND MOTIVATION FOR USING POWERS AND EXPONENTS 12.5!:Thus if we didn't have exponents, all of the separate multiplications would have to be written out. For example a^19 as part of the previously mentioned termwould have to bewritten out a multiplied and so we would be writing 19 "a's". This would make avery long expression and so exponents help us alot in condesning our writing.

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II. The nth power of the mth power of a base is the lanth powerof that base.III. (a) The mth power of a product of bases is the product ofthe mth poivers of those bases.(fS) The mth power of the quotient of two bases is the quotientof the mth powers of those bases.*The proof of these laws depends merely on the definition ofan integral power, viz. thata^ = axaxa'X. . . m factors,

and on the laws of association and commutation for multiplicationand division.To prove I. (a), let us consider first a special case, saya^ X a^ X a^. By the definitions of a^, a^, a^ we havea^ xa^ xa^ = (axa)x(axaxa)x{ax a),by the law of association==axaxaxaxaxaxa,where there are 2 + 3 + 2 factors ; hence, finally, by the definitionof a powerThe general proof may be stated thusa'^xa^xai^x. . .

by the definitions of a'"-, etc.= {ax ax . . . m factors)x(ax ax . . . n factors)x{ax ax . . . p factors)/N Jby the law of association= axax . .xaxax...v_^^^_^^^^_ factors ; xax ax . . 'Pn by the definition of a power

__ jm+?i+2>+ . '^* The beginner may be cautioned against the freqiient error of confusingthese laws with one another ;

NOTE ON PROOF OF LAW I OF EXPONENT ADDITION BY USING THE LAW OF ASSOCIAITON: Thus by using the law of association for mutlipclation this law can be proved.


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To prove I. (/3), consider first a particular case, say a^-â^.We havea^ -r a-.by tlie definitions of a^ and a^= (axax ax ax a)-^{axa) iby the law of association= a X a X a X a X a-â-â ',by the law of commutation= a X ax ax a-âx a-â ;by the mutual inverseness of multiplication and division= ax ax a X {5 2 factors);bv the definition of an indexIn general, by definition of a power

a"^-7-a"' = {ax a X . . . m factors)-^{ax ax . . . 7i factors);by law of association= ax a X . . . m factors-^ -^ a -r . . . n divisions.If now m>n, we have, by law of commutationa'" -â'^ = a X a X . . . m n factorsX a-â X a-âx . . . n pairs ;by the mutual inverseness of multi23lication and division

= a X a X . . . m n factors ;by the definition of a powerIf TYKUj there are more divisions than multiplications, andwe have(jin -âî = a-â X a-âx . . . m pairs-^ rt -f- rt . . . n m divisions= \-â-â . . . n - m divisions ;by the law of association= 1 -^ ((t X a X . . . 11 -m factors) j

NOTE ON PROOF OF THE LAW FOR DIVIDING FACTORS AND SUBTRACTING EXPONENTS 12!: ......


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by the definition of a power= 1 -^a-.To prove II., consider the special case (a^)^ Since (a^)^ bythe definition of a power means a^ x a^, we have(a3)2 = a^ X a^,= (a X a X a) X (a X X a) ;by law of association= ax axa . . . 3x2 factors ;by definition of a powerIn generalby definition of a power= a"* X a'" X . . . n factors ;by definition of a power= (a X rt X . . . m factors) \x{a X ax . . . m factors) > n rows;X . . . j

by law of association= a X a X . . . mn factors ;by definition of a powerAs the reader has probably now grasped the simple principlesinvolved, we give the general proof of III. (a) and III. (^) atonce.We have, by the definition of a power{axbxc . . .)"* = (axhx ex . . .))X {a xb X c X . . .) >m rows ;X . . . jby the laws of association and commutation

= {ax ax . . . m factors)X (6 X 6 X . . . m factors)X (c X c X . . . ??. factors)

NOTE ON PROOF OF RASING A POWER TO ANOTHER POWER BY MUTLIPLYING THE EXPONENTS 12!:...

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where we have in effect turned the columns of tlie first schemeinto rows in the second. Hence, fiucilly, by the definition of a


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power{axbxcx . . .)" = a"^ x Z/'" x c'" x . . .Again, by the definition of a power(a -^ 6)'" = {a-^b)x{a-r-h) X . . . m pairs;by the huv of association= a-^h X a-^hx . . . m pairs ;by the law of commutation= a X rt X . . . m factors-i- 6 -r i -^ . . . m divisions ;by the law of association= {a X ax . . . m factors) -^-Qjxhx . . . m factors) ;by the definition of a power 30. Let us now return to the monomial(3a9) X i2af ^ {ahcf x {tC-f x (~ V.Using the laws of indices, we haveby IIL (a), {2af = 2^a^ = Sa^ ;by IIL (a), {abcf = a%h^]

by II, (a2)3 = a6.by IIL (ft (^y = (a - hf =a4 -f- h\Hence(3a9) X {2cif^{ahcf x {a^f x f^Y= (3 X a9) X (8 X a3)-j-(a3 xPx c^) x a^ x {a^-^) ;by the law of association= 3 X a^ X 8 X a^ -^ cr -^h^-^c^x a^ xa^~h^ \by the law of commutation= 3 X 8 X a^ X a^ X a


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After the fundamental principles have become perfectly familiar, muchof the work can be safely carried out mentally and the writtencalculation much shortened ; but the learner should never fall intothe bad habit of quoting formulae or rules whose connection withfundamental principles lie does not perfectly understand ; by so doinghe will strain his memory and retard his ultimate progress.

NOTE IMPORTANT IMPLCTN FR SLF STDY ON REVIEW FUNDAMENTAL PURPOSES IF FALL INTO DOUBT OR PERPLEXITY 12.5!: Thus saysif ever fallen into doubt or perplexity, this model of deriving everything fromthe fundamental principles shouldbe followed. After the fundamental principles have become familiar, much of thework can be shortened.1.1 However do not fall into the bad habit of quoting formulae or rules whose connection with fundamental principlies is notperfectly understood, else will strain memory and retard ultimate progress.

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in reckoning,' the de


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The degree of the product of two integral monomials in any givenset of letters is the sum of the degrees of the two factors in thoseletters.If the quotient of one integral monomial hy another he integral,the degree of the quotient in any given set of letters is the differencebetween the degrees of the divisor and dividend in those lettei'S.Ex. The degree of lahx^y^z' in x, y, z is 9, and the degree ofSab-x^y^z in x, y, z is 6 ; the degree of the product, viz. Qa%^x^y~:^ inir, 2/, 2 is 15 = 9 + 6 ; the quotient, viz. ( ^ Jxyz, is integral so far asx, y, z is concerned, and its degree in these letters is 3 = 9-6.The theory of degree is of great importance in Algebra ; infact, degree will be found in many respects to play the samepart in Algebra as absolute magnitude does in Arithmetic.

NOTE MYBE IMPORTANT FOR UNDERSTANDING ALGBRA 12!: Says the notion and theory ofdegree is of great importance in Algebra and wil be found in many respects to play the same part in Algebra as absolute magnitude (abolsute value I think) doesin arithmetic.

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CHAPTER IVfundamental laws of algebraThe Law of Distribution 33. The primitive meaning of multiplication is repeatedaddition. Thus 8 x 3 is a contraction for 8 + 8 + 8.In our discussion of the laws of commutation and associationfor multiplication and division we considered only the casewhere the operands are absolute, i.e. merely arithmeticalquantities. The further points that arise when the operandsare algebraical quantities that is to say, absolute quantities withthe signs + or - attached are most conveniently consideredin connection with the Law of Distribution, which is the last ofthe three fundamental laws of algebraical operation.

NOTE: Thus says before we considered the commutative and associative laws without having to worry to much about the algebraic notion of the + and - signs beingattached to the numerical quantities. However when considering the law of distribution, we have to consider them algebraic quantities with the + and - signs attached.

Reverting to 8x3, let us write the product more fully as( + 8) X ( + 3), and notice that we mayalsowrite8 + 8 + 8 more fullyin the form + 8 + 8 + 8, or if we choose +8x1 + 8x1 + 8x1.

NOTE ON CONSIDERING MULTIPLICATION REPEATED ADDITION DO NOT THINK COMMUTATIVE LAW 12!: Thus yes the commutative law holds for multiplication, but when thinkingof multiplication asrepeated addition, it's best to remember the terms multiplicand and multiplier.1.1 In this instance we have (+8) * (+3). The only way to interpret this in terms of repeated addition is 8 + 8 + 8, and NOT 3 + 3..., because we're taking8 first (multiplicand) and adding it 3 times (multiplier).1.2 One of the main reasons for this distinction becomes apparent when regardingmultiplying by negative numbers as repeated subtraction.


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Remembering that + 3 is a contraction for + 1 + 1 + 1, we maytherefore write the equation 8x3 = 8 + 8 + 8 in the forms ( + 8)x( + 3)=+( + 8) + ( + 8) + (+8)= +8 + 8 + 8=+24 (1);or ( + 8) x(+l + l + l) =+8x1+8x1+8x1 (2).We thus look upon multiplication by a positive multiplieras a contraction for repeated addition.

NOTE ON BY REMEMBERING ASSOCIATIVE LAW WE EASILY GET TO DISTRIBUTIVE LAW 13!: Thus remembering that 3 >is< a contraction for (1+ 1+ 1), we easily get that8*3 = ( + 8) x (+l + l + l) = +8x1+8x1+8x1. Thus this is a very good arithmetical demonstration of the distributive law.

In like manner, it isnatural to regard multiplication by a negative multiplier as acontraction for repeated subtraction. Taking this view, we have (+8)x(-3)=-(+8)-(+8)-( + 8),= -8-8-8= -24 (3);or ( + 8)( -1-1-1) =-8x1-8x1-8x1 (4).

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Also ( - 8) X ( + 3) = + ( - 8) + ( - 8) + ( - 8),= _8.-8-8= -24 (5);or (-8X +1 + 1 + 1) =-8x1-8x1-8x1 (6).(-8)x(-3)=-(-8)-(-8)-(-8),= +8 + 8 + 8=+ 24 (7);or (-8)(-l -1-1)= +8x 1 + 8x1+8x1 (8).

NOTE ON MULTIPLICATION BY A NEGATIVE NUMBER AS A CONTRACTION FOR REPEATED SUBTRACTION AND WHY MULTIPLICATION OF TWO NEGATIVE NUMBERS EQUALS A POSITIVE 13!: Thus we see if the multiplier (the second number in the multiplication), is negative, thenwe take repeated subtractions. However if the multiplicand (the first number inthe multiplication expression) is negative, then it is repeated addition of that negative quantity.1.1 IMPORTANT ON WHY A NEGATIVE MULTIPLIED BY A NEGATIVE IS POSITIVE 13!: The expression (-8)x(-3) shows cleary by the distributive property why a negative multiplied by a negative is positive.

Thus we see that it is equal to -(-8)-(-8)-(-8) which is equal to +8 + 8 + 8 because we're subtracting negatives.

34. Consider now the case wliere the multiplier is analgebraic sum, say +8-5. To multiply + 8 by +8-5 may,according to our present view, be taken to mean : add + 8 eighttimes and subtract + 8 five times that is to say, using multipli-cation -by positive and negative multipliers as before to denoterepeated additions and subtractions respectively, we have


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( + 8)x(+8-5)=+8x8-8x5 (9).In like manner(-8)x(+8-5)=-8x8 + 8x5 (10).The equations (1), (3), (5), and (7) suggest the laws for thesign of the product of two algebraic quantities, viz. the productis positive if the factors have the same signs, negative if they haveopposite signs.

NOTE ON WHEN INSTEAD OF A SINGLE NUMBER THE MULTIPLIER IS AN ALGEBRAIC SUM 13!:Thus sayssince the multiplier positive is repeated addition, and egative is repeated subraction, in the case of +8 * +8-5, this would mean add+8 eight times, and subtract 8 five times.

IMPORTANT ON WHY A NEGATIVE MULTIPLIED BY A NEGATIVE IS POSITIVE 13!: The expression (-8)x(-3) shows cleary by the distributive property why a negative multiplied by a negative is positive.Thus we see that it is equal to -(-8)-(-8)-(-8) which is equal to +8 + 8 + 8 because we're subtracting negatives.1.1 DEF 13!: Notice that this rule of signs was illustrated with single numbers. In the case of the multiplier being an algebraic sum then, we would have to d

etermine if the result of the algebraic sum is positive or negative depending onwhichof the numbers is larger. This can be done with numbers, but when we move to letters we can't finally determine this.

The equations (2), (4), (6), (8), (9), and (10) suggest the follow-ing rule for multiplying any algebraic quantity by an algebraicsum. TFrite down all the partial products formed by multiplyingthe multiplicand hy each term of the multiplier, and determine thesign of each partial product hy the law of signs just given. Or, ingeneral symbols { + A)( + a-h-c + d)= +Aa-Ah-Ac + Ad;

(-A)( + a-6-c + rf)= -Aa + Ab + Ac-Ad (11).This process we call Distributing the Multiplier.

NOTE ON DISTRIBUTIVE LAW STATED AND THE LOGIC OF HOW AND WHY IT WORKS 12.5!: Thus this is simply saying the distributive law and how it works.Basically if the multiplier is an algebraic sum, we take one of the multipliersat a time and multiply it with the multiplicand anddetermine each partial product by the law of signs taking into account each multiplier as a separate case.1.1 Thus this is the logic of how the distributive law works with arithmetic.

The order of ideas which we are now following suggests thatthe multiplicand may also be distributed. For we have, by themeanings attached to positive and negative multipliers ( + 7-5)x( + 3)=+(+7-5) + ( + 7-5) + ( + 7-5),= +7-5 + 7-5 + 7-5,= +7 + 7 + 7-5-5-5,by the laws of association and commutation for algebraic sums.Hence, finally, by the primitive signification of multiplication


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(+7-5)x( + 3)= +7 x3-5 x3

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In like manner ( +7- 5) x(- 3)= -7x3 + 5x3 (13).Hence to multi^ily an algebraic sum by an algebraic quantitywrite down all the partial ^products obtained by multiplying eachterm of the multiplicand by the multiplier, and determine the signof each piartial product by the law for the sign of the product of twoalgebraic quantities. Or, in general symbols ( + A-B + C-D)( + a)= +Aa-Ba + Ca-I)a;(-|-A-B + C-D)(-a)= -Aa + Ba-Ca + Ba (14).

NOTE ON THE MULTIPLICAND MAY ALSO BE DISTRIBUTED 11.5!:......

35. Consider finally the product of two algebraic sums,say ( + A - B + C) X ( + a - 6). We may, in the first instance,consider + A - B + C as associated into a single operand+ ( + A B + C). If we distribute the multiplier, we have{+( + A-B + C)}( + a-&)=+( + A-B + C)a-( + A-B + C)?;.Since we may also distribute the multiplicand, we have,reading (4-A-B + C)a as ( + A- B + C)( + a),( + A-B + C)a= +Aa-Ba+Ca.Also ( + A-B + G)6= +A6-B6 + C6;and -( + A-B + C)6= -Ab + Bb-Ck

Hence, finally ( + A-B + C)( + a-&)= i- Aa -Ba + Ca - Ab + Bb - Cb (15).This last result suggests the Law of Distribution in its fullform, viz. to multiply one algebraic sum by another write downthe algebraic sum of all the partial products obtained by multiplyingeach term of the multiplicand by each term of the multiplier, deter-mining the sign of each partial -product by the law that the productof two terms having the same sign is to have the signproduct of two terms having opposite signs the sign .

NOTE ON THE LAW OF DISTRIBUTION FOR TWO ALGEBRAIC SUMS AND STATED IN ITS FULL FO

RM 13!:.......

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It will be observed that in the product of an algebraic sumof m terms into an algebraic sum of n terms there are mn partialproducts, if they are all written down directly in accordancewith the law, without any collection of like or suppression of


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mutually destructive terms. This rule will sometimes enable thebeginner to correct a mistake in his calculations.

NOTE ON THE AMOUNT OF TERMS IN AN ALGEBRAIC SUM IS MN AND CAN HELP CORRECT MISTAKES IN CALCULATIONS FOR GEBINNERS 12!:.........

36. It will be seen that the law of distribution formultiplication has been suggested to us by the consideration ofarithmetical operations with integral numbers. A little thoughtwill convince the learner that the law holds whether the operandsbe integral or fractional. Thus, for example, the arithmeticaltruth of the following equations(i-|)x(|-J)=(4-|)x|-(f-l)xi5w6 3 K/ 5 5wl I 3 w 1as arithmetical statements will be readily seen ; and they aresimply particular applications of the law of distribution. Wetherefore lay down this law as one of the fundamental principlesof Algebra in the assurance that it agrees with the fundamental

principles of ordinary arithmetic ; beyond this, all we have toconsider is merely the mutual consistency or non-contradictionof the various laws we adopt and of the consequences thatfollow therefrom. On this latter point the learner will graduallyacquire conviction as he jjroceeds witli the study of the subjectand of its applications.

NOTE ON CONVICTION OF MUTUAL CONSISTENCY OF THE LAWS WILL COME WITH FURTHER STUDY 12!.....

In the meantime we remark that, as in the case of the otherlaws, and in Algebra generally, we shall not confine the operandsto be arithmetical or even algebraic quantities in a reducedform, the operands may be complex functions of other quantities.Thus/x-l X \ /x+l\V^T2 ~ ^^ly ^ \x'+i)- \x+2) "" \x^+i) \x-ij "" W+i;'where x is not specifically assigned, is a particular case of thelaw of distribution.

37. Law of Distribution for Division. The law ofdistribution has a limited application to division.

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In the first place, we may point out that the laws of signsfor the division of algebraic quantities follow from the corre-sponding laws for multiplication. For example, since+ {a-^h)x{-b)= - {{a^b)xh],hy law of signs for multiplica-tion,= - a,hj the mutual inverseness of multiplica-tion and division,it follows, if b be any finite quantity differing from 0, that+ (a^b)x(-b)^(-h) = (-a)-^{-h).Hence, suppressing the mutually destructive operationsX ( 6) -^ ( 6) on the left, and interchanging the two sides ofthe equation, we get{-a)^{-b)=+{a^byIn like manner we establish all the four cases { + a)î + b)= +{a^b), ( + a)-^{-b)= -(a-6),(_,)^( + 6)=-(a--6), (_a)-^(-6)=+(^6) (i).

NOTE ON THE LAW OF DISTRIBUTION HAS LIMITED APPLICATION TO DIVISION 13!: Thus says the law of distribution has limited application todivision.1.1 However the law of signs for division is the same for multiplicaiton and fol

lows from multiplication.

Again, by the law of distribution for multiplication, if Abe any finite quantity or operand diff"ering from 0, we have(a-Â-&-Â-f-cÂ-(^Â)x(-l-A)= -|-{(aÂ)xA}-{(6Â)xA} + {(cÂ)xA}-{{dÂ)xA}= +a-hi-c-d. (2).If now we divide both sides of (2) by -f A, and suppress themutually destructive operations x (-f A)-f-(-f A) on the left, weget, after interchanging the two sides of the equation { + a-b + c-d)^{ + A)= +(a-Â)-(6-Â) + (cÂ)-((Z-Â) (3).

And we could in the same way deduce that{ + a-h + c-d)-^{-A)= -{aÂ) + (bÂ)-{c-Â) + (d---A) (4).Equations (3) and (4) are evidently particular cases of thefollowing general law.To divide an algebraic sum by any algebraic quantity writedown all the ^partial quotients obtained by dividing each term of thedividend by the divisor, attaching the sign + if the term and thedivisor have like signs, the sign if they have opjjosite signs.

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We may express this result briefly by saying that thedividend may be distributed. The same is not true of thedivisor, at least not as a general rule of algebraic operation ;and it is only with such rules that we are now concerned. Toestablish this it is sufficient to advance a single arithmeticalexception. If the divisor could be distributed, then we shouldhave 3-^(2 + l) = 3-^2 + 3-rl; in other words, 1 = 4 J, which isfalse.


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NOTE IMPORTANT ON THE DIVIDEND CAN BE DISTRIBUTED BUT THE DIVISOR CANNOT BE DISTRIBUTED 13!: Thus when using the distributive rule for division,the divisor has to be a single whole number, however the dividend can be an algebraic sum. The divisor cannot be an algebraic sum.1.1 Remember this counterexample of 3/(2+1) is not equal to 3/2 + 3/1.

38. Distributive Properties of 0. If a be any finitequantity, and h any finite quantity or 0, then we have, by thelaws already established ( + a-a)x6= + (a&) - (a&) ;that is to say 0x6 = (1);and, in particular 0x0 = (2).Again, if h be any finite quantity, excluding 0, we also have( + a - a) -^ 6 = + (a 4- 6) - (a -f 6) ;in other words -^ 6 = (3).The equations (1), (2), and (3) may be called the distributiveproperties of zero.

NOTE ON THE DISTRIBUTIVE PROPERTY IN THE CASES OF "0" 11!:...

39. Excepted Operands. In laying down the laws ofAlgebra we have assumed throughout that all operands arefinite definite quantities. Otherwise the operands so far mayhave any finite value in the series of real quantity, except onlythat may not be a divisor. It may be of interest to satisfythe reader that the admission of division by as a generalalgebraic operation would lead to contradiction. Let ussuppose that a is any finite quantity whatever ; then, if divisionby is to be admissible as an algebraic operation, a-^0 must

have some finite value, h say. We should then havea-^0 = 6 (1).Again, we should deduce from (1) thata-^ 0x0 = 6x0 (2).If is to be admitted as an ordinary operand, a -^ x is, bythe mutual relation of multiplication and division, simply a.On the other hand, 6 x 0, since 6 is finite, is 0. Hence (2)

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asserts that a=0, wliicli contradicts our hypothesis that a isany finite quantity whatever.

NOTE ON JUSTIFICATION OF DIVISION BY "0" LEADS TO INCONSISTENCY OF ALGEBRAIC RULES INSTEAD OF TYPICAL REASON THAT "IT DOESN'T MAKE SENSE" 13!: Thus says thatif we allowed division by 0, then by mutual relation of division and multiplicat


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ion a * 0 / 0 would equal "a", which would lead to contradictions.1.1 These contradictions are important because if we want to generalize to letters, we have to make sure for certain that these things make sense with concretearithmetical quantiites and are consistent.

It is a very common beginner's mistake to suppose that -^is an admissible algebraic operation, and that its value is 1.No such operation can be admitted as we have seen ; but it isperhaps well that it should be seen that, even were it to beadmitted, it could not be asserted that the result is 1 in allcases. This will be seen from the consideration of the threequotients, x^-^x^ 2x-^x,x-i- x^. If it were the case that -^ isadmissible and equivalent to 1 in all cases, it would followthat as x is made smaller and smaller each of these quotientsshould approach more and more nearly to 1 ; whereas it isobvious that the first becomes more and more nearly ; thesecond is 2 ; and the third becomes greater and greater, andcan be made to exceed any given quantity whatever.

NOTE ON WHY 0/0 IS NOT "1" AND PRELUDE ON SHOWING THIS THROUGH AN UNNAMED PRELUDE TO CALCULUS 12.5!: Thusshows logically how making these approach 0, that tey approach different quantit

ies.

40. As we have now discussed all the fundamental lawsof Algebra, it will be convenient to give a synoptic table ofthem for convenience of reference.For the sake of brevity we have condensed the statementsby the use of double signs. Thus, instead of writing allthe different cases +( + a + &)=+( + a) + ( + &), +( + a-6) =+ ( + a) + ( - 6), etc., we have written (a + h)= + ( a) ( 6),the understanding being that the signs are to be taken from

corresponding places on both sides.

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41. Meaning of the Sign =, Identical and ConditionalEquations. The reader should here mark the exact significa-tion of the sign = as hitherto used. It means " is transform-abla.into by applying the laws of Algebra and the definitions of

the symbols or functions involved, without any assumptionregarding the operands in\ rived."

Any " equation " which i.: true in this sense is called an"Identical Equation," or an "Identity" ; and must, in the first


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Notes on Algbra G Chrystal

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