ZXTape! 0Created with Ramsoft MakeTZXmh0 lkg!l0:6:6 maths help tape 0  G C E Tutoring  Dec 1983  r(10 ) getinp=9450$ 5:"MATHS help mh0"::"ARITHMETIC & NUMBER SYSTEMS" :" G C E Tutoring Dec 1983"  o=0 2l:"Hello. What's your name ?":n$:n$:"OK, ";n$;", I'll begin by telling you how to use this maths tape" 4f:"This tape covers 14 subjects which all come under the headingof arithmetic and number systems" 7u:"You can either work through the lessons in order, or choose which you want to do by typing the lesson name" :>:"Press the key marked C so I can type the next screenfull" <a$=:a$="c"62> = 60< >:z=1100d:z AG:"Thanks. From now on when I want you to press the C key, I'll say:" D:" Hit C to continue" F1:"Here's the list of subjects on this tape": H"fractions square roots decimals logarithms rounding areas accuracy volumes indices number sets std. form bases interest modulo" K2:"Which subject do you want to try ?":s$: L%s$>3s$=s$(13) M 96` Ns$="fra"100d Os$="squ"550& Ps$="dec"340T Qs$="are"582F Rs$="acc"458 Ss$="vol"582F Ts$="ind"975 Us$="num"622n Vs$="rou"410 Ws$="log"8259 Xs$="std"s$="sta"1062& Ys$="bas"750 Zs$="int"650 [s$="mod"700 _ 70F `Fj=1̱(s$):(s$(jj))<97as$(jj)=(32 +(s$(jj))) aj: c d!"Although many mesurements are" f""made in the metric system, and" h"the Country has had decimal" j"coinage since 1971, fractions are still essential for many branches of mathematics, These include geometry, trigonometry and basic algebra" l 9900& n: x,"Here is a simple example of a fraction" z:,1:,"-":,2: |U"It means ONE PART OF SOMETHING TAKEN FROM TWO and can also be written like this" ~ ,1;"/";2: `"Similarly, 1/7 means one part taken from seven, and 8/8 means eight parts taken from eight" 9900& "Taking the last example again, ( 8/8 ) think what that means. It means that all the parts have been taken. So if a cake was slices into eight pieces, for example, 8/8 means the whole cake." :"So 8/8 = A WHOLE = 1" 6:"And similarly 2/2 = 1, 7/7 = 1 and 343/343 =1": @"See if you understand this by trying the folowing examples" 9900& egc=0 :9800H& Rr(1);"/";r(2);" means ";r(1);" parts taken from how many ?" exp=r(2):getinp:a=ans ar(2)egw=1 %a=r(1)+r(2)9700% a=r(1)9710% egc=egc+1 egc<3151 : Ҋ"A fraction is not changed if thetop number ( numerator) and the bottom number ( denominator ) are multiplied by the same number" : !"For example : 4/8 = 2/4 = 1/2" : ,"This is called CANCELLING a fraction"  9600% egc=0 9900& 9800H& ^"Is ";r(1);"/";r(2);" = ";r(1)*r(3);" / ";r(2)*r(3);" ?" 9630% a$="t"a$="true"9720% Pa$="f""Yes they are ! Both the top and bottom are multiplied by ";r(3) egc=egc+1 egc<3227 9900& : R"To add or subtract fractions they must have the same denominator.": ""1/2+2/7 does NOT equal 3/9 !": X"To add them, make the denominators equal by finding a COMMON DENOMINATOR": ]"Do this by multiplying the denominators together. The example above becomes"   " 7/14 + 4/14 = 11/14":  egc=0  C"Notice how the numerators changeto keep the fractions the same."  9900&  9600%  9800H& D"Type the numerator, then enter, then the denominator, then enter" ?r(1);"/";r(2);"+";r(3);"/";r(4);"= ?"  9650% %f1=(((n/d)*1000))/1000 Mf2=(((r(1)/r(2)+r(3)/r(4))*1000))/1000 f1=f29720% f1f2"No." egc=egc+1 egc<3270  9900& "Multiplying fractions is very easy. Simply multiply the numerators together, and multiply the denominators together" "x:r(4);"/";r(5);" * ";r(6);"/";r(7);" = ";r(4)*r(6);"/";r(5)*r(7): $/"It is good to cancell the answerif you can." ' 9900& )d:"To divide a fraction by another,turn the second fraction upsidedown, and then multiply." , ."1 2 1 7 7" 1"- / - = - * - = -" 3"2 7 2 2 4" 6 9600% 7 egc=0 9 9900& ::r(4);"/";r(5);" / ";r(7);"/";r(6);" = ";r(4);"/";r(5);" * ";r(6);"/";r(7);" = ?": ; 9650% <#f1=((1000*n/d))/1000 =Y f22=(r(4)/r(5))*(r(6)/r(7)):f2=((f22*1000))/1000 @f1=f29720% Af1f2"No." Begc=egc+1 C 9800H& Eegc<3314: Jn:"OK, that finishes fractions. If you think you need to try this section again then hit r ( for repeat )" L 9900& Oa$="r"100d P T:"Decimals":"________": V"Decimals are fractions with the denominators not written. Something else special about the denominators is that they are always one followed by noughts" W Y"For example 0.2 = 2/10 " [" and 0.04 = 4/100": ^"1/10 = 0.1 1/100 = 0.01": `o"The number of digits after the decimal point is the same as thenumber of noughts in the denominator" c 9900& eZ:"To change a fraction into a decimal, divide the denominator into the numerator": h "4 1.00" i " " j " 0.25" k mj"It is good if you can learn a few decimals off by heart"::"1/2 = 0.5":"1/4 = 0.25 ":"1/8 = 0.125" o 9900& r"Multplying a decimal by 10, 100, 1000, 10000 etc. simply means moving the decimal point either 1, 2, 3, or 4 places to the right." u:"eg. 0.00234 * 10 = 0.0234" w"and 0.0234 * 100 = 2.34" y:"Dividing decimals is also easy. To divide by 10, move the decimal point 1 place to the right. To divide by 100, move the decimal point 2 places to the left, and so on for 1000 etc." |:9600% }9900&:egc=0 ~ 9800H&  Cd1=r(1)/10 +r(2)/100d+r(3)/1000 d1;" * ";100d;" = "; >exp=d1*100d:exp=((1000*exp))/1000:getinp )egc=egc+1:egc<3382~   egc=0 O9800H&:d1=r(1)/10 +r(2)/100d+r(3)/1000 d1;" / ";10 ;" = "; =exp=d1/10 :exp=((1000*exp))/1000:getinp )egc=egc+1:egc<3398 f:"That enough decimals. Hit r if you want to repeat that section"::9900&:a$="r"340T  "Rounding":"--------": ]"To ROUND OF a decimal to a certain number of DECIMAL PLACES, follow the following rules :"  9900& ;:"1 split the last place from the rest of the decimal" #:" eg. 0.413 becomes 0.41 3": Y"2 if the last digit is less than 5, drop it and leave the rest of the decimal"  :" eg. 0.41 3 becomes 0.41" X:"3 if the last place is more than 5, drop it, and add one to the new last place": 7" eg. 0.56 7 becomes 0.56 which becomes 0.57"  9900& >:"Rounding can be to a certain number of DECIMAL PLACES" y:"eg. 3.14159 5 places":" 3.1416 4 places":" 3.142 3 places":" 3.14 2 places":" 3.1 1 place"  9600%:9900&  egc=0  9500% 0exp=(((dec+0.005y# =)*100d))/100d &:"What is ";dec;" to 2 dec. places" getinp )egc=egc+1:egc<3440 ?:"That finishes rounding. To repeat this section hit r"  9900& a$="r"410  :"Limits of accuracy" ̮:"A number can be expressed to a certain number of SIGNIFICANT FIGURES. Below is the example from the section on rounding changed to show significant figures." e:" 3.142 4 sg.fgs.":" 3.14 3 sg.fgs.":" 3.1 2 sg.fgs.":" 3 1 sg.fg." ~:"To round off a number to a certain number of significant figures, count the total number of digits in the result."  9900& l:"To express a whole number to a certain number of significant figures, look at the example below:" V:" 864 3 sg.fogs. 860 2 sg.fgs. 900 1 sg.fg." ۢ:"Remember that the 0s before the first non-zero digit in a decimal, and the 0s after the last non-zero digit in a whole number, are not significant." 9:"If this doesn't make sense then hit r ( for repeat)" 9900&:a$="r"458 9600%:9900&  egc=0  9500% $:"What is ";int;" to 2 sg.fgs. ?" *exp=(((int+5)/10 ))*10 getinp ans=exp9720% )egc=egc+1:egc<3484  9900&  r:"The accuracy of a measurement depends on how many significant figures it is expressed to. For example:" !:"3 means between 2.5 and 3.5" P:"Because any number between thesetwo values rounds off to 3. Similarly:" +:"2.72 means between 2.715 and 2.725" D:"Because any number between thesetwo values rounds off two 2.72" w:"In these examples the two numbers are called the LIMITS OFACCURACY. Another way of showingthem is like this" 9:"3 = 3 +- 0.5 2.72 = 2.72 +-0.005"  9900& f:"The PERCENTAGE ERROR can be worked out from the maximum error and the measurement itself" 9:"eg. % error = error/measurement * 100"  #:"Which in the last example is:"  (:"100*0.005/2.72 = ";0.184~:"eg. square root of 64 = 8 since 8 squared = 64" 09600%:9900& 2 egc=0 5 9500% 8!:"What is the square root of:" :r(1)*r(1);" ? "; <exp=r(1) ?getinp B)egc=egc+1:egc<35655 D 9900& E FC:"Here are some formulae for calculating areas and volumes" L :"areas": N"triangle 1/2 b x h" P"parallelogram b x h" S"circle pi x r x r" V"sphere 4 pi x r x r" X :"volumes" Z ]""sphere 4/3 pi x r x r x r" `"cylinder pi x r x r x h" b$"cone 1/3 x pi x r x r x h" d"cuboid l x b x h" g!:"l=length b=breadth h=height" jd:"If there are any of these formulae you don't know, learn one each time you use the tape" l 9900& m n:"The symbols in our number systemare called DIGITS. They are : 0,1,2,3,4,5,6,7,8,9 and go to makeup numbers which can be split into the following sets:" o 9900& qA:"Integers"::"these are whole numbers such as -3 12 and 245" tD:"Natural numbers"::"are positive integers such as 1,2,3,4,5" vw:"Rational numbers"::"are numbers which can be expressed as ratios or fractionseg. 1/3 1/2 1.7 ( = 17/10 )" x 9900& {"irrational numbers"::"are numbers which cannot be expressed as fractions or ratios. Pi and the square roots of numbers such as 2,3,5 etc are inexpressable as ratios ( 22/7 is an APPROXIMATION of pi correct to only 2 d.p. )" ~n:"prime numbers"::"are numbers which are divisible by only 1 and themselves. eg. 2 3 5 7 11 13 etc"  9900& a:"real numbers"::"are numbers which fall into one of the categories already mentioned" h:"That finishes the section on number sets. Hit r if you want to repeat this section":9900& a$="r"625q  :"Interest" :"When money is loaned or borrowedthe amount is known as the PRINCIPAL. There is a charge for the privelege of borrowing, which is called the INTEREST. The percentage at which interestis charged is known as the RATE." F:"The total amount repaid is the principal plus the total interest" L:"There are two types of interest:simple interest and compound interest."  9900& :"Simple interest"::"Simple interest is what it sounds - simple. The interest is calculated at the end of the loan period."::"eg. P = `200 R = 20% per year T = 3 years"  "Total interest = 20% per year x 3 years = 60% of the premium = `200 x 60% = 200 x 60/100 = `120"  9900& :"Remember that this is just the charge - the original amount still has to be added"::"So the total amount repaid is `200 + ` 120 = `320" :"Unfortunately this is not how things work in the real world. Instead, the amount owed is calculated at the end of each year"  9900& g:"Compound interest"::"Taking the previous example, here is how compound interest works out:" :"year 1 = `200 + 20% of `200 = `240 year 2 = `240 + 20% of `240 = `288 year 3 = `288 + 20% of `288 = `345.60" :"Which is `25.60 more expensive than simple interest. Simple interest gives a rough idea though, and is quicker to work out."  9900& 0:"Try these questions on simple interest": 9600%:egc=0  9500% %:p=int:t=r(4):r=r(5) "What is the total amount repaid on a principal of `";p;" over ";t;" years at ";r;" % per year":exp=p+(p*r*t/100d):exp=(((100d*exp))/100d) getinp )egc=egc+1:egc<3685  9900& N:"That finishes interest. Hit r to repeat this section if you want to." 9900&:a$="r"648  :"Modulo arithmetic": "Is normal arithmetic but with a special set of numbers. Modulo 4, for example, means arithmetic with the numbers 0,1,2 and 3. It is sometimes called CLOCK ARITHMETIC as the numbers can be imagined as being drawn around a clockface:"  9900& :" 0 3 1 2" %43+,47/+88X,23  q"When adding, go round the clockface clockwise; when subtracting, go round anticlockwise" 3:"So 2 + 2 = 0 in modulo 4 and 1 - 2 = 3"  9900& {:"Multiplication is just repeated addition. For example 3 x 3 is simply 3+3+3, so go round the clock 3 units 3 times" :" 0 3 1 2" .43+,47/+88X-39',23  :"so 3 x 3 = 3+3+3 = 2+3 = 1" %:"Try a few questions":9900&  egc=0  9500% 8r(1) :" 3" @"eg. 10 = 1000" C-:"In this case, base=10 and power=3" F8:"Logs with a base of 10 are called COMMON LOGS" H 9900& J K "Here are some logs to base 2" L:"log number": M-z=19 :z;" ";2^z:z P:"Logs make arithmetic easier. To multiply two numbers, simply add the logs, and convert the result back using the log table." Q 9900& R:"log number": T-z=19 :z;" ";2^z:z Wy:"eg. to work out 8*16, find the logs of 8 and 16, and add them together. Look up the ANTILOG to find the answer." X5"The antilog means THE NUMBER WHOSE LOG IS ... " Z 9900& \:"log number": ^-z=19 :z;" ";2^z:z a:"log 8 = 3" d"log 16 = 4" f"log 8 + log 16 = 3 + 4 = 7" h8"The number whose log is 7 is 128so the answer is 128" k 9900& n:"log number": p-z=19 :z;" ";2^z:z r:"Similarly, logs make division easier by turning the problem intosubtraction, and, by the laws of indices, turn the problem of raising a number to a power intomultiplication." sM9900&::"log number"::z=19 :z;" ";2^z:z t:"ie. 4 to the power 3 is solved by finding the log of 4 (2), multiplying by 3 (6), and finding the antilog (64)." u 9900& x:"log number": z-z=19 :z;" ";2^z:z |Y:"To divide 128 by 16, find the logs of the two numbers:"::"log 128=7 and log16=4"  9900& :"log number": -z=19 :z;" ";2^z:z >:" 7-4=3":" antilog 3 = 8":"so the answer to 128/16 is 8"  9900& F:"Normally, common logs are used to solve arithmetic problems. " 1:"Here are some common logs"::"number log": V"1 0":"2 0.301":"3 0.477":"4 0.602":"5 0.699":"6 0.778" -"7 0.845":"8 0.903":"9 0.954" 9900&: V"1 0":"2 0.301":"3 0.477":"4 0.602":"5 0.699":"6 0.778" -"7 0.845":"8 0.903":"9 0.954" :"Since these logs are to base 10,it is easy to work out the logs for 10,20,30 etc. and 100,200,300 etc. by simply adding the power of 10 required." 6"The added power of 10 is called the CHARACTERISTIC" G9900&::"so log 20 = 1.301 log 200 =2.301 etc"  9900& <:"Here is an example of logs in an arithmetic problem." @:"eg. 20 x 300 find the logs : log 20 = 1.301 log 300 = 2.477 add logs 1.301 + 2.477 = 3.778 find antilog charactersitic = 3 leaving 0.778 antilog 0.778 = 6" :"so the answer is 6 x 10^3"  9900& #:"Here are a few questions : ":  9500% y"To multiply using logs, do you A multiply them B add them C subtract them ?": $s$:s$:s$="b"s$="B"9720% s$"b"s$"B""No, it is B" "To divide using logs, do you A multiply them B add them C subtract them ?": $s$:s$:s$="c"s$="C"9720% s$"c"s$"C""No, it is C" "To find a power using logs, do you A multiply them B add them C subtract them ?": $s$:s$:s$="a"s$="A"9720% s$"a"s$"A""No, it is A" ::"That finishes logs. Hit r to repeat this section."   :"indices" D:"An INDEX is a POWER. 7 x 7 x 7 is called 7 TO THE POWER OF 3," -:" 3":" and is written 7" +:"In this case, 7 is CALLED the base."  9900& 4:" 2 3 ":"7 x 7 = 7 x 7 x 7 x 7 x 7" ^:"If you count the times signs, notice that"::" 2 3 2 + 3 5":"7 x 7 = 7 7 = 7" B:"So to multiply numbers with the same bases, add the indices." =:"Expressed algebraically, the rule looks like this : " !:" b c b + c":"a x a = a"  9900&  1045 7:" 2 3 2 2 2 6":"(7 ) = 7 x 7 x 7 = 7" 8:" 2 3 2 + 2 + 2 6":"(7 ) = 7 = 7" N:"so to raise a number to a power,multiply the index by the power required" =:"Expressed algebraically,the rule looks like this : " :" b c b * c"  "(a ) = a "  9900& Q:"A very easy mistake to make is to confuse :"::" 3 3":"(2x) = 8x" (:"and"::" 3 3":"2(x ) = 2x" _:"Don't fall into that trap ! "::"Always work out what's inside the brackets first."  egc=0  9900& ":"Here are a few questions :":   9500% *a=r(1):b=r(2):c=r(3) -" ";b;" ";c;" ";" ?":a;" x ";a;" = ";a  exp=b+c  getinp *egc=egc+1:egc<31030  9900&  1062& Y:"Conversely, subtracting indices allows the division of numbers with the same base." =:"Expressed algebraically, the rule looks like this : " !:" b c b - c":"a / a = a" U:"From this, it follows that :"::" b b b - b 0":"a / a = a a = a = 1" "2:"This is very important - REMEMBER IT." $ 9900& % &:"Standard form": )L"Standard form is a quick way of writing very large or very smallnumbers." ,::"In standard form, a number is written like this : " .:" n":"a x 10" 0M:"so 256, for example would be written : "::" 2":"2.56 x 10" 3 9900& 6|:"The first number is called the MANTISSA, and is always chosen so that the decimal point comes after the first digit." 8S:"Here are some more examples : "::"in these examples ^ means TO THE POWER OF" ::"0.03937 = 3.937 x 10^-2 5280 = 5.28 x 10^3 31 536 000 = 3.1536 x 10^7 18 446 000 000 000 000 000 = 1.8446 x 10^19" = 9900& @:"Standard form is a use of indices. Not suprisingly, it follows that the index laws apply. "::"To multiply, ADD the indices To raise to a power, MULTIPLY the index." B=:"so 1.2 x 10^3 x 1.1 x 10^4 = 1.32 x 10^7" D 9900& E egc=0 G":"Here are a few questions :": L':"express these numbers normally:": NF9500%::r(3);".";r(2);" x 10^";r(1);" = ? "; O:exp=(r(3)+r(2)/10 )*10 ^r(1) P exp=exp Qgetinp T*egc=egc+1:egc<31102N V 9900& W egc=0 X":"Here are some more questions" [ ^?9500%:"10^";r(1);" x 10^";r(2);" = 10^ ? "; `#exp=r(1)+r(2):getinp b*egc=egc+1:egc<31118^ e?:"That finishes standard form. Hitr to repeat this section." h 9900&:a$="r"1062& i a$ (a$) z=1100d:z  2000 f1f2"No." 1s$"a"s$"A""No, it is A" #' $ $ $getinp $ $?dp=0:err=0:u$:u$:(u$)>10 err=1 $(u$)=09460$ $[j=1̱(u$):(u$(jj))<45-ů(u$(jj))>579ů(u$(jj))=47/err=1 $%9480%:j:u$="-."err=1 $Cerr=1dp>1"type the answer as a number":9460$ $Gans=(u$):(exp-ans)>0.005y# =o=0"No, the answer is ";exp $.(ans-exp)<0.005y# =o=09720% $ $ % %$(u$(jj))=46.dp=dp+1 % ,j>1Ư(u$(jj))=45-err=1 %  %k9800H&:d1=r(1)/10 +r(2)/100d+r(3)/1000:dec=d1:int=1000*dec %& %A"See if you understand this by trying the following examples" % 9800H& % %"True or false ?" %a$ %a$ %Ha$"t"a$"f"a$"true"a$"false""answer true or false":9630% % %o=1:getinp:n=ans:"--":getinp:o=0:d=ans:d=0"don't divide by 0 !":"try the whole fraction again.":9650% % % 9630% %h"No. Don't add the numbers together. It's just the second number, so the answer is ";r(2) % %K"No, its the second number, not the first, so the answer is ";r(2) % %%9720%+((*9 )+1): % "Good": % "Correct": %"Right again": %"OK": %"That's it": %"Brilliant !": %"Well done": &"Very good": &"Genius !": &HWz=110 :r(z)=(*9 )+1:z:r(1)r(2)9800H&: &R#r(1)r(2)9800H& &\ &z "trap": & &"Hit C to continue" &a$= &a$="c"9950& &a$="r"9950& & 9920& & &z=1100d:z & etin$oN &A 5  mh1 @?&@6:6:0: maths help tape 1  G C E Tutoring  Jan 1984  2000 r(10 ) Lscrhdr=9300T$:qsthdr=9350$:endsec=9400$:getans=9450$ ':"MATHS help mh1"::"BASIC ALGEBRA" :" G C E Tutoring Jan 1984" 2o:"Hello. What's your name ?":n$:n$:"OK, ";n$;",":"I'll begin by telling you how touse this maths tape." 4T:"This tape covers 8 subjects which all come under the headingbasic algebra." :>:"Press the key marked C so I can type the next screenfull" <a$=:a$="c"62> = 60< >:z=1100d:z AG:"Thanks. From now on when I want you to press the C key, I'll say:" D:" Hit C to continue" F1:"Here's the list of subjects on this tape": H"multiplication and division indices algebraic laws use of brackets quadratic expressions factors simultaneous equations solving quadratics" K2:"Which subject do you want to try ?":s$: L'(s$)>3s$=s$(13) M 96` Ns$="mul"s$="div"100d Os$="ind"200 Ps$="alg"300, Qs$="use" 400 Rs$="qua"500 Ss$="fac"600X Ts$="sim"800  Us$="sol"700 _ 70F `Fj=1̱(s$):(s$(jj))<97as$(jj)=(32 +(s$(jj))) aj: c d f imultiplication and div l n!:"multiplication and division" o p"minus times minus is plus plus times plus is plus minus times plus is minus plus times minus is minus" q s"-a x -b = +ab a x b = +ab -a x b = -ab a x -b = -ab" v xk"A easy rule is that two UNLIKE signs give a MINUS answer, and two LIKE signs give a PLUS answer." zscrhdr }=:"The rule for division is the same as multiplication."   "-a / -b = a/b a / b = a/b -a / b = -a/b a / -b = -a/b"  k"The easy rule is that two UNLIKEsigns give a MINUS answer, and two LIKE signs give a PLUS answer."  qsthdr  "-a x - b = ?" 'a$:a$:a$="ab"a$="+ab"9720% "a$"ab"a$"+ab""No, it is ab"  "a x -b = ?" a$:a$ a$="-ab"9720% a$"-ab""No, it is -ab"  "-a / b = ?" a$:a$ a$="-a/b"9720% a$"-a/b""No, it is -a/b" Dstart=100d:return=172:n$="multiplication anddivision" :endsec    indices  P:"The word indices is the plural of index, which is another word for power." :" 3" "eg. a = a x a x a"  :" This is called a to the power 3"  6"In this example, the index is 3 and the base is a." 3:"Multiplying and dividing indicesis very easy." scrhdr "Multiplication"  ""To multiply, add the indices :"  " 3 4" "a x a = axaxa x axaxaxa"  " 3 + 4" " = a"  " 7" " = a"  "This is really common sense, as the index represents the number of times the base is written in the long way of writing the problem" scrhdr "division"  :"To divide, subtract the second index from the first :"   " 4 2"  "a / a = axaxaxa / axa"    " 4 - 2" " = a " :" 2":" = a"  "Again, this is really common sense, as the index represents the number of times the base is written in the long way of writing the problem" scrhdr  qsthdr: 69500%:a=r(1):b=r(2):c=r(3) '" ";b;" ";c;" ";" ?":"a x a = a"  exp=b+c "loop=279:return=285 getans  scrhdr :"Powers"::"Raising an expression woth an index to a power is also easy. Simply multiply th original index by the required power:"   !" 4 3 12" " "(a ) = a " # *9n$="indices":start=200:return=299+:endsec + , . 1algebraic laws 4 6:"algebraic laws" 8 ;P"There are three laws that deal with simple algebraic expressions" >U:"1) COMMUTATIVE 2) ASSOCIATIVE 3) DISTRIBUTIVE" @.:"Each of these laws will be takenin turn." Bscrhdr D"The commutative law" E H3"a + b = b + a a x b = b x a": J>"a - b DOES NOT EQUAL b - a a / b DOES NOT EQUAL b / a" Ln:"so the commutative law applies for addition and multiplication,but not for subtraction and division" Oscrhdr R"The associative law" T V="(a + b) + c = a + (b + c) (a x b) x c = a x (b x c)" Yq:"(a / b) / c DOES NOT EQUAL a / (b / c) (a - b) - c DOES NOT EQUAL a - (b - c)" \n:"so the associative law applies for addition and multiplication,but not for subtraction and division" ^scrhdr `"The distributive law": c"a(b + c) = ab + ac" f hqsthdr j:"Do the commutative and laws apply to :"::"A multiplication and division B division and subtraction C addition and multiplication" m&:a$:a$:a$="c"a$="C"9720% na$"c"a$"C""No, it is A" p r@start=300,:return=372t:n$="algebraic laws":endsec sa$"c"a$"C""No, it is A" t    brackets  :"case 1": -"-(a - b) implies -1 x (a - b) = - a + b"  6"this sign change simplifies expressions like :" M:"(3a + 4b) - (2a + 3b) = 3a + 4b - 2a 3b NB sign change= a + b" scrhdr  "case 2": )"(a + b) (c + d) = ac + ad + bc + bd": 6"this principle simplifies expressions like :"  T"(a + 2) (a + 3) 2 =a + 3a + 2a + 6" " 2":"=a + 5a + 6" !:"which is called a QUADRATIC"  An$="use of brackets":return=445:start=400:endsec    quadratic expressions  /:"A quadratic has the general formula :" :" 2":"ax + bx + c" 8:"There are two special cases of the quadratic :" P:"1) the perfect square 2) the difference between two squares" scrhdr  "1) The perfect square"  +:" 2":"(a + b) = (a + b) (a + b)"  5:" 2 2":" = a + 2ab + b" *:" 2":"(a - b) = (a - b)(a - b)" 5:" 2 2":" = a - 2ab - b" 8:"are both the expansion of perfect squares. " scrhdr +:"These forms are called perfect squares, as they can beimagined as the area of a square with sides of length a + b or a - b. " ,scrhdr .$"2)difference between two squares" 0?:"The difference between two squares looks like this :" 6 <:"(a - b)(a + b)" >=:" 2 2 2 2":"= a - ab + ba + b = a - b" ?D:"This can be used to simplify quite complicated expressions." BU:"See if you can simplify the following expressions using these two forms" Cscrhdr D"(x + 7)(x - 7) + 49" FJ:"is this :"::"A) the difference of two squaresB) a perfect square ?" G :a$:a$ Ha$="A"a$="a"9720% Ia$"A"a$"a""No, it is A" J;:"using the difference of two squares, it becomes :" L(:" 2 2 2":"x - 7 + 49 = x" Mscrhdr N" 2 2":"x + 6x + 19 - 10" OJ:"is this :"::"A) the difference of two squaresB) a perfect square ?" P :a$:a$ Qa$="B"a$="b"9720% RKa$"b"a$"B""No, it is B":"using a perfect square, this becomes :" S.:" 2 2":"x + 6x + 9 = (x + 3)" Tscrhdr U?n$="quadratic expressions":start=500:return=598V V W X Z ] factorising ` b:"Factorising is a way of simplifying algebraic expressions. It makes use of thedistributative law, which if you've forgotten is :" d:"a(b + c) = ab + ac" gp:"In this equation, a has been distributed. Notice that it is the common term in the right hand side." j=:"In general, when factorising look for common terms. " kscrhdr l:" 2":"eg. 3a + 6a" m:" " n?"The common term is 3a, as both terms can be divided by 3a." q$:"The factorised expression is :" t:"3a(a + 2)" vZ:"Sometimes, with complex expressions, factorising takes more than one step :" xscrhdr {:"eg. 2ac - ad + 2bc - bd" ~:"= a(2c - d) + b(2c - d)" :"= (a + b)(2c - d)" :"As well as making expressions simpler, factorising can help solve equations. This is shown in the section on quadratics." qsthdr  >"what must you look for to factorise an expression ? "  a$:a$ aright=0:a$="a common term"a$="common term"a$="common terms"right=1:9720% 2right=0"No, you look for a common term."  <"what is the factorised version of :"::" 3ab + 6abc ? "  a$:a$ kright=0:a$="3ab(1 + 2c)"a$="3ab(1+2c)"a$="3ab(2c+1)"a$="3ab(2c + 1)"right=1:9720% (right=0"No, it is 3ab(1 + 2c)"  -"which law does factorising make use of ? "  a$:a$ iright=0:a$="the distributive law"a$="distributive"a$="distributive"right=1:9720% 2right=0"No, it is the distributive law."  1start=600X:return=682:n$="factors" endsec    quadratic equations  9:"A quadratic equation has the general formula :": " 2":"ax + bx + c = 0" Q:"There are two ways of finding the solutions to equations of this sort :" B:"1) factorising 2) using the general formula" ;:"At O'level it is commonest to solve by factorising." scrhdr "1) factorising": /" 2":"To factorise x + 7x + 10" ?:"1) write down two pairs of brackets like this :": " (x )(x )": 5"2) find the factors of 10 "::" ie. 1,2,5 & 10"  W"3) find two factors which i) multiply to make 10 ii) add to make 7" scrhdr ?:"4) put these factors inside the brackets like this :": d" (x + 5)(x + 2)"::" It doesn't matter which way around you put the 5 and the 2." 1:"5) the equation now looks like this :": " (x + 5)(x + 2) = 0" scrhdr " (x + 5)(x + 2) = 0"::"6) For the above expression to equal 0, either x + 5 = 0 or x + 2 = 0, as only 0 times another number = 0.": 9" so x = -5 and x = -2 are the solutions." scrhdr :"Not all equations work out this easily. If the sign of the number term is negative, then the brackets look like this :": "(x + )(x - )" @:"So there are more possibilities when adding the factors.": scrhdr  "eg. ": " 2":"x + 3x - 10 = 0"  E"the factors of 10 must add up to 3, so +5 and -2 are correct.": "The equation becomes :": 1"(x + 5)(x - 2) = 0"::"solutions = -5 and +2" scrhdr "2) general formula":  m"The general formula solves all quadratic equations. Simply put the values of a and b into the equation."  *:" 2":"(-b (b - 4ac ) )/2a" P:"For one solution use the plus sign, and for the other use the minus." scrhdr :"In the equation used in the first example, a=1, b=7, c=10. The solutions are : (-7 (49 - 40))/2 = (-7 -3)/2 = -5 and (-7 + 3)/2 = -2" scrhdr C:"give the most negative solution to :"::" 2":"x + 5x + 6 ?" 7:egc=3:exp=-3:return=789:getans (:"how about "::" 2":"x + 3x - 4": 5egc=3:return=791:exp=-4:getans scrhdr Mn$="quadratic equations":start=700:return=796:endsec     " %simultaneous equations ( *:"simultaneous equations" , /0"An equation like this has one variable :": 2"x + 5 = 10": 4M"If there are two variables, thentwo equations are needed for thesolution." 6!:"eg. x + y = 10, x + 2y = 15" 8scrhdr 9o:"Solving simultaneously means solving both equations at the same time. "::"Here is how its done :": i"the bottom equation has been subtracted from the top, and in doing this, only 1 variable is left." ?scrhdr @:"Now that the value of y is known, all that is needed is to find the value of x. This done by SUBSTITUTING the value for y in either one of the original equations." CP:"eg. x + y = 10 becomes x + 5 = 10 so x = 5." F6:"The solutions are therefore x = 5 and y = 5." Hscrhdr J^:"sometimes it is necessary to multiply, one of theequations before subtracting it.": M5"eg. 2x + y = 12 x + 2y= 21": P"Subtracting one equation from the other will not remove one ofthe variables. Instead, first multiply the bottom equation by two :": Qscrhdr Rb"2(x + 2y = 21) = 2x + 4y = 42 - 2x + y = 12 = 3y = 30 " T.:"so y = 10, and by substitution, x = 1" W:"The two equations that go to make up a simultaneous equation are the equation of lines. The solution is where they meet." Z:"Try a question"::scrhdr \P" x + 2y = 5 -(x + y = 3) y = ?" ^4egc=3:exp=2:return=863_:getans dU"so what is the value of x ?"::egc=3:exp=1:return=869e:getans kscrhdr nEn$="simultaneous equations":start=800 :return=880p oendsec p  z=07 row:""+z,row z 'z=07:row:""+z,row:z E7,4,4,4,8,104h,40(,16 I16,16,124|,16,16,0,124|,0   #' $T $V scrhdr() $Y $\9900&:: $ $ qsthdr() $ $'9600%:9900&:egc=0: $ $ $endsec(start,n$) $ $5"That finishes ";n$:"Hit r to repeat this section" $ 9900& $a$="r"start $return $ $ $getans(loop,exp) $ $"dp=0:err=0:u$:u$ $(u$)=09460$ $[j=1̱(u$):(u$(jj))<45-ů(u$(jj))>579ů(u$(jj))=47/err=1 $9480%:j $Cerr=1dp>1"type the answer as a number":9460$ $&ans=(u$):ansexp"No, it is ";exp $ans=exp9720% $$egc=egc+1:egc<3loop $return $ % %$(u$(jj))=46.dp=dp+1 % $u$(jj)="-"j>1err=1 %  %k9800H&:d1=r(1)/10 +r(2)/100d+r(3)/1000:dec=d1:int=1000*dec %& %A"See if you understand this by trying the following examples" % 9800H& % %"True or false ?" %a$ %a$ %a$"t"a$"f"9650% % %n:n;"/";:d:d: % 9630% %h"No. Don't add the numbers together. It's just the second number, so the answer is ";r(2) % %K"No, its the second number, not the first, so the answer is ";r(2) % %%9720%+((*9 )+1): % "Good": % "Correct": %"Right again": %"OK": %"That's it": %"Brilliant !": %"Well done": &"Very good": &"Genius !": &HWz=110 :r(z)=(*9 )+1:z:r(1)r(2)9800H&: &R#r(1)r(2)9800H& &\ &z "trap": & &"Hit C to continue" &a$= &a$="c"9950& &a$="r"9950& & 9920& & &z=1502:z & ed>o5 crhdT$sthd$ndse$etan$NAc mh2 HE H6:6:0: maths help tape 2  G C E Tutoring  Feb 1984 !w(20,8):2000 r(10 ) Lscrhdr=9300T$:qsthdr=9350$:endsec=9400$:getans=9450$ Laxes=9100#:triangle=9150#:diagram=218:praise=9720% &:"MATHS help mh2"::"TRIGONOMETRY" :" G C E Tutoring Feb 1984" 2l:"Hello. What's your name ?":n$:n$:"OK, ";n$;", I'll begin by telling you how to use this maths tape" 4V:"This tape covers 9 subjects which all come under the headingof trigonometry." :>:"Press the key marked C so I can type the next screenfull" <a$=:a$="c"62> = 60< >:z=1100d:z AF:"Thanks. From now on when I want you to press the C key, I'll say" D:" Hit C to continue" F1:"Here's the list of subjects on this tape": H"sine cosine tangent special angles 3D trigonometry functions through 360 graphs identities rules: sine & cosine" K2:"Which subject do you want to try ?":s$: Lc$=s$ N'(s$)>3s$=s$(13) Os$="sin"100d Ps$="cos"200 Qs$="tan" 300, Ss$="spe"600X Ts$="3D "700 Us$="fun"800  Vs$="gra"900 Ws$="ide"1000 Xs$="rul"1100L ^gs$"sin"s$"cos"s$"tan"s$"tab"s$"spe"s$"3D "s$"fun"s$"gra"s$"ide"s$"rul"9200# _ 70F c d f isine l n:" Sine" p6:"The diagram below shows a right angled triangle." s triangle vX"In trigonometry the name SINE isgiven to the ratio of the opposite/hypoteneuse" xscrhdr zZ:"Sine is a function. The sine of the angle at A gives the ratio opposite/hypoteneuse" } triangle #"so sin A = opposite/hypoteneuse" scrhdr <"Here is an example of trigonometry in action.:" ^"In the right angled triangle below, the side AC is 10cm long and the angle at A is 40." triangle #"What is the length of side BC ?" scrhdr "sin 40 = opposite/hypoteneuse = opposite/10 = BC/10 therefore 10 x sin 40 = BC" A:"sin 40 = 0.643 so BC = 10 x 0.643 = 6.43cm" qsthdr &:"Which of these ratios is sin A ?" ]:"1. opposite/adjacent 2. opposite/hypoteneuse 3. adjacent/hypoteneuse"  a$:a$ a$="2"praise a$"2""No, it's 2" .start=100d:return=172:n$="sine" endsec     cosine  :"cosine" 6:"The diagram below shows a right angled triangle." triangle X"In trigonometry the name COSINE is given to the ratio of the adjacent/hypoteneuse" scrhdr `:"Cosine is a function. The cosineof the angle at A gives the ratio adjacent/hypoteneuse" triangle #"so cos A = adjacent/hypoteneuse" scrhdr <"Here is an example of trigonometry in action.:" ^"In the right angled triangle below, the side AC is 10cm long and the angle at A is 40." triangle #"What is the length of side AB ?" scrhdr "cos 40 = adjacent/hypoteneuse = adjacent/10 = AB/10 therefore 10 x cos 40 = AB" A:"cos 40 = 0.766 so BC = 10 x 0.766 = 7.66cm" qsthdr &:"Which of these ratios is cos A ?" ]:"1. opposite/adjacent 2. opposite/hypoteneuse 3. adjacent/hypoteneuse"   a$:a$ a$="3"praise  a$"3""No, it's 3"  0start=200:return=272:n$="cosine" endsec  + , . 1 tangent 4 6 :"Tangent" 86:"The diagram below shows a right angled triangle." ; triangle >U"In trigonometry the name TANGENTis given to the ratio of the opposite/adjacent" @scrhdr Bb:"Tangent is a function. The tangent of the angle at A gives the ratio opposite/adjacent" E triangle H "so tan A = opposite/adjacent" Jscrhdr L<"Here is an example of trigonometry in action.:" O^"In the right angled triangle below, the side AB is 10cm long and the angle at A is 40." R triangle T#"What is the length of side BC ?" Vscrhdr Y"tan 40 = opposite/adjacent = opposite/10 = BC/10 therefore 10 x tan 40 = BC" \A:"tan 40 = 0.839 so BC = 10 x 0.839 = 8.39cm" ^qsthdr `&:"Which of these ratios is tan A ?" c]:"1. opposite/adjacent 2. opposite/hypoteneuse 3. adjacent/hypoteneuse" f h a$:a$ ja$="1"praise ma$"1""No, it's 1" p1start=300,:return=372t:n$="tangent" rendsec t    tables       W X Z ]special angles ` b:"special angle": dy"some angles have sines cosines and tangents that can be remembered as ratios rather thandecimals. They are:": g"sin 60 = 3/2 tan 60 = 3 sin 45 = 1/2 cos 45 = 1/2 cos 30 = 3/2 tan 30 = 1/3" jscrhdr l-"see if you can remember a few ratios.": n3"is sin 60 = 3/2 ? (true or false)" q a$:a$ ta$="true"praise va$"true""No, it's true" x:"is cos 45 = 1/2 ?" { a$:a$ ~a$="true"praise a$"true""No, it's true" 8start=600X:return=648:n$="special angles" endsec      3D trig  :"3D trigonometry": "Problems in 3 dimensions can be solved by reducing them to a number of simpler 2 dimensional problems. As well as perspectivedrawings, PLAN and ELEVATION allow a 3 dimensional object to be depicted in 2 dimensions." u:"The PLAN view is from directly overhead."::"The ELEVATION view is seen either from one side or one end." scrhdr 1n$="3D trig":start=700:return=725 endsec     " %functions through 360 ( *:"Functions through 360": ,"Although until now only angles between 0 and 90 have been considered, sin, cos and tan do have values for angles outside this range.": /["There are two stages in finding the sin,cos or tan of a value between 90 and 360:": 26"1. modify the angle 2. decide the sign" 3scrhdr 4d:"Modifying the angle means changing it in a certain way so that it is between 0 and 90." 6Z:"Deciding the sign means working out whether the value should be positive or negative" 9_:"To do either of these two steps it is necessary to know which QUADRANT the angle is in." :scrhdr <:"There are 4 quadrants:": >"0 to 90 1st quadrant 90 to 180 2nd quadrant 180 to 270 3rd quadrant 270 to 360 4th quadrant" @:"If the angle whose sin, cos or tan is required is A, then look up the angle A'. A' is easily calculated from the table below:" C:"If A is in 1st quadrant A'=A if A is in 2nd quadrant A'=180-Aif A is in 3rd quadrant A'=A-180if A is in 4th quadrant A'=360-A" Fscrhdr H"The sign of the value is worked out by looking up the quadrant in the table below. The table gives the function (sin,cos or tan) which is positive:": J"If A is in 1st quadrant ALL if A is in 2nd quadrant SINE if A is in 3rd quadrant TANGENT if A is in 4th quadrant COSINE" M:"To remember which function is positive in the 4 quadrants, thephrase ""All Silly Tom Cats"" is useful, as the initial letters are ASTC." Pscrhdr R,"Let's look at the two stages again:": T"If A is in 1st quadrant A'=A if A is in 2nd quadrant A'=180-Aif A is in 3rd quadrant A'=A-180if A is in 4th quadrant A'=360-A" W:"If A is in 1st quadrant ALL if A is in 2nd quadrant SINE if A is in 3rd quadrant TANGENT if A is in 4th quadrant COSINE" Z:"so to find sin 170, (2nd quadrant) look up sin (180-170) ie. sin 10. Looking at the 2nd table, you'll see that the sign is positive." \scrhdr ^qsthdr a d2"If A is in the 2nd quadrant, do you look up:": fp"1. 360-A 2. A-180 3. 180-A 4. A ?": ha$:a$="3"praise ia$"3""No, it's 3" k2"If A is in the 4th quadrant, do you look up:": np"1. 360-A 2. A-180 3. 180-A 4. A ?": pa$:a$="1"praise ra$"1""No, it's 1" uscrhdr x@n$="functions through 360":start=800 :return=892| zendsec |    graphs  :"Graphs": |"This section consists of graphs of the functions sin, cos and tan. Their values are plotted for the range 0 to 360" :"If you learn the shapes of the three curves, you will be able to make good guesses at the values of the functions for any angle. This will make it easy to spot big errors in your working." scrhdr  "Sine": atype=0:axes 30,90Z +t=02*-0.06|u\0.06|u\ )2,502*((t+0.06|u\)-(t)) t z=16::z scrhdr  "cosine": atype=0:axes 30,140 +t=02*-0.06|u\0.06|u\ )2,502*((t+0.06|u\)-(t)) t z=16::z scrhdr  "tangent": atype=1:axes 30,90Z +t=0̧/2-0.65&fff0.06|u\ #y=502*((t+0.06|u\)-(t))  2,y <t:30+20+90Z*200/360h,25 @t=/2+0.65&fff3*/2-0.65&fff0.06|u\ #y=502*((t+0.06|u\)-(t))  2,y Ft:30+10 +20+270*200/360h,22 5t=3*/2+0.65&fff2*0.06|u\ #y=502*((t+0.06|u\)-(t))  2,y t  :scrhdr 0n$="graphs":start=900:return=998 endsec      identities  :"Identities": "An identity is a formula which is always true. For example: 0x = 0 for all values of x Identities allow complicated expressions to be simplified.": I"Probably the most important of all trigonometric identities is:": "sinA + cosA = 1": X"This is true for all values of A. It can be proved by Pythagorus' theorem." scrhdr X" AC = AB + BC so (AB/AC) + (BC/AC) = 1 so cosA + sinA = 1"  triangle scrhdr $"Another important identity is:":  "sin A/cos A = tan A": "This is proved like this:":  triangle `"sin A/cos A = (BC/AC) / (AB/AC) = (BC/AC) x (AC/AB) = BC/AB = tan A" scrhdr D"Here is an example of the use of the identity sin+cosA=1:": P"If sin 15 = 0.259 what is 1. cos 15 2. tan 15 ?"  :"First calculate cos 15:": " sin15 + cos15 = 1 so (0.259)+cos15 = 1 so 0.067+cos15 = 1 so cos15 = 1-0.067 = 0.933 so cos 15 = 0.933 = 0.966" "e:"since tan15 = sin15/cos15 so tan15 = 0.259/0.966 = 0.268" #scrhdr $qsthdr &:"sinA+cosA= ?" )-return=1070.:exp=1:egc=3 ,getans .:"sinA/cosA = ?" 0 a$:a$ 3a$="tanA"a$="tan A" praise 6-a$"tanA"a$"tan A""No, it equals tan A" 86start=1000:return=1085=:n$="identities" :endsec = K L N Qsine and cos rules T V:"sine and cosine rules": Xm"These rules apply to triangles that do not have the right angleessential to the previous sections." [:"The sine rule allows the calculation of all the angles and side lengths in a triangle from 2 known angles and one known side length." ^:"The cosine rule allows the calculation of all the angles and side lengths in a triangle from one known angle and 2 known side lengths." `scrhdr bL"In the triangle below the sine rule is:"::"a/sinA = b/sinB = c/sinC": e"The cosine rule is also in threeparts:"::"a=b+c-2bc cosA b=c+a-2ac cosB c=a+b-2ab cosC" hN30,20:61=,0:-30,502:-30,-502 j.12 ,8;"A":18,7;"a" l/19,2;"B":15,10 ;"b" o19,12 ;"C" r15,4;"c" s:: tscrhdr v<start=1100L:return=1148|:n$="sine & cos rules" yendsec |  x$="": o=1̱x$ n=18 #w(o,n):x$(oo)+n-1,w(o,n) n o  H24,16,16,32 ,16,16,24,0 C24,8,8,4,8,8,24,0 F0,0,62>,64@,64@,64@,62>,0 E0,0,124|,2,2,2,124|,0 G0,66B,66B,66B,66B,66B,44,,0 G0,44,,66B,66B,66B,66B,66B,0  G0,0,60<,64@,124|,64@,60<,0 I240,8,120x,128,248,0,0,0 C0,0,0,4,126~,4,0,0 I248,136,136,136,0,0,0,0 C4,4,228,4,0,0,0,0 G240,8,240,8,240,0,0,0 G28,20,16,16,8,8,40(,568  E4,68D,228,68D,0,0,0,0 "G16,16,16,0,0,480,72H,480 %E12 ,18,18,12 ,0,0,0,0 (E7,4,4,8,8,144,80P,32 4 )::(40(*3.14159Iρ/180) #' # # #axes # #_30,90Z:210,0:30,90Z:0,557:0,-110n #10 ,2;"0" #f11 ,9 ;"90":11 ,15;"180":11 ,22;"270":11 ,29;"360" #atype=09124# #atype=19130# #.4,2;"1":16,1;"-1" #07,1;".5":13 ,0;"-.5" # #.7,2;"1":13 ,1;"-1" #.4,2;"2":16,1;"-2" # # # #draw triangle # #N30,60<:100d,0:0,60<:-100d,-60< #15,6;"adjacent" # 8,1;"hypoteneuse" #11 ,18;"opposite" #6123{,60<:0,7:7,0 #H41),60<:0,2:-3,2:-2,0 #14,2;"A" #14,17;"B" #6,17;"C" #::::::::: # # #)::9200#+((*6)+1): #"Berk ! What's ";c$;" ?": #'"Wally ! ";c$;" isn't on the tape.": ##"Moron ! Try spelling straight": #"sorry, ";n$;" ? ": #"Are you tired, ";n$;" ?": #"Don't wind me up, ";n$: $T $V scrhdr() $Y $\9900&:: $ $ qsthdr() $ $'9600%:9900&:egc=0: $ $ $endsec(start,n$) $ $5"That finishes ";n$:"Hit r to repeat this section" $ 9900& $a$="r"start $return $ $ $getans(loop,exp) $ $?dp=0:err=0:u$:u$:(u$)>10 err=1 $(u$)=09460$ $[j=1̱(u$):(u$(jj))<45-ů(u$(jj))>579ů(u$(jj))=47/err=1 $%9480%:j:u$="-."err=1 $Cerr=1dp>1"type the answer as a number":9460$ $=ans=(u$):(exp-ans)>0.005y# ="No, the answer is ";exp $$(ans-exp)0.005y# =9720% $return $ % %$(u$(jj))=46.dp=dp+1 % ,j>1Ư(u$(jj))=46.err=1 %  %k9800H&:d1=r(1)/10 +r(2)/100d+r(3)/1000:dec=d1:int=1000*dec %& %A"See if you understand this by trying the following examples" % 9800H& % %"True or false ?" %a$ %a$ %a$"t"a$"f"9650% % %n:n;"/";:d:d: % 9630% %h"No. Don't add the numbers together. It's just the second number, so the answer is ";r(2) % %K"No, its the second number, not the first, so the answer is ";r(2) % %%9720%+((*9 )+1): % "Good": % "Correct": %"Right again": %"OK": %"That's it": %"Brilliant !": %"Well done": &"Very good": &"Genius !": &HWz=110 :r(z)=(*9 )+1:z:r(1)r(2)9800H&: &R#r(1)r(2)9800H& &\ &z "trap": & &"Hit C to continue" &a$= &a$="c"9950& &a$="r"9950& & 9920& & &z=1502:z & % >@@@>||BBBBB,,BBBBB<@|@<x~(8DD0H0  P X 5 crhdT$sthd$ndse$etan$xe#riangl#iagrarais%* mh3 86-`86:6:0: maths help tape 3  G C E Tutoring  Jan 1984 !w(10 ,8):2000 r(10 ) Lscrhdr=9300T$:qsthdr=9350$:endsec=9400$:getans=9450$ :"MATHS help mh3"::"SETS" :" G C E Tutoring Jan 1984" 2l:"Hello. What's your name ?":n$:n$:"OK, ";n$;", I'll begin by telling you how to use this maths tape" 4N:"This tape covers 5 subjects which all come under the headingof sets." :>:"Press the key marked C so I can type the next screenfull" <a$=:a$="c"62> = 60< >:z=1100d:z AG:"Thanks. From now on when I want you to press the C key, I'll say:" D:" Hit C to continue" F1:"Here's the list of subjects on this tape": H"terms set laws Venn diagrams De Morgan's laws practical applications" K2:"Which subject do you want to try ?":s$: Lc$=s$ N'(s$)>3s$=s$(13) Os$="ter"100d Ps$="set"200 Qs$="Ven"s$="ven" 300, Rs$="De "s$="de "500 Ss$="pra"600X ULs$"ter"s$"set"s$"Ven"s$"ven"s$"de "s$"De "s$"pra"9200# _ 70F c d f iterms l n :"terms": p"There are a lot of words used inthe mathematics of sets that make it seem harder than it really is. The obvious one to start with is the word SET." s:"A set is a collection of things called ELEMENTS. The elements can be almost any thing; for example they might be letters of the alphabet or colours of the spectrum." uscrhdr v$:"This is how to write a set :": x"A = a,b,c" z{:"A set is represnted by a capitalletter, and the elements are enclosed in curly brackets when they are listed." { }3"The symbol means ""is an element of "" " 3:"so b A means that b is an element in A." scrhdr :"Two sets are equal if they have the same elements. If one set has all the elements of another and some others as well, it is called a SUPERSET.": "A =1,2,3 B = 1,2,3,4,5": "so B is a superset of A. This iswritten :"::"B A"::"Another way of writing it is by saying that A is a subset of B. This is written :"::"A B" scrhdr l"If the elements of two sets are grouped together to make one bigset, the UNION of two sets is formed. " :"eg. C = a,b D = c,d" :"C D = a,b,c,d" B:"The INTERSECTION of two sets arethe elements common to both." :"eg. C D = " :"There are no elements in both C and D, so the intersection is an empty set. Two sets with no common elements are called DISJOINT." scrhdr J"The number of elements in a set is written like this :"::"n(C) = 2": D"Notice that the brackets around the name of the set are ordinary" .:"See if you've learnt the symbols." scrhdr *"What does the symbol mean ?":a$:a$ a$="subset"9720% $a$"subset""No, it means subset" ,:"What does the symbol mean ?":a$:a$ a$="intersection"9720% 0a$"intersection""No, it means intersection" ,:"What doea the symbol mean ?":a$:a$ a$="superset"9720% (a$"superset""No, it means superset" 7start=100d:return=188:n$="terms":endsec     set laws  :"set laws": d"The commutative, associative anddistributive laws apply to sets.They take the following form :": H"Commutative law :"::"A B = B A and A = B A" b:"Notice the similarity with the commutative law when applied to addition and multiplication." scrhdr "associative law": !"(A B) C = A (B C) and" "(A B) C = A (B C)" Y:"Again there is a similarity withthe laws as applied to addition and multiplication." : "distributive law"  c"A (B C) = (A B) (A C) and A (B C) = (A B) (A C)" c:"These laws can be verified by the use of Venn diagrams. This is shown in the next section." scrhdr :start=200:return=250:n$="set laws":endsec  + , . 1venn diagrams 4 6:"A Venn diagram is used to simplify operations on sets. Sets are represented as ballons within a rectangle - the rectangle representing all possible elements." 7scrhdr 8.:"Here is a set shown as a Venn diagram." 9315;:322B ;`502,140:100d,0:0,-62>:-100d,0:0,62> >100d,110n,20 @6,12 ;"A" A B:::::scrhdr D E&"A subset of A is shown like this": H 315; J100d,103g,10 L8,12 ;"B" M::: Oscrhdr R("The union of two sets looks likethis" S340T:346Z T 315; V80P,95_,15 Y10 ,9 ;"B": Z:: \9"The union is represented by the whole area of A and B" ^scrhdr `@:"The intersection of two sets is represented like this :" c 340T f%85U,100d:0,9 g%89Y,94^:0,11 h::: jK"The intersection is the overlap of the two sets, and is shown shaded." mscrhdr pL:"Disjoint sets, in which no intersection occurs, look like this :" r 315; t67C,95_,11 w9 ,8;"B" z::: |scrhdr ~q:"The commutative, associative anddistributive laws can all be verified by the use of Venn diagrams.": "The associative law will be taken as an example. Venn diagrams for both sides of the equation will be presented." scrhdr :"Associative law :": "(A B) C = A (B C)" _0,120x:100d,0:0,-62>:-100d,0:0,62> a105i,120x:100d,0:0,-62>:-100d,0:0,62> x502,80P,20:502-12 ,80P+15,20:502+12 ,80P+15,20 105i+502,80P,20:105i+502-12 ,80P+15,20:105i+502+12 ,80P+15,20 ,8,4;"A":8,8;"B" 13 ,6;"C" ^8,4+13 ;"A":8,8+13 ;"B":13 ,6+13 ;"C" '502,112p:0,-30 -502-3,83S:0,24 %535,83S:0,24 x=0:y=0 415:425 )30+x,80P+y:40(,0 R32 +x,86V+y:36$,0:32 +x,73I+y:36$,0 R37%+x,92\+y:26,0:37%+x,68D+y:26,0 )43++x,98b+y:14,0  13 ,6;"C" -x=105i-12 :y=14:415 8,17;"A" (152,100d:0,-17 0152+5,100d:0,-22 9152+10 ,100d-3:0,-22 9152+16,100d-6:0,-19 ::::::"The area shaded both vertically and horizontally is the same in each diagram. This shows that the two sides of the equation are equal." scrhdr %n$="Venn diagrams":start=300, return=452 endsec     De Morgan's laws  :"De Morgan's laws deal with the simplification of set operationsinvolving COMPLEMENTS. The complement of a set is the set of all elements it does not contain. In a venn diagram this is represented as the area outside the ballon of the set, but within the rectangle." 0:"The complement of a set A is written A'" scrhdr A:"There are two De Morgan laws. "::"Here is the first one"  "(A B)' = A' B'"  A:"This can be used to simplify quite complicated problems."  ::scrhdr N:"The second law is similar to thefirst, but with the signs swapped."  "Here it is :": "(A B)' = A' B'" \:"This can be used in many occaisions to perform the same function as the first." scrhdr Bn$="De Morgan's laws":start=500:return=545!:endsec ! W X Z ]large example ` bw:"Having read the previous sections on sets it may be hard to see how these laws may be used practically." d~:"Here is an example of a questionwhich at first sight seems very complicated, but can be solved quite easily using sets." escrhdr g:"In a class of 30 pupils 10 studyphysics, 23 study maths and 16 study art. 14 pupils study art and maths, 6 study art and physics, and 9 study maths and physics. 3 pupils study maths physics and art." h:"(a) how many pupils do not study any of these three subjects.(b) how many study art and maths but not physics. (c) how many study art only." iscrhdr j:"The first thing to realise is that the statement that 10 pupilsstudy physics DOES NOT imply that they study only physics." lV:"This is how the information in the question looks written in set notation :": n"class size n(E) = 30 maths n(M) = 23 physics n(P) = 13 art n(A) = 19 maths & physics n(M P) = 9 maths & art n(M A) = 14 art & physics n(A P) = 6 all 3 n(M P A) = 3" qscrhdr rF:"First fill in the number of pupils that study all 3 subjets." t 630v u 650 va20,35#:0,100d:160,0:0,-100d:-160,0 x$75K+2,70F,30 {-65A+502-2,70F,30 ~%95_,70F+30,30 12 ,12 ;"3" 6,12 ;"P" /15,8;"M":15,15;"A"  :::: scrhdr a"Next, fill in the remaining overlaps, subtracting 3 from each to allow for the centre."  655  665  630v 14,11 ;"11" 010 ,13 ;"3":10 ,10 ;"6"  48,24;"AP":9 ,24;"6-3=3" 95,24;"AM=14":6,24;"14-3=11" 611 ,24;"MP":12 ,24;"9-3=6"  :::: scrhdr a"Now fill in the remaining spacesso that the totals for each subject are the values given."  674  682  655 /13 ,8;"2":13 ,15;"3" -8,12 ;"1":8,4;"1" 5,23;"13-6-3-3":6,23;"=1":8,23;"19-11-3-3":9 ,23;"=2":11 ,23;"23-11-3-6":12 ,23;"=3"  ;:::::"It is now possible to answer the question." scrhdr W"The diagram shows 1 person outside the ballons representingthe three subjects."  674 (:::::"so the answer to (a) is 1." scrhdr A"The overlap between art and maths, but not physics, is 11"  674 *::::::"so the answer to (b) is 11" scrhdr E"The area of the art baloon with no overlap on other subjects is 3"  674 )::::::"so the answer to (c) is 3" scrhdr Kstart=600X:return=720:n$="the practical example":endsec   x$="": o=17 n=18 #w(o,n):x$(oo)+n-1,w(o,n) n o  H24,16,16,32 ,16,16,24,0 C24,8,8,4,8,8,24,0 F0,0,62>,64@,64@,64@,62>,0 E0,0,124|,2,2,2,124|,0 G0,66B,66B,66B,66B,66B,44,,0 G0,44,,66B,66B,66B,66B,66B,0  G0,0,60<,64@,124|,64@,60<,0  #' #)::9200#+((*6)+1): #"Berk ! What's ";c$;" ?": #'"Wally ! ";c$;" isn't on the tape.": ##"Moron ! Try spelling straight": #"sorry, ";n$;" ? ": #"Are you tired, ";n$;" ?": #"Don't wind me up, ";n$: $T $V scrhdr() $Y $\9900&:: $ $ qsthdr() $ $'9600%:9900&:egc=0: $ $ $endsec(start,n$) $ $5"That finishes ";n$:"Hit r to repeat this section" $ 9900& $a$="r"start $return $ $ $getans(loop,exp) $ $ ans:ans $ans=exp9720% $$egc=egc+1:egc<3loop $return $ % %k9800H&:d1=r(1)/10 +r(2)/100d+r(3)/1000:dec=d1:int=1000*dec %& %A"See if you understand this by trying the following examples" % 9800H& % %"True or false ?" %a$ %a$ %a$"t"a$"f"9650% % %n:n;"/";:d:d: % 9630% %h"No. Don't add the numbers together. It's just the second number, so the answer is ";r(2) % %K"No, its the second number, not the first, so the answer is ";r(2) % %%9720%+((*9 )+1): % "Good": % "Correct": %"Right again": %"OK": %"That's it": %"Brilliant !": %"Well done": &"Very good": &"Genius !": &HWz=110 :r(z)=(*9 )+1:z:r(1)r(2)9800H&: &R#r(1)r(2)9800H& &\ &z "trap": & &"Hit C to continue" &a$= &a$="c"9950& &a$="r"9950& & 9920& & &z=1502:z &   >@@@>||BBBBB,,BBBBB<@|@<X 5 crhdT$sthd$ndse$etan$