ࡱ> `b_#` <bjbjmm 7@ F(F(F(F($j(LF())));;;EFFFFFF$GhI%Fi;a7;;;%F)) F%<%<%<;)@)E%<;E%<%<rDTE)( sEF(;#EEF0F1ExJ;vxJExJE0;;%<;;;;;%F%F<;;;F;;;;$"($"( Rational and Irrational Numbers Terminating decimals e.g.:  EQ \f(1,8)  =  EQ \f(125,1000)  = 0.125 Recurring decimals e.g.:  EQ \f(2,7)  = 0.285714285714. An Integer is any positive or negative whole number including zero. A Rational Number can be expressed as  EQ \f(a,b)  where a and b are both integers. e.g.: 5 =  EQ \f(5,1)  0.25 =  EQ \f(1,4)  0.666666 =  EQ \f(2,3)  "36 = 6 =  EQ \f(6,1)  An Irrational number is not rational (cannot be written as a fraction), does not terminate or have a recurring pattern. Famously this includes  (3.141592654& & .) and any square root that is not an integer like "35 The set of all Rational and Irrational numbers are called Real numbers. ------------------------------------------------------------------------------------------------------- Question 1 Which of these numbers are rational and which are Irrational? (Put an R or I next to each one.)   Question 3 Write down a Rational number between 3 and 4 (show and explain how you got this answer) Question 4 Write down an Irrational number between 3 and 4 (show and explain how you got this answer) Question 5 Write down a number that is greater than 17 and less than 18, and has a rational square root. (show and explain how you got this answer) Question 6 An Irrational number is multiplied by another Irrational number. Write down an example to show that the answer could be a rational number. Show one way where two of the same numbers are used, and a way where two different numbers are used. HINT: Use the square root rule "10 x "2 = "20 Question 7 Show a way where a rational number x irrational number = Irrational number HINT: use  Question 8 Show 2 different Irrational numbers ( a and b) where  EQ \f(a,b)  is a Rational number. HINT: use square roots HOMEWORK SHEET RATIONAL/IRRATIONAL NUMBERS   Remember Pythagoras theorem?  _____________________________________________________________________ Exam Question 2002 Right-angled triangles can have sides with lengths which are a rational or irrational number of units. Give an example of a right-angled triangle to fit each description below. Explain your answers carefully for full marks. All sides are Rational     The hypotenuse is Rational and the other two sides are Irrational.    The hypotenuse is Rational and the other two sides are Rational  The hypotenuse and one of the other sides are Rational and the remaining side is Irrational.   Changing decimals to fractions Terminating decimals are easily changed to fractions. Always keep going to the simplest form. E.g.: 0.6 =  EQ \f(6,10)  =  EQ \f(3,5)  and 0.63 =  EQ \f(63,100)  and 0.648 =  EQ \f(648,1000)  =  EQ \f(324,500)  =  EQ \f(162,250) =  EQ \f(81,125)  _________________________________________________________________ Question 1 Change these decimals to fractions. (a) 0.85 (b) 0.408 (c) 0.0256 (d) 0.0125 Recurring decimals can also be converted into fractions but the method is far more ingenious! Example Change 0.8 recurring to a fraction (0.88888888.)     3 3.14  3  EQ \f(1,7)   EQ \f(142,45)  "5 "6 "10 0.6 (" 5)3  EQ \f(4,9)  0.4 Cos 60 2 Question 2 Using numbers from the cloud write down two numbers whose SUM is Rational (b) Using numbers from the cloud write down two numbers whose PRODUCT is Rational. a a b a2 + b2 = c2 c R R I R I R 5 3 4 32 + 42 = 9 + 16 = 25 and  !,7;=>JKQRcdnw   + , 4 5 : ; O S c d p q   . 0 H ^ f ; h8h5CJ\mH sH h8h6]mH sH jh8hUmH sH h8h8mH sH h8h5\mH sH h8hmH sH h8h>*CJmH sH < !mnO 2 8 B 2  I J ! " $ % $a$S< " # 0 1 N W xz)*,/01WXg  PR;?CHKLYZ`h8h8mH sH jh8hUmH sH h8h5CJ\mH sH h8h5\mH sH !jh8h8CJUmH sH h8hmH sH !jh8hCJUmH sH :% & ' ( ) * + , - . / 0 2 3 4 5 6 A     "$&(*,.D)+,0TUVWYZg & F    PSTUVWXY:;9: & F`amnx|Fj?@DEFGNRS ":<TVtvzԺԩԩԩԩԢjh5U\h5CJ\ h5\!jh8hCJUmH sH h8ht;dmH sH jh8hUmH sH h8hmH sH h8h5\mH sH #jh8h5U\mH sH 5:Fjk?ABCDFKLMNSxz  (*6.6066788V8Z888889@9z9~999999:4:Z: ;;(;*;&<@<R<f<<<<<<<<<<踴h8hmH sH jhUhPv/h56\] hCJ h6]U hH* h5\jh5U\hh5CJ\h5CJH*\;bcdh`h & F  6$6&666476788V8X8Z88899R9T99"25 = 5 Let f stand for the fraction which is equivalent to 0.8 recurring. If we multiply f by 10, each figure 8 moves one place to the left. Now subtract the first equation from the second. The infinite string of 8s above one another will cancel out. Then we divide both sides by 9. f = 0.888888 10f = 8.888888 9f = 8.000000 so the fraction f =  EQ \f(8,9)  Question 3 But what if only part of the decimal recurs? Like in 0.433333333333333333333 Save the 0.4 for later and work out the 0.03 as a fraction first. Later, add the 0.4 ( EQ \f(4,10) ) on to the fraction found for 0.03 Question 2 Use this method to change the following recurring decimals to fractions. 0.7777777777& (b) 0.4848484848& (hint: use 100f) (c) 0.731731731731& (hint: use 1000f) 999999:\:^:r;t;v;x;;;$<&<B<<<<<<h^h & F501h:p8/ =!"#$% @@@ NormalCJ_HaJmH sH tH @@@ Heading 1$$@&a$5\:@: Heading 2$@&5\>@> Heading 3$@& 5CJ\H@H Heading 4$$@&a$ 5>*CJ\DA@D Default Paragraph FontViV  Table Normal :V 44 la (k(No List 4B@4 Body Text5\6P@6 Body Text 2CJ+ B:[s      !')*+,10+ B:[  s@ !mnOY  rs~YZ[\]^i_`abcdefgrc e|$%8  2 4 9 = > ? @ C D ; < > @ _ {  # $ % 1 2 r s  WXY \]123]^45'Kpqt0p0000000000000000000000s0s0s0s0s0s0s0s0s0s0s0s0s0s0s0s0s0s0s0s0s0s000000000^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^000000000(00|0|0|0|0|0|0|0|0|0|0|00%0%0%0%0% 0%0%0%0%0%0%0%0%0% 0%0%0%0%0%0%0%0%0% 0%0%0%0%0%0%0%0% 0%0%0%0%80%0@ 0@ 0@ 0@ 0@ 0@ 0@ 0@ 0@ 0@ 0@ 0@ 0@ 0@ 0@ 0@ 0@ 0@ 0@ 0@ 0@ 0@ 0@ 0@ 0@ 000000000 0+@0+@0+@0+0+0+0+0+p0+0+0+0+0+0+0+0+0+0+0+0+0+0+0+0+0+0+ 0+0+0+0+0+0+0+0+0+0+0+0+0+0+0+0+0+0+0+0+`0+0+0+0+0+0+0+0+0+0+0+0+0+0+0000000000000 00000t0 `<% :9<<=JQc+cp@M   * < @ Q U f i y s1111111111111111.;HX11111812@R $1 (  B  VE`FNQ&UVW))? XX6381-D81^ DS &{'LO^ D+ YL^0L8]T+ YL7Gn2H+IJ7GI:9]T:I:Q= qR&QJ 7JJ >:*;9>:+$.+] x!+] 6381$ 3-D^ D %D^0L8]TH+ YL^0L8]T7G@8Cn2H+IJI:B,= qR&N7#Q7JK J 7J>:8*;9+ +$ x!+ ] x!+$(,`C0*0*ITNT0*0* BCCloud"H  #  H  #  (b  H   #  H   #  H   #  H   #  H   #  (b   (b  (b  (b  N  3  N  3     N  3    N  3    H  #   H  #   (b   N  3  N  3   N  3  N   3   N ! 3   (b " (b # (b % Z ' S 'GnHg  T ) C GH#  Z * S *G]H " T + C GH0 ! N , 3 , B 0    B 1   B S  ??0') +2 4 5 6 7 9 : ; @ A < > s!t/l t;%t.t d4t t 6t nt HVt!X tPtt$ tT@ t b~t t8bT~tdK4gtK t t"st 3 Ot8t t#WCtp{\ gt%dPt$ ?+t'x !t, t)!t1H%#t0K#t+_ t*  t t t7;OS qt tŪ<qh\V[@\Qv2o^`o(()^`.pLp^p`L.@ @ ^@ `.^`.L^`L.^`.^`.PLP^P`L.^`o(()^`.pLp^p`L.@ @ ^@ `.^`.L^`L.^`.^`.PLP^P`L.808^8`0o(()^`.pLp^p`L.@ @ ^@ `.^`.L^`L.^`.^`.PLP^P`L. ^`o(^`.pLp^p`L.@ @ ^@ `.^`.L^`L.^`.^`.PLP^P`L.\VQvq         .n        \i        B        Pv/h 3t;dH8l8@kd sP@P P@PP$@PP4@P6UnknownGz Times New Roman5Symbol3& z Arial"heCeC  !9r4d 2QHX)?Pv/2Rational and Irrational NumbersRM Administrator    Oh+'0 ( H T `lt| Rational and Irrational NumbersRM Normal.dotAdministrator2Microsoft Office Word@F#@fsE@fsE ՜.+,0 hp|   '  Rational and Irrational Numbers Title  "#$%&'(*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNPQRSTUVXYZ[\]^aRoot Entry F2sEcData !1Table)JWordDocument7@SummaryInformation(ODocumentSummaryInformation8WCompObjq  FMicrosoft Office Word Document MSWordDocWord.Document.89q