{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 128 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 128 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 128 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 128 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 128 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 128 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 128 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 0 128 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 277 "" 0 1 0 128 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 128 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 0 1 0 128 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 281 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 282 "" 0 1 0 128 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 283 "" 0 1 0 128 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 285 "" 0 1 0 128 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 286 "" 0 1 0 128 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 287 "" 0 1 192 192 192 1 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 288 "" 0 1 128 128 128 1 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "N ormal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 257 50 "High School Modul es > Algebra by Gregory A. Moore " }}{PARA 3 "" 0 "" {TEXT -1 4 " \+ " }{TEXT 256 21 "The Nature of Numbers" }}{PARA 0 "" 0 "" {TEXT -1 109 "\nAn exploration of the various subsets and properties of the rea l numbers, and interesting features of each.\n" }}{PARA 0 "" 0 "" {TEXT 258 153 "[Directions : Execute the Code Resource section first. \+ Although there will be no output immediately, these definitions are us ed later in this worksheet.]" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 38 " 1. Natural Numbers and Whole Numbers " }}{PARA 0 "" 0 "" {TEXT -1 228 "\nMankind began as most children do. ..counting objects they see. Numbers are the first level of abstractio n in mathematics because there is nothing in common with five apples, \+ five rocks, and five days... except the number five!\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "`\\nSome Natural Numbers ` = \{k $ \+ k = 1..100\};" }}}{PARA 0 "" 0 "" {TEXT -1 137 "\nIt took many centuri es before mankind introduced a new number ...0... a number which signi fies no objects at all. This was a huge step.\n" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 45 "`\\nSome Whole Numbers ` = \{k $ k = 0..100 \};" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 29 " 2. How Big Can Numbers Get?" }}{PARA 0 "" 0 " " {TEXT -1 14 "\nPretty big!\n\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "3^2; `\\nThis has this many digits :` ,length(%);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "3^20; `\\nThis has thi s many digits :` ,length(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "3^200; `\\nThis has this many digits :` ,length(%);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "3^2000; `\\nThis has thi s many digits :` ,length(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "3^20000; `\\nThis has this many digits :` ,length(%);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 178 "\nLets t ry to roughly compute some common numbers we may encounter. Would you \+ guess that there are more blades of grass on a football field, or prin ted characters in a library?\n\n\n" }{TEXT 260 24 " Little Question 2.1 " }{TEXT -1 105 ": A movie theatre has 40 rows with 26 seat per \+ row. How many people are there when the theatre is full?\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "40*26;" }}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 261 24 " Little Question 2.2 " }{TEXT -1 148 ": How \+ many characters are there on the page of a book?\n (count the number of lines per page and the average number of characters per lin e)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "52*80;" }}}{PARA 0 "" 0 "" {TEXT -1 2 "\n\n" }{TEXT 262 24 " Little Question 2.3 " } {TEXT -1 53 ": How many characters are there in a 350 page book?\n" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "350*52*80;" }}}{PARA 0 "" 0 "" {TEXT -1 2 "\n\n" }{TEXT 263 24 " Little Question 2.4 " } {TEXT -1 116 ": How many books are there in the library? (42 bookcase s, with 6 rows of book per shelf, with 32 books per shelf).\n" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "42*6*32;" }}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 264 23 " Little Question 2.5 " }{TEXT -1 54 " : How many printed characters are there in a library?\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "(350*52*80) *(42*6*32) ;" }}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 265 23 " Little Question 2.6 " } {TEXT -1 136 ": If there are 19 blades of grass in a square inch of la wn, how many blades of grass are there on a football field 350 feet by 125 feet?" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "19 * (12 * 350 ) * (12 * 125);" }}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 259 24 " \+ Little Question 2.7 " }{TEXT -1 190 ": If there are 400 grains of san d in a teaspoon of sand that would cover a square inch if spreadout, h ow many grains of sand are there on the top surface of a beach 200 fee t wide, by 3 mile?" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "400 * \+ (12*12) * 200 * 3* 5280;" }}}{PARA 3 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 14 " 3. Integers" }}{PARA 0 "" 0 "" {TEXT -1 15 "\nThe integers, " }{TEXT 266 1 "Z" }{TEXT -1 82 ", consist of t hree components : natural numbers, zero, and negatives of naturals.\n " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "`\\nSome Integers ` = \{ k $ k = -50..50\};" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "`\\n Positive Natural Numbers ` = \{k $ k = 1..50\};\n`\\nZero ` = \{k $ k \+ = 1..50\};\n`\\nNegative Natural Numbers ` = \{k $ k = -50..-1\};" }}} {PARA 0 "" 0 "" {TEXT -1 300 "\n\nSome people like to think of a bank \+ statement. Deposits are positive numbers, and checks or withdrawals ar e negative numbers. Two deposits add to a greater balance, while two w ithdrawals add to be equivalent to a greater withdrawal. This principl e is seen when adding two numbers of the same sign.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "(123) + (456);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "(-123) + (-456);" }}}{PARA 0 "" 0 "" {TEXT -1 333 "\n\nHowever adding numbers of opposite signs causes a net transaction that depends on which is larger. A deposit and withdrawal at the same time can result in a net deposit if the deposit is bigger, or a net w ithdrawal if the withdrawal is bigger. We compute the result by subtra cting the two numbers and taking the sign of the larger." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "(123) + (-456);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "(-123) + (456);" }}}{PARA 0 "" 0 "" {TEXT -1 263 "\n\n\nNote that negative numbers have this reversal property w hen it comes to greater than. 3 is less than 4, but -3 is greater than -4. Thats because its better to have $4 in the bank rather than $3, a nd its also better to be only $3 in debt rather than $4 in debt." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "3 < 4 ; -3 > -4;" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 22 " 4. Rational Numbers" }}{PARA 0 "" 0 "" {TEXT -1 105 "\nRational num bers are defined as the set of all fractions of integers where the den ominator is non-zero.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "Q := \{ a/b \};" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "`\\nSome \+ Rational Numbers ` = [ seq( seq( i/j, i = 1..j-1), j = 1..10)];" }}} {PARA 0 "" 0 "" {TEXT -1 5 "\n " }{TEXT 267 45 "Converting from Rat ional to Repeating Decimal" }{TEXT -1 229 "\n\nALL rational numbers ca n be expressed as numbers with repeating decimals. Some numbers (whose quotients are powers of 2 and 5) have terminating decimals ... or you can also think of them as numbers with repeating zero decimals." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "1/64: % = evalf(%,70);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "1/625: % = evalf(%,70);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "1/2500: % = evalf(%,70);" }} }{PARA 0 "" 0 "" {TEXT -1 135 "\nOther rational numbers have infinitel y repeating periods of digits. Can you state how many digits are in th e repeating period of eac/\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "2/3: % = evalf(%,70);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "1/14: % = evalf(%,70);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "10/13: % = evalf(%,70);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "5/39: % = evalf(%,70);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "8/19: % = evalf(%,70);" }}}{PARA 0 "" 0 "" {TEXT -1 8 "\n\n\n \+ " }{TEXT 268 45 "Converting from Rational to Repeating Decimal" } {TEXT -1 162 "\n\n\nIt is not obvious, but EVERY number with a repeati ng sequence of digits must be a rational number. That means it can be \+ expressed as a fraction.\n\n " }{TEXT 269 10 "Example 1 " }{TEXT -1 2 ": " }{XPPEDIT 18 0 "4.52380952380952380952380952380952380 952380952380952380952381;" "6#-%&FloatG6$\"gn\"Q_4Q_4Q_4Q_4Q_4Q_4Q_4Q_ 4Q_4Q_%!#f" }{TEXT -1 2 "\n\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "secret := evalf( 95/21, 60);" }}}{PARA 0 "" 0 "" {TEXT -1 279 " \nWe notice that there is a period of six repeating digits. So if we \+ \"slide\" the number over by a facor of 10^6, the periods will still b e lined up. Then if we subtract the original from this one, all of the infinite part of the decimal will cancel - leaving only an integer pa rt." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "A := round(secret*10^ 6 - secret);" }}}{PARA 0 "" 0 "" {TEXT -1 122 "\nThis integer is equal to (10^6 - 1) times the original number S. We can now solve for S, as the quotient of two integers." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "S*10^n - S = S*(10^n - 1);\na = S*(10^n - 1); solve(%,S);" }}} {PARA 0 "" 0 "" {TEXT -1 58 "\nLets execute this idea ....to find the \+ original fraction." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "A/(10^ 6 - 1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "`Check : `,% \+ = evalf( %, 40);" }}}{PARA 0 "" 0 "" {TEXT -1 30 "\n\nLets try another example. \n " }{TEXT 270 23 " Example 2 " }{TEXT -1 2 ": \+ " }{XPPEDIT 18 0 ".227272727272727272727272727272727272727272727272727 27272727;" "6#-%&FloatG6$\"fnFFFFFFFFFFFFFFFFFFFFFFFFFFFFF#!#f" } {TEXT -1 79 "\n\n\nThis one has a period of only two digits. However i ts a little more tricky.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "secret := evalf( 19/22, 60);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "A := round(secret*10^4 - secret*10^2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "S = A/(10^4 - 10^2);" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 18 " 5. Real Numbers" }}{PARA 0 "" 0 "" {TEXT -1 229 "\nReal numbers include both rational and irrational numb ers. Irrational numbers can not be expressed as fractions. Here are so me examples. You can try looking for repeating decimals if you wish, b ut don't waste too much time. ;->\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "Pi: % = evalf(%,1000);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "sqrt(2): % = evalf(%,500);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "sqrt(Pi): % = evalf(%,500);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 25 "exp(1): % = evalf(%,500);" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 49 "sqrt( 2 + sqrt( 1 + sqrt( 3))); % = evalf(%, 500);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "2^(1/3); % = evalf (%,500);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "1/(Pi + sqrt(5) + exp(1)); % = evalf(%,500);" }}}{PARA 0 "" 0 "" {TEXT -1 2 "\n " }}} {SECT 0 {PARA 4 "" 0 "" {TEXT -1 27 " 6. Properties of Numbers" }} {PARA 0 "" 0 "" {TEXT -1 89 "\nThere are a number of properties of rea l numbers which underly all that we do in math.\n\n" }{TEXT 272 7 " \+ " }{TEXT 273 22 "1. Identity Properties" }{TEXT 274 193 "\n\nFor t he operations of addition and multiplication there are special numbers which are identities - they don't change other numbers. The identity \+ for addition is 0 and for multiplication is 1." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "a + 0; 0 + a; 0 + 12324;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 23 "a*1; 1*a;; 1* 12324;" }}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 275 7 " " }{TEXT 276 21 "2. Inverse Prope rties" }{TEXT 277 249 "\n\nIn regard to both addition and multiplicati on, each number has an inverse - a number that cancels out the origina l number and transforms it into the inverse. The additive inverse for \+ a is -a, and the multiplicative inverse for a (non-zero) is 1/a." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "a + (-a); 3 + (-3); (3 24234/2306491) + (-324234/2306491);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "a*(1/a); 234 * (1/234); (345/671) * (671/345); " }}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 278 7 " " }{TEXT 279 26 "3. Associative Properties" }{TEXT 280 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 " 3 + ( 4 + 5); (3 + 4) + 5;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "50723423 + ( 1812388 + 99993211 ); \n( 50723423 + 1812388 ) + 99993211 ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "3 + ( 4 + 5); (3 + 4) + 5;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "50723423 * ( 1812388 * 99993211 );\n( 50723423 * \+ 1812388 ) *99993211 ;" }}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 281 7 " " }{TEXT 282 25 "4. Commutative Properties" }{TEXT 283 1 "\n " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "a + b = b + a;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "79234091912091147546 + 48567 65464338;\n4856765464338 + 79234091912091147546;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "a * b = b* a;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "1299934 * 3692010; 3692010*1299934;" }}}{PARA 0 "" 0 "" {TEXT -1 2 "\n\n" }{TEXT 284 7 " " }{TEXT 285 26 "5. Distri butive Property\n\n" }{TEXT 286 150 "This is the only property which i nvolves both operations at the same time. All of the previous rules in volved addition or multiplication but not both." }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 20 "a*(b+c) = a*b + a*c;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "7105871 * ( 10344392 + 47183017439 );\n( 710587 1 * 10344392) + (7105871 * 47183017439 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "(10000)*(333*x + 777777);" }}}{PARA 0 "" 0 "" {TEXT -1 2 "\n\n" }{TEXT 271 1 " " }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT 287 4 "\n " }{TEXT 288 56 " \251 2002 Waterloo Maple I nc all rights reserved." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "0 \+ 1" 7 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }