MATHEMATICS
CONTENTS 1 NUMBER SYSTEMS 2 POLYNOMIALS 3 FACTORISATION OF POLYNOMIALS 4 LINEAR EQUATIONS IN TWO VARIABLES 5 COORDINATE GEOMETRY 6 INTRODUCTION TO EUCLID’S GEOMETRY 7 LINES AND ANGLES 8 TRIANGLES 9 CONGRUENCE OF TRIANGLES AND INEQUALITIES IN A TRIANGLE 10 QUADRILATERALS 11 AREAS OF PARALLELOGRAMS AND TRIANGLES 12 CIRCLES 13 GEOMETRICAL CONSTRUCTIONS 14 AREAS OF TRIANGLES AND QUADRILATERALS 15 VOLUME AND SURFACE AREA OF SOLIDS 16 PRESENTATION OF DATA IN TABULAR FORM 17 BAR GRAPH, HISTOGRAM AND FREQUENCY POLYGON 18 MEAN, MEDIAN AND MODE OF UNGROUPED DATA 19 PROBABILITY INTRODUCTION WE HAVE LEARNT ABOUT VARIOUS TYPES OF NUMBERS IN OUR EARLIER CLASSES LET US REVIEW THEM AND LEARN MORE ABOUT NUMBERS NATURAL NUMBERS COUNTING NUMBERS ARE CALLED NATURAL NUMBERS THE COLLECTION OF NATURAL NUMBERS IS DENOTED BY N AND IS WRITTEN AS { , , , , , ,…} N 1 2 3 4 ? REMARKS (I) THE LEAST NATURAL NUMBER IS 1 (II) THERE ARE IN? NITELY MANY NATURAL NUMBERS WHOLE NUMBERS ALL NATURAL NUMBERS TOGETHER WITH 0 FORM THE COLLECTION W OF ALL WHOLE NUMBERS, WRITTEN AS { , , , , , ,…} W 0 1 2 3 ? REMARKS (I) THE LEAST WHOLE NUMBER IS 0 (II) THERE ARE IN? NITELY MANY WHOLE NUMBERS (III) EVERY NATURAL NUMBER IS A WHOLE NUMBER (IV) ALL WHOLE NUMBERS ARE NOT NATURAL NUMBERS, AS 0 IS A WHOLE NUMBER WHICH IS NOT A NATURAL NUMBER INTEGERS ALL NATURAL NUMBERS, 0 AND NEGATIVES OF NATURAL NUMBERS FORM THE COLLECTION OF ALL INTEGERS IT IS REPRESENTED BY Z AFTER THE GERMAN WORD ‘ZAHLEN’ MEANING ‘TO COUNT’ THUS, WE WRITE {…, , , , , , , , , , , ,…} Z 1 0 1 2 3 ? ? ? ? ? ? REMARKS (I) 0 IS NEITHER NEGATIVE NOR POSITIVE (II) THERE ARE IN? NITELY MANY INTEGERS (III) EVERY NATURAL NUMBER IS AN INTEGER (IV) EVERY WHOLE NUMBER IS AN INTEGER REPRESENTATION OF INTEGERS ON NUMBER LINE A NUMBER LINE IS A VISUAL REPRESENTATION OF NUMBERS ON A GRADUATED STRAIGHT LINE TO REPRESENT INTEGERS ON THE NUMBER LINE, DRAW A LINE XY WHICH EXTENDS ENDLESSLY IN BOTH THE DIRECTIONS, AS INDICATED BY THE ARROWHEADS IN THE DIAGRAM BELOW NUMBER SYSTEMS SECONDARY SCHOOL MATHEMATICS FOR CLASS TAKE ANY POINT O ON THIS LINE LET THIS POINT REPRESENT THE INTEGER 0 (ZERO) NOW, TAKING A FIXED LENGTH, CALLED UNIT LENGTH, SET OFF EQUAL DISTANCES TO THE RIGHT AS WELL AS TO THE LEFT OF O ON THE RIGHT-HAND SIDE OF O, THE POINTS AT DISTANCES OF 1 UNIT, 2 UNITS, 3 UNITS, 4 UNITS, 5 UNITS, ETC, FROM O DENOTE RESPECTIVELY THE POSITIVE INTEGERS 1, 2, 3, 4, 5, ETC SIMILARLY, ON THE LEFT-HAND SIDE OF O, THE POINTS AT DISTANCES OF 1 UNIT, 2 UNITS, 3 UNITS, 4 UNITS, 5 UNITS, ETC, FROM O DENOTE RESPECTIVELY THE NEGATIVE INTEGERS –1, –2, –3, –4, –5, ETC SINCE THE LINE CAN BE EXTENDED ENDLESSLY ON BOTH SIDES OF O, IT FOLLOWS THAT WE CAN REPRESENT EACH AND EVERY INTEGER BY SOME POINT ON THIS LINE FOR INSTANCE, STARTING FROM O AND MOVING TO ITS RIGHT, AFTER 836 UNITS, WE GET A POINT WHICH REPRESENTS THE INTEGER 836 SIMILARLY, STARTING FROM O AND MOVING TO ITS LEFT, A POINT AFTER 750 UNITS, REPRESENTS THE INTEGER ‘–750’ THUS, EACH AND EVERY INTEGER CAN BE REPRESENTED BY SOME POINT ON THE NUMBER LINE RATIONAL NUMBERS THE NUMBERS OF THE FORM , Q P WHERE P AND Q ARE INTEGERS AND , Q ! ARE KNOWN AS RATIONAL NUMBERS THE COLLECTION OF RATIONAL NUMBERS IS DENOTED BY Q AND IS WRITTEN AS · : , , ARE INTEGERS Q Q P P Q Q ! ?( ‘RATIONAL’ COMES FROM THE WORD ‘RATIO’ AND Q COMES FROM THE WORD ‘QUOTIENT’ THUS, , , , , ? ETC, ARE ALL RATIONAL NUMBERS REMARKS (I) THERE ARE IN? NITELY MANY RATIONAL NUMBERS (II) THERE IS NO LEAST OR GREATEST RATIONAL NUMBER (III) 0 IS A RATIONAL NUMBER, SINCE WE CAN WRITE, · ? (IV) EVERY NATURAL NUMBER IS A RATIONAL NUMBER SINCE WE CAN WRITE, , , , ? ? ? ETC (V) EVERY INTEGER IS A RATIONAL NUMBER SINCE AN INTEGER A CAN BE WRITTEN AS , A 1 EG, , ? ?? ? AND · ? HENCE, RATIONAL NUMBERS INCLUDE NATURAL NUMBERS, WHOLE NUMBERS AND INTEGERS SCHOOL MATHEMATICS HIGH NUMBER SYSTEMS EQUIVALENT RATIONAL NUMBERS RATIONAL NUMBERS DO NOT HAVE A UNIQUE REPRESENTATION IN THE FORM , Q P WHERE P AND Q ARE INTEGERS AND Q ! THUS, … … … ? ? ? ? ? ? ? ? ? THESE ARE KNOWN AS EQUIVALENT RATIONAL NUMBERS SIMPLEST FORM OF A RATIONAL NUMBER A RATIONAL NUMBER Q P IS SAID TO BE IN ITS SIMPLEST FORM, IF P AND Q ARE INTEGERS HAVING NO COMMON FACTOR OTHER THAN 1 (THAT IS, P AND Q ARE CO-PRIMES) AND Q ! THUS, THE SIMPLEST FORM OF EACH OF , , , , 5 ETC, IS · SIMILARLY, THE SIMPLEST FORM OF 6 IS 2 AND THAT OF 95 IS · EXAMPLE WRITE FOUR RATIONAL NUMBERS EQUIVALENT TO · SOLUTION WE HAVE # # # # # # # # ? ? ? ? & · ? ? ? ? THUS, FOUR RATIONAL NUMBERS EQUIVALENT TO 4 ARE , , 16 AND · REPRESENTATION OF RATIONAL NUMBERS ON REAL LINE DRAW A LINE XY WHICH EXTENDS ENDLESSLY IN BOTH THE DIRECTIONS TAKE A POINT O ON IT AND LET IT REPRESENT 0 (ZERO) TAKING A FIXED LENGTH, CALLED UNIT LENGTH, MARK OFF OA ? UNIT THE MIDPOINT B OF OA DENOTES THE RATIONAL NUMBER · 1 STARTING FROM O, SET OFF EQUAL DISTANCES EACH EQUAL TO OB ? UNIT FROM THE POINT O, ON ITS RIGHT, THE POINTS AT DISTANCES EQUAL TO OB, 2OB, 3OB, 4OB, ETC, DENOTE RESPECTIVELY THE RATIONAL NUMBERS , , , , 4 ETC SIMILARLY, FROM THE POINT O, ON ITS LEFT, THE POINTS AT DISTANCES EQUAL TO OB, 2OB, 3OB, 4OB, ETC, DENOTE RESPECTIVELY THE RATIONAL NUMBERS , , ? ? , , ? ? ETC THUS, EACH RATIONAL NUMBER WITH 2 AS ITS DENOMINATOR CAN BE REPRESENTED BY SOME POINT ON THE NUMBER LINE NEXT, DRAW THE LINE XY TAKE A POINT O ON IT REPRESENTING 0 LET OA ? UNIT DIVIDE OA INTO THREE EQUAL PARTS WITH OC AS THE FIRST PART THEN, C REPRESENTS THE RATIONAL NUMBER · FROM THE POINT O, SET OFF EQUAL DISTANCES, EACH EQUAL TO OC ? UNIT ON BOTH SIDES OF O THE POINTS AT DISTANCES EQUAL TO OC, 2OC, 3OC, 4OC, ETC, FROM THE POINT O ON ITS RIGHT DENOTE RESPECTIVELY THE RATIONAL NUMBERS , , , , 4 ETC SIMILARLY, THE POINTS AT DISTANCES EQUAL TO , , , , OC OC OC OC ETC, FROM THE POINT O ON ITS LEFT DENOTE RESPECTIVELY THE RATIONAL NUMBERS , ? , , , ? ? ? ETC THUS, EACH RATIONAL NUMBER WITH 3 AS ITS DENOMINATOR CAN BE REPRESENTED BY SOME POINT ON THE NUMBER LINE (OR, THE REAL LINE) PROCEEDING IN THIS MANNER, WE CAN REPRESENT EACH AND EVERY RATIONAL NUMBER BY SOME POINT ON THE LINE EXAMPLE REPRESENT (I) 3 AND (II) ? ON REAL LINE SOLUTION DRAW A LINE XY AND TAKING A ? XED LENGTH AS UNIT LENGTH, REPRESENT INTEGERS ON THIS LINE SCHOOL MATHEMATICS HIGH NUMBER SYSTEMS (I) ON THE RIGHT OF O, TAKE OA ? UNIT THEN, OB ? UNITS DIVIDE THE 3RD UNIT BC INTO 8 EQUAL PARTS BP REPRESENTS 3 OF A UNIT THEREFORE, P REPRESENTS · (II) ON THE LEFT OF O, TAKE OD ? UNIT DIVIDE THE 2ND UNIT DE INTO 7 EQUAL PARTS DQ REPRESENTS 5 OF A UNIT THEREFORE, Q REPRESENTS · ? EXAMPLE 8 AND (II) ? ON THE NUMBER LINE SOLUTION DRAW A LINE XY AND TAKING A ? XED LENGTH AS UNIT LENGTH, REPRESENT INTEGERS ON THIS LINE (I) · ? ON THE RIGHT OF O, TAKE OA ? UNIT DIVIDE THE 2ND UNIT AB INTO 5 EQUAL PARTS AP REPRESENTS (II) · ? ?? ON THE LEFT OF O, TAKE OD ? UNIT DIVIDE THE 2ND UNIT DE INTO 7 EQUAL PARTS THEN, DQ REPRESENTS 4 OF A UNIT THEREFORE, Q REPRESENTS · ? FINDING RATIONAL NUMBERS BETWEEN TWO GIVEN RATIONAL NUMBERS METHOD 1 SUPPOSE WE ARE REQUIRED TO ? ND ONE RATIONAL NUMBER BETWEEN TWO RATIONAL NUMBERS X AND Y SUCH THAT X Y ? THEN, ( ) X Y ? IS A RATIONAL NUMBER LYING BETWEEN X AND Y EXAMPLE FIND A RATIONAL NUMBER LYING BETWEEN (I) 1 AND ; 1 (II) 2 AND · ? SOLUTION (I) LET X ? AND · Y ? ? REQUIRED RATIONAL NUMBER LYING BETWEEN X AND Y · ( ) X Y # ? ? ? ? ? ? B L HENCE, 5 IS A RATIONAL NUMBER LYING BETWEEN · AND (II) LET X ?? ? REQUIRED RATIONAL NUMBER LYING BETWEEN 2 AND ? ( ) X Y ? ? ? ?? ? ? B B L L < F · # ? ? ?? B L HENCE, ? IS A RATIONAL NUMBER LYING BETWEEN FIND THREE RATIONAL NUMBERS BETWEEN ? AND ? [2014] SOLUTION A RATIONAL NUMBER LYING BETWEEN ? IS [( ) ( )], ? ?? IE, · ? NOW, A RATIONAL NUMBER LYING BETWEEN ? IS ( ) , ? ?? B L < F IE, , # ? B L IE, · ? AND, A RATIONAL NUMBER LYING BETWEEN ? IS , ( ) ? THUS, WE HAVE ? ? ? ? ? ? ? ? HENCE, THREE RATIONAL NUMBERS BETWEEN ? ARE , ? ? AND · ? METHOD 2 SUPPOSE WE ARE REQUIRED TO ? ND N RATIONAL NUMBERS BETWEEN TWO RATIONAL NUMBERS, X AND Y WITH LIKE DENOMINATORS THEN, WE CONVERT THE GIVEN RATIONAL NUMBERS INTO EQUIVALENT RATIONAL NUMBERS BY MULTIPLYING THE NUMERATOR AND DENOMINATOR BY A SUITABLE NUMBER, USUALLY ( ) N ? NOW, THE REQUIRED RATIONAL NUMBERS MAY BE MANUALLY CHOSEN SCHOOL MATHEMATICS HIGH NUMBER SYSTEMS FIND ? VE RATIONAL NUMBERS BETWEEN · AND [2015] SOLUTION LET N ? WE CONVERT 3 AND 4 INTO EQUIVALENT RATIONAL NUMBERS BY MULTIPLYING THE NUMERATOR AND DENOMINATOR BY ( ), N ? IE, 6 THUS, # # ? ? AND · # # ? ? CLEARLY, WE HAVE ? ? ? ? ? ? OR · ? ? ? ? ? ? HENCE, ? VE RATIONAL NUMBERS BETWEEN , , 11 AND · FIND SIX RATIONAL NUMBERS BETWEEN 3 AND 4 [2015] SOLUTION LET N ? WE CONVERT 3 AND 4 INTO EQUIVALENT RATIONAL NUMBERS USING ( ) N ? ? AS MULTIPLYING FACTOR THUS, # # ? ? ? AND · # # ? ? ? NOW, ? ? ? ? ? ? ? OR ? ? ? ? ? ? ? HENCE, SIX RATIONAL NUMBERS BETWEEN 3 AND 4 ARE , , , , · AND INSERT 10 RATIONAL NUMBERS BETWEEN ? AND · SOLUTION WE HAVE · ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? HENCE, 10 RATIONAL NUMBERS BETWEEN 6 ARE , , , , , , , , ? ? ? ? AND · INSERT 100 RATIONAL NUMBERS BETWEEN SOLUTION CLEARLY, WE HAVE · ( ) AND # # # # ? ?? ?? ? ? NOW, … … ? ? ? ? ? ? ? ? ? ? ? ? · ? ? HENCE, 100 RATIONAL NUMBERS BETWEEN 7 ARE , ,…, , , , , ,…, · ? ? ? ? METHOD 3 SUPPOSE WE ARE REQUIRED TO ? ND N RATIONAL NUMBERS BETWEEN TWO GIVEN RATIONAL NUMBERS X AND Y (ESPECIALLY THOSE WITH UNLIKE DENOMINATORS) SUCH THAT X Y ? LET · ( ) ( ) D N Y X ? ? ? THEN, N RATIONAL NUMBERS LYING BETWEEN X AND Y ARE ( ), ( ), X D X D ? ? ( ), , ( ) X D X ND … ? ? REMARK THERE ARE IN? NITELY MANY RATIONAL NUMBERS BETWEEN ANY TWO GIVEN RATIONAL NUMBERS EXAMPLE INSERT ? VE RATIONAL NUMBERS BETWEEN SOLUTION LET X ?? AND · Y ? CLEARLY, X Y N ? ? LET · ( ) D N Y X # ? ? ? ? ? ?? ? ? ? ? B B L L SO, THE ? VE RATIONAL NUMBERS BETWEEN 3 ARE ( ), ( ), ( ), ( ) ( ), AND X D X D X D X D X D ? ? ? ? ? IE , , , AND ? ? ? ? ? ? ? ? ? ? B B B B B L L L L L IE , , , , AND ? ? ? IE , , , · AND ? ? ? SCHOOL MATHEMATICS HIGH NUMBER SYSTEMS FIND NINE RATIONAL NUMBERS BETWEEN 0 AND 01 SOLUTION HERE , AND X Y N ? ? ? ? ( ) ( ) ( ) ( ) D N Y X ? ? ? ? ? ? ? ? HENCE, THE REQUIRED NUMBERS BETWEEN 0 AND 01 ARE ( ), ( ), ( ), ( ), ( ), ( ), ( ), X D X D X D X D X D X D X D ? ? ? ? ? ? ? ( ), ( ), X D X D ? ? IE 001, 002, 003, 004, 005, 006, 007, 008 AND 009, IE , , , , , , , 8 AND , IE , , , , , , , HENCE, NINE RATIONAL NUMBERS BETWEEN 0 AND 01 ARE , , , , , , , · AND F EXERCISE 1A 1 IS ZERO A RATIONAL NUMBER? JUSTIFY 2 REPRESENT EACH OF THE FOLLOWING RATIONAL NUMBERS ON THE NUMBER LINE: (I) 5 (II) 8 (III) 3 FIND A RATIONAL NUMBER BETWEEN (I) AND 2 (II) 13 AND 14 (III) –1 AND (IV) AND ? ? (V) AND [2015] 4 FIND THREE RATIONAL NUMBERS LYING BETWEEN 3 AND · HOW MANY RATIONAL NUMBERS CAN BE DETERMINED BETWEEN THESE TWO NUMBERS? [2011] 5 FIND FOUR RATIONAL NUMBERS BETWEEN [2010] 6 FIND SIX RATIONAL NUMBERS BETWEEN 2 AND 3 7 FIND ? VE RATIONAL NUMBERS BETWEEN 8 INSERT 16 RATIONAL NUMBERS BETWEEN 21 AND 22 9 STATE WHETHER THE FOLLOWING STATEMENTS ARE TRUE OR FALSE GIVE REASONS FOR YOUR ANSWER (I) EVERY NATURAL NUMBER IS A WHOLE NUMBER (II) EVERY WHOLE NUMBER IS A NATURAL NUMBER (III) EVERY INTEGER IS A WHOLE NUMBER (IV) EVERY INTEGER IS A RATIONAL NUMBER (V) EVERY RATIONAL NUMBER IS AN INTEGER (VI) EVERY RATIONAL NUMBER IS A WHOLE NUMBER ANSWERS (EXERCISE 1A) 1 YES, BECAUSE 0 CAN BE WRITTEN AS 0 WHICH IS OF THE FORM , Q P WHERE P AND Q ARE INTEGERS AND Q ! 3 (I) 31 (II) 135 (III) ? (IV) ? (V) 4 , , 5 , , , 6 , , , , , 7 , , , , 9 (I) TRUE (II) FALSE; 0 IS A WHOLE NUMBER WHICH IS NOT A NATURAL NUMBER (III) FALSE; NEGATIVE INTEGERS ARE NOT WHOLE NUMBERS (IV) TRUE; EVERY INTEGER CAN BE WRITTEN IN THE FORM , Q P WHERE P AND Q ARE INTEGERS AND Q ! (V) FALSE; FRACTIONAL NUMBERS ARE NOT INTEGERS (VI) FALSE; FRACTIONAL NUMBERS ARE NOT WHOLE NUMBERS SCHOOL MATHEMATICS HIGH NUMBER SYSTEMS DECIMAL REPRESENTATION OF RATIONAL NUMBERS EVERY RATIONAL NUMBER Q P CAN BE EXPRESSED AS A DECIMAL ON DIVIDING P BY Q, TWO POSSIBILITIES ARISE (I) THE REMAINDER BECOMES ZERO AND THE DIVISION CONCLUDES AFTER A ? NITE NUMBER OF STEPS IN THIS CASE, THE DECIMAL EXPANSION OBTAINED ALSO TERMINATES OR ENDS (II) THE REMAINDER NEVER BECOMES ZERO AND A REPEATING STRING OF REMAINDERS IS OBTAINED IN THIS CASE, WE GET A DIGIT OR A BLOCK OF DIGITS REPEATING IN THE DECIMAL EXPANSION THUS, WE HAVE TWO TYPES OF DECIMAL EXPRESSIONS: 1 TERMINATING DECIMAL A DECIMAL THAT ENDS AFTER A ? NITE NUMBER OF DIGITS IS CALLED A TERMINATING DECIMAL EXAMPLES WE HAVE (I) , ? (II) , ? (III) ? ? ) ( 4 ) ( 8 ) ( 5 13 # – # THUS, EACH OF THE NUMBERS , 5 AND 3 CAN BE EXPRESSED AS A TERMINATING DECIMAL IMPORTANT RULE A RATIONAL NUMBER Q P IS EXPRESSIBLE AS A TERMINATING DECIMAL ONLY WHEN PRIME FACTORS OF Q ARE 2 AND 5 ONLY EXAMPLES EACH ONE OF THE NUMBERS , , , 13 IS A TERMINATING DECIMAL SINCE THE DENOMINATOR OF EACH HAS NO PRIME FACTORS OTHER THAN 2 AND 5 EXAMPLE WITHOUT ACTUAL DIVISION, ? ND WHICH OF THE FOLLOWING RATIONAL NUMBERS ARE TERMINATING DECIMALS: (I) 11 (III) · SOLUTION (I) DENOMINATOR OF 5 IS 32 AND, ? ? 32 HAS NO PRIME FACTORS OTHER THAN 2 SO, 5 IS A TERMINATING DECIMAL (II) DENOMINATOR OF 11 IS 24 AND, 3# ? THUS, 24 HAS A PRIME FACTOR 3, WHICH IS OTHER THAN 2 AND 5 ? 11 IS NOT A TERMINATING DECIMAL (III) DENOMINATOR OF 27 IS 80 AND, 4# ? THUS, 80 HAS NO PRIME FACTORS OTHER THAN 2 AND 5 ? 27 IS A TERMINATING DECIMAL 2 REPEATING (OR RECURRING) DECIMALS A DECIMAL IN WHICH A DIGIT OR A SET OF DIGITS IS REPEATED PERIODICALLY, IS CALLED A REPEATING, OR A RECURRING, DECIMAL IN A RECURRING DECIMAL, WE PLACE A BAR OVER THE FIRST BLOCK OF THE REPEATING DIGITS AND OMIT THE OTHER REPEATING BLOCKS EXAMPLES WE HAVE (I) … ? ? (II) … ? ? … ) ( 3 … ) ( 11 SCHOOL MATHEMATICS HIGH NUMBER SYSTEMS (III) 2 142857142857… ? (IV) 1 8333… ? 2 ? 1 ? … ) ( 7 15 … ) ( 6 11 YOU MUST HAVE NOTICED A REPEATING STRING OF REMAINDERS IN EACH OF THE ABOVE CASES IN (I), IT IS 2, 2, …; IN (II), IT IS 8, 3, 8, 3, … IN (III), IT IS 1, 3, 2, 6, 4, 5, 1, …; IN (IV), IT IS 5, 2, 2, 2, … KINDLY NOTE THAT THE NUMBER OF ENTRIES IN THE REPEATING STRING OF REMAINDERS IS LESS THAN THE DIVISOR IN , 2 ONLY ONE NUMBER 2 REPEATS ITSELF AND THE DIVISOR IS 3 IN , 15 A SET OF 6 DIGITS, NAMELY 132645, REPEATS ITSELF AND THE DIVISOR IS 7 THUS, IF THE DIVISOR IS N THEN THE MAXIMUM NUMBER OF ENTRIES IN THE REPEATING BLOCK OF DIGITS IN THE DECIMAL EXPANSION OF N 1 IS ( ) N ? LENGTH OF PERIOD REPEATED NUMBER OF DECIMAL PLACES IN A RATIONAL NUMBER IS CALLED THE LENGTH OF ITS PERIOD EXAMPLE ? SO, THE LENGTH OF ITS PERIOD IS 6 SPECIAL CHARACTERISTICS OF RATIONAL NUMBERS (I) EVERY RATIONAL NUMBER IS EXPRESSIBLE EITHER AS A TERMINATING DECIMAL OR AS A NONTERMINATING RECURRING DECIMAL (II) A NUMBER WHOSE DECIMAL EXPANSION IS TERMINATING OR NONTERMINATING RECURRING IS RATIONAL S O L V E D E X A M P L E S EXAMPLE EXPRESS 1 IN DECIMAL FORM SOLUTION WE HAVE · ? BY ACTUAL DIVISION, WE HAVE ) ( 8 25 000 × ? ? EXAMPLE 2 IN DECIMAL FORM SOLUTION BY ACTUAL DIVISION, WE HAVE … ) ( 11 2 0 … ? ? EXAMPLE WRITE 3 IN DECIMAL FORM AND SAY WHAT KIND OF DECIMAL REPRESENTATION IT HAS [2010] SCHOOL MATHEMATICS HIGH NUMBER SYSTEMS SOLUTION BY ACTUAL DIVISION, WE HAVE ) ( 13 3 0 … ? ? CLEARLY, 3 HAS A NONTERMINATING RECURRING DECIMAL REPRESENTATION EXAMPLE FIND THE DECIMAL EXPANSION OF · 1 CAN YOU PREDICT WHAT THE DECIMAL EXPANSIONS OF , , , , 6 ARE, WITHOUT ACTUALLY DOING THE LONG DIVISION? IF SO, HOW? SOLUTION BY LONG DIVISION, WE HAVE … ) ( 7 … ? ? CLEARLY, ; # # ? ? ? ; # # ? ? ? # # ? ? ? EXAMPLE WHAT CAN THE MAXIMUM NUMBER OF DIGITS BE IN THE REPEATING BLOCK OF DIGITS IN THE DECIMAL EXPANSION OF 1 ? PERFORM THE DIVISION TO CHECK YOUR ANSWER SOLUTION BY LONG DIVISION, WE HAVE ) ( 17 1 00 CLEARLY, … ? 0 ? LENGTH OF PERIOD = 16 HENCE, THE MAXIMUM NUMBER OF DIGITS IN THE REPEATING BLOCK OF DIGITS IN THE DECIMAL EXPANSION OF 1 IS LESS THAN 17, IE, 16 EXAMPLE EXPRESS 0 3 IN THE FORM , Q P WHERE P AND Q ARE INTEGERS AND Q ! [2014] SOLUTION LET X ? THEN, … X ? (I) SINCE THE REPEATING BLOCK HAS ONLY ONE DIGIT 3, WE MULTIPLY X BY 10 TO GET … X ? (II) SUBTRACTING (I) FROM (II), WE GET · X X + ? ? ? HENCE, · 0 EXPRESS 1 4 AS A FRACTION IN SIMPLEST FORM SOLUTION LET X ? THEN, X 1 444 … ? (I) SINCE THE REPEATING BLOCK HAS ONLY ONE DIGIT 4, WE MULTIPLY X BY 10 TO GET X 14 444 … ? (II) SUBTRACTING (I) FROM (II), WE GET · X X + ? ? ? HENCE, · 1 EXPRESS 09999 AS A FRACTION IN SIMPLEST FORM SOLUTION LET X 0 9999… ? … (I) SINCE THE REPEATING BLOCK HAS ONLY DIGIT 9, WE MULTIPLY X BY 10 TO GET X 9 9999… ? … (II) SUBTRACTING (I) FROM (II), WE GET X X + ? ? HENCE, … ? EXPRESS 0 57 IN THE FORM , Q P WHERE P AND Q ARE INTEGERS AND Q ! [2014] SOLUTION LET X ? THEN, X 0 575757 … ? (I) SINCE THE REPEATING BLOCK 57 HAS TWO DIGITS, WE MULTIPLY X BY 100 TO GET X 57 5757 … ? (II) SUBTRACTING (I) FROM (II), WE GET · X X EXPRESS 0 001 AS A FRACTION IN SIMPLEST FORM [2011] SOLUTION LET X ? THEN, X 0 001001001… ? (I) SINCE THE REPEATING BLOCK 001 HAS 3 DIGITS, WE MULTIPLY X BY 1000 TO GET X 1 001001001… ? … (II) SUBTRACTING (I) FROM (II), WE GET · X X + ? ? HENCE, · 0 EXPRESS 2 4178 IN THE FORM , Q P WHERE P AND Q ARE INTEGERS AND Q ! [2015] SOLUTION LET X ? THEN, X 2 4178178178… ? (I) SINCE THE REPEATING BLOCK 178 HAS 3 DIGITS, WE MULTIPLY X BY 1000 TO GET X 2417 8178178178… ? (II) SUBTRACTING (I) FROM (II), WE GET · X X 2415 & ? ? ? ? F EXERCISE 1B 1 WITHOUT ACTUAL DIVISION, ? ND WHICH OF THE FOLLOWING RATIONAL NUMBERS ARE TERMINATING DECIMALS (I) (II) (IV) (V) 2 WRITE EACH OF THE FOLLOWING IN DECIMAL FORM AND SAY WHAT KIND OF DECIMAL EXPANSION EACH HAS (I) (VI) (VII) (VIII) 3 EXPRESS EACH OF THE FOLLOWING DECIMALS IN THE FORM , Q P WHERE P, Q ARE INTEGERS AND Q ! (I) 0 [2014] (II) 0 [2010] (III) 2 [2010, ’15] (IV) [2014] (V) 0 235 [2010] (VI) 0 (VII) 1 [2010] (VIII) 0 [2014] (IX) [2011] (X) 0 [2014] 4 EXPRESS ? AS A FRACTION IN SIMPLEST FORM [2011] 5 EXPRESS IN THE FORM OF : Q P ? [2015] ANSWERS (EXERCISE 1B) 1 (I) AND (V) 2 (I) 0625; TERMINATING (II) 028; TERMINATING (III) ; 0 27 NONTERMINATING RECURRING (IV) ; 0 384615 NONTERMINATING RECURRING (V) ; 0 4583 NONTERMINATING RECURRING (VI) 06525; TERMINATING (VII) ; 0 3696 TERMINATING (VIII) ; 2 416 NONTERMINATING RECURRING 3 (I) 2 (II) 53 (III) 97 (IV) 610 (V) 235 (VI) 131 (VIII) 635 (IX) 318023 (X) IRRATIONAL NUMBERS IRRATIONAL NUMBER A NUMBER WHICH CAN NEITHER BE EXPRESSED AS A TERMINATING DECIMAL NOR AS A REPEATING DECIMAL, IS CALLED AN IRRATIONAL NUMBER THUS, NONTERMINATING, NONREPEATING DECIMALS ARE IRRATIONAL NUMBERS EXAMPLES OF IRRATIONAL NUMBERS TYPE (I) CLEARLY, 001001000100001 … IS A NONTERMINATING AND NONREPEATING DECIMAL AND THEREFORE, IT IS IRRATIONAL SIMILARLY, (II) 002002000200002 … IS IRRATIONAL, 003003000300003 … IS IRRATIONAL, AND SO ON (III) 012112111211112 … IS IRRATIONAL, 013113111311113 … IS IRRATIONAL, AND SO ON (IV) 054554555455554 … IS IRRATIONAL, 064664666466664 … IS IRRATIONAL, AND SO ON TYPE IF M IS A POSITIVE INTEGER WHICH IS NOT A PERFECT SQUARE THEN M IS IRRATIONAL THUS , , , , , , , , 11 ETC, ARE ALL IRRATIONAL NUMBERS TYPE IF M IS A POSITIVE INTEGER WHICH IS NOT A PERFECT CUBE THEN M IS IRRATIONAL THUS, , , , , , , , , ETC, ARE ALL IRRATIONAL NUMBERS THE NUMBER ?: ? IS A NUMBER WHOSE EXACT VALUE IS NOT · IN FACT ? HAS A VALUE WHICH IS NONTERMINATING AND NONREPEATING SO, ? IS IRRATIONAL, WHILE 22 IS RATIONAL PROPERTIES OF IRRATIONAL NUMBERS 1 IRRATIONAL NUMBERS SATISFY THE COMMUTATIVE, ASSOCIATIVE AND DISTRIBUTIVE LAWS FOR ADDITION AND MULTIPLICATION 2 (I) SUM OF TWO IRRATIONALS NEED NOT BE AN IRRATIONAL EXAMPLE EACH ONE OF ( ) ? AND ( ) ? IS IRRATIONAL BUT, ( ) ( ) , ? ? ? ? WHICH IS RATIONAL (II) DIFFERENCE OF TWO IRRATIONALS NEED NOT BE AN IRRATIONAL EXAMPLE EACH ONE OF ( ) ? IS IRRATIONAL BUT, ( ) ( ) , ? ? ? ? WHICH IS RATIONAL (III) PRODUCT OF TWO IRRATIONALS NEED NOT BE AN IRRATIONAL EXAMPLE 3 IS IRRATIONAL BUT, , # ? WHICH IS RATIONAL SCHOOL MATHEMATICS HIGH NUMBER SYSTEMS (IV) QUOTIENT OF TWO IRRATIONALS NEED NOT BE AN IRRATIONAL EXAMPLE EACH ONE OF 2 3 AND 3 IS IRRATIONAL BUT, , ? WHICH IS RATIONAL 3 (I) SUM OF A RATIONAL AND AN IRRATIONAL IS IRRATIONAL (II) DIFFERENCE OF A RATIONAL AND AN IRRATIONAL IS IRRATIONAL (III) PRODUCT OF A RATIONAL AND AN IRRATIONAL IS IRRATIONAL (IV) QUOTIENT OF A RATIONAL AND AN IRRATIONAL IS IRRATIONAL REMARK IF A IS RATIONAL AND B IS IRRATIONAL THEN EACH ONE OF ( ), ( ), A B A B A B ? ? AND B A IS IRRATIONAL EXAMPLES: EACH ONE OF ( ), ( ), ? ? AND 3 IS IRRATIONAL AN IMPORTANT RESULT IF A AND B ARE TWO DISTINCT POSITIVE RATIONAL NUMBERS THEN 1 A B ? IS A RATIONAL NUMBER LYING BETWEEN A AND B 2 AB IS AN IRRATIONAL NUMBER LYING BETWEEN A AND B S O L V E D E X A M P L E S EXAMPLE ARE THE SQUARE ROOTS OF ALL POSITIVE INTEGERS IRRATIONAL? IF NO, GIVE TWO EXAMPLES SOLUTION THE SQUARE ROOT OF EVERY POSITIVE INTEGER IS NOT ALWAYS IRRATIONAL EXAMPLES (I) , ? WHICH IS RATIONAL (II) , ? WHICH IS RATIONAL EXAMPLE ARE THE CUBE ROOTS OF ALL POSITIVE INTEGERS IRRATIONAL? IF NO, GIVE TWO EXAMPLES SOLUTION THE CUBE ROOT OF A POSITIVE INTEGER IS NOT ALWAYS IRRATIONAL EXAMPLES (I) , WHICH IS RATIONAL (II) , WHICH IS RATIONAL EXAMPLE WRITE THREE NUMBERS WHOSE DECIMAL EXPANSIONS ARE NONTERMINATING AND NONREPEATING SOLUTION WE KNOW THAT IF M IS A POSITIVE INTEGER WHICH IS NOT A PERFECT SQUARE, THEN M IS IRRATIONAL ALSO, WE KNOW THAT THE DECIMAL EXPANSION OF AN IRRATIONAL NUMBER IS NONTERMINATING AND NONREPEATING SO, THE REQUIRED NUMBERS ARE , 3 AND 5 EXAMPLE INSERT A RATIONAL AND AN IRRATIONAL NUMBER BETWEEN 3 AND 4 SOLUTION A RATIONAL NUMBER BETWEEN 3 AND 4 IS ? ? AN IRRATIONAL NUMBER BETWEEN 3 AND 4, IS , # IE, FIND TWO IRRATIONAL NUMBERS BETWEEN 1 AND · SOLUTION BY ACTUAL DIVISION, WE KNOW THAT ? AND ? NOW, CONSIDER THE NUMBERS A 0 16016001600016… ? AND B 0 21211211121111… ? CLEARLY, A AND B ARE IRRATIONAL NUMBERS SUCH THAT · A B ? ? ? HENCE, A AND B ARE THE REQUIRED NUMBERS EXAMPLE FIND TWO RATIONAL AND TWO IRRATIONAL NUMBERS BETWEEN 2 AND [2014, ’15] SOLUTION WE HAVE 1 41421356… ? AND 1 732050807 … ? IF WE CONSIDER THE NUMBERS 15 AND 16, WE ? ND THAT BOTH OF THEM ARE RATIONAL NUMBERS SUCH THAT AND 16 NOW, CONSIDER THE NUMBERS A 1 5050050005… ? AND B 1 6161161116… ? CLEARLY, A AND B ARE IRRATIONAL NUMBERS SUCH THAT A B ? ? ? HENCE, A AND B ARE TWO IRRATIONAL NUMBERS BETWEEN 2 AND 3 EXAMPLE FIND TWO RATIONAL NUMBERS IN THE FORM Q P BETWEEN 0343443444344443 … AND 0363663666366663 … [2010] SOLUTION LET A 0 343443444344443… ? AND B 0 363663666366663… ? CONSIDER THE NUMBERS C ? ? ? AND · D ? ? ? CLEARLY, C AND D ARE RATIONAL NUMBERS SUCH THAT A C D B ? ? ? HENCE, 7 AND 9 ARE THE REQUIRED RATIONAL NUMBERS SCHOOL MATHEMATICS HIGH NUMBER SYSTEMS F EXERCISE 1C 1 WHAT ARE IRRATIONAL NUMBERS? HOW DO THEY DIFFER FROM RATIONAL NUMBERS? GIVE EXAMPLES 2 CLASSIFY THE FOLLOWING NUMBERS AS RATIONAL OR IRRATIONAL GIVE REASONS TO SUPPORT YOUR ANSWER (I) (IV) 1 (VIII) 1232332333 … (IX) 3040040004 … (X) 2356565656 … (XI) 6834834 … 3 LET X BE A RATIONAL NUMBER AND Y BE AN IRRATIONAL NUMBER IS X Y ? NECESSARILY AN IRRATIONAL NUMBER? GIVE AN EXAMPLE IN SUPPORT OF YOUR ANSWER [2010, ’14] 4 LET A BE A RATIONAL NUMBER AND B BE AN IRRATIONAL NUMBER IS AB NECESSARILY AN IRRATIONAL NUMBER? JUSTIFY YOUR ANSWER WITH AN EXAMPLE 5 IS THE PRODUCT OF TWO IRRATIONALS ALWAYS IRRATIONAL? JUSTIFY YOUR ANSWER 6 GIVE AN EXAMPLE OF TWO IRRATIONAL NUMBERS WHOSE [2015] (I) DIFFERENCE IS AN IRRATIONAL NUMBER (II) DIFFERENCE IS A RATIONAL NUMBER (III) SUM IS AN IRRATIONAL NUMBER (IV) SUM IS A RATIONAL NUMBER (V) PRODUCT IS AN IRRATIONAL NUMBER (VI) PRODUCT IS A RATIONAL NUMBER (VII) QUOTIENT IS AN IRRATIONAL NUMBER (VIII) QUOTIENT IS A RATIONAL NUMBER 7 EXAMINE WHETHER THE FOLLOWING NUMBERS ARE RATIONAL OR IRRATIONAL (I) ? (II) ? (III) # (V) # 8 INSERT A RATIONAL AND AN IRRATIONAL NUMBER BETWEEN 2 AND 25 9 HOW MANY IRRATIONAL NUMBERS LIE BETWEEN 2 AND 3? FIND ANY THREE IRRATIONAL NUMBERS LYING BETWEEN 2 AND 3 [2010] 10 FIND TWO RATIONAL AND TWO IRRRATIONAL NUMBERS BETWEEN 05 AND 055 11 FIND THREE DIFFERENT IRRATIONAL NUMBERS BETWEEN THE RATIONAL NUMBERS 5 AND · 12 FIND TWO RATIONAL NUMBERS OF THE FORM Q P BETWEEN THE NUMBERS 02121121112… AND 02020020002… 13 FIND TWO IRRATIONAL NUMBERS BETWEEN 016 AND 017 14 STATE, IN EACH CASE, WHETHER THE GIVEN STATEMENT IS TRUE OR FALSE (I) THE SUM OF TWO RATIONAL NUMBERS IS RATIONAL (II) THE SUM OF TWO IRRATIONAL NUMBERS IS IRRATIONAL (III) THE PRODUCT OF TWO RATIONAL NUMBERS IS RATIONAL (IV) THE PRODUCT OF TWO IRRATIONAL NUMBERS IS IRRATIONAL (V) THE SUM OF A RATIONAL NUMBER AND AN IRRATIONAL NUMBER IS IRRATIONAL (VI) THE PRODUCT OF A NONZERO RATIONAL NUMBER AND AN IRRATIONAL NUMBER IS A RATIONAL NUMBER (VII) EVERY REAL NUMBER IS RATIONAL (VIII) EVERY REAL NUMBER IS EITHER RATIONAL OR IRRATIONAL (IX) ? IS IRRATIONAL AND 22 IS RATIONAL ANSWERS (EXERCISE 1C) 2 (I) IRRATIONAL (II) RATIONAL (III) IRRATIONAL (IV) RATIONAL (V) IRRATIONAL (VI) RATIONAL (VII) RATIONAL (VIII) IRRATIONAL (IX) IRRATIONAL (X) RATIONAL (XI) RATIONAL 3 YES 4 YES 5 NO 6 (I) ( ) ? AND ( ) ? (II) ( ) ? (III) ( ) ? (IV) ( ) ? (V) ( ) ? (VI) ( ) ? (VII) 18 AND (VIII) 27 AND 7 (I) IRRATIONAL (II) IRRATIONAL (III) RATIONAL (IV) RATIONAL (V) RATIONAL (VI) RATIONAL 8 225; 9 IN? NITE; 15050050005 …, 16161161116 …, 170770770777… 10 051 AND 053; 05151151115 … AND 05455455545… 11 0727222722227 …, 07577577757 …, 08080080008 … 12 , 13 016116111611116 …, 01686686668666 … 14 (I) TRUE (II) FALSE (III) TRUE (IV) FALSE (V) TRUE (VI) FALSE (VII) FALSE (VIII) TRUE (IX) TRUE SCHOOL MATHEMATICS HIGH NUMBER SYSTEMS REAL NUMBERS REAL NUMBERS A NUMBER WHOSE SQUARE IS NON-NEGATIVE, IS CALLED A REAL NUMBER IN FACT, ALL RATIONAL AND ALL IRRATIONAL NUMBERS FORM THE COLLECTION OF ALL REAL NUMBERS EVERY REAL NUMBER IS EITHER RATIONAL OR IRRATIONAL CONSIDER A REAL NUMBER (I) IF IT IS AN INTEGER OR IT HAS A TERMINATING OR REPEATING DECIMAL REPRESENTATION THEN IT IS RATIONAL (II) IF IT HAS A NONTERMINATING AND NONREPEATING DECIMAL REPRESENTATION THEN IT IS IRRATIONAL THE TOTALITY OF RATIONALS AND IRRATIONALS FORMS THE COLLECTION OF ALL REAL NUMBERS COMPLETENESS PROPERTY ON THE NUMBER LINE, EACH POINT CORRESPONDS TO A UNIQUE REAL NUMBER AND, EVERY REAL NUMBER CAN BE REPRESENTED BY A UNIQUE POINT ON THE REAL LINE DENSITY PROPERTY BETWEEN ANY TWO REAL NUMBERS, THERE EXIST IN? NITELY MANY REAL NUMBERS ADDITION PROPERTIES OF REAL NUMBERS (I) CLOSURE PROPERTY THE SUM OF TWO REAL NUMBERS IS ALWAYS A REAL NUMBER (II) ASSOCIATIVE LAW ( ) ( ) A B C A B C ? ? ? ? ? FOR ALL REAL NUMBERS A, B, C (III) COMMUTATIVE LAW A B B A ? ? ? FOR ALL REAL NUMBERS A AND B (IV) EXISTENCE OF ADDITIVE IDENTITY CLEARLY, 0 IS A REAL NUMBER SUCH THAT A A A ? ? ? ? FOR EVERY REAL NUMBER A 0 IS CALLED THE ADDITIVE IDENTITY FOR REAL NUMBERS (V) EXISTENCE OF ADDITIVE INVERSE FOR EACH REAL NUMBER A, THERE EXISTS A REAL NUMBER ( )A ? SUCH THAT ( ) ( ) A A A A ?? ?? ? ? A AND ( )A ? ARE CALLED THE ADDITIVE INVERSE (OR NEGATIVE) OF EACH OTHER MULTIPLICATION PROPERTIES OF REAL NUMBERS (I) CLOSURE PROPERTY THE PRODUCT OF TWO REAL NUMBERS IS ALWAYS A REAL NUMBER (II) ASSOCIATIVE LAW ( ) ( ) AB C A BC ? FOR ALL REAL NUMBERS A, B, C (III) COMMUTATIVE LAW AB BA ? FOR ALL REAL NUMBERS A AND B (IV) EXISTENCE OF MULTIPLICATIVE IDENTITY CLEARLY, 1 IS A REAL NUMBER SUCH THAT A A A · · ? ? FOR EVERY REAL NUMBER A 1 IS CALLED THE MULTIPLICATIVE IDENTITY FOR REAL NUMBERS (V) EXISTENCE OF MULTIPLICATIVE INVERSE FOR EACH NONZERO REAL NUMBER A, THERE EXISTS A REAL NUMBER A B L SUCH THAT A A A A · · ? ? A AND A 1 ARE CALLED THE MULTIPLICATIVE INVERSE (OR RECIPROCAL) OF EACH OTHER (VI) DISTRIBUTIVE LAWS OF MULTIPLICATION OVER ADDITION WE HAVE ( ) A B C AB AC ? ? ? AND ( ) A B C AC BC ? ? ? FOR ALL REAL NUMBERS A, B, C SOME MORE RESULTS ON REAL NUMBERS FOR ALL POSITIVE REAL NUMBERS A AND B, WE HAVE (I) AB A B # ? , (II) · B A B A ? S O L V E D E X A M P L E S EXAMPLE ADD ( ) ? SOLUTION WE HAVE ( ) ( ) ( ) ( ) ? ? ? ? ? ? ? ( ) ( ) ? ? ? ? ( ) ? ? EXAMPLE MULTIPLY 5 11 BY SOLUTION WE HAVE ( ) # # # # # # ? ? ? EXAMPLE DIVIDE 15 15 BY ' # # ? ? ? EXAMPLE SIMPLIFY (I) ( )( ) ? ? (II) ( )( ) ? ? (III) ( ) ? (IV) ( )( ) ? ? SOLUTION WE HAVE (I) ( )( ) # # # # ? ? ? ? ? ? ? ? ? ? (II) ( )( ) ( ) ( ) ? ? ? ? [ ( )( ) ] A B A B A B A ? ? ? ? ( ) ? ? ? SCHOOL MATHEMATICS HIGH NUMBER SYSTEMS (III) ( ) ( ) ( ) # # ? ? ? ? [ ( ) ] A B A B AB A ? ? ? ? ? ? ? ? ? (IV) ( )( ) ( ) ( ) ? ? ? ? [ ( )( ) ] A B A B A B A ? ? ? ? ? ? ? EXAMPLE SIMPLIFY · ? ? [2014] SOLUTION WE HAVE # # ? ? ? ? ? · · · · ? ? ? # # # # ? ? ? ? ? ? · ( ) ? ? ? ? EXAMPLE SIMPLIFY · A B B A B A A A B A B B ' ? ? ? ? ? ? ? ? SOLUTION WE HAVE A B B A B A A A B A B B A B B A A B A B B A A B ' # ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ( ) ( ) ( ) ( ) A B B A A B ? ? ? ? ? · ( ) ( ) A B B A A B A B B A A B A B ? ? ? ? ? ? ? ? ? ? ? F EXERCISE 1D 1 ADD (I) ( ) ( ) AND ? ? (II) ( ) ( ) AND ? ? ? ? (III) AND ? ? ? ? B B L L 2 MULTIPLY (I) BY (II) BY (III) BY (IV) BY (V) BY (VI) BY 3 DIVIDE (I) BY 4 SIMPLIFY (I) ( )( ) ? ? [2014] (II) ( )( ) ?? ?? [2014] (III) ( ) ? (V) ( )( ) ? ? (VI) ( )( ) ? ? 5 SIMPLIFY ( )( ) ? ? [2010] 6 EXAMINE WHETHER THE FOLLOWING NUMBERS ARE RATIONAL OR IRRATIONAL: (I) ( )( ) ? ? (II) ( ) ? (III) ? ? 7 ON HER BIRTHDAY REEMA DISTRIBUTED CHOCOLATES IN AN ORPHANAGE THE TOTAL NUMBER OF CHOCOLATES SHE DISTRIBUTED IS GIVEN BY ( )( ) ? ? [2014] (I) FIND THE NUMBER OF CHOCOLATES DISTRIBUTED BY HER (II) WRITE THE MORAL VALUES DEPICTED HERE BY REEMA 8 SIMPLIFY (I) ? ? ? [2010] (II) ? ? (III) ? ? [2014] ANSWERS (EXERCISE 1D) 1 (I) ( ) ? (II) ? ? (III) ? ? 2 (I) 3 (I) (II) 3 5 (III) ? ? ? (VI) ? ? ? 5 ? ? ? SCHOOL MATHEMATICS HIGH NUMBER SYSTEMS 6 (I) RATIONAL (II) IRRATIONAL (III) RATIONAL (IV) IRRATIONAL 7 (I) (II) TO HELP THE POOR AND NEEDY AND TO MAKE THE DEPRIVED CHILDREN HAPPY 8 (I) REPRESENTING IRRATIONAL NUMBERS ON THE NUMBER LINE EXAMPLE REPRESENT EACH OF THE NUMBERS , 3 AND 5 ON THE REAL LINE SOLUTION LET X OX L BE A HORIZONTAL LINE, TAKEN AS THE X-AXIS AND LET O BE THE ORIGIN LET O REPRESENT 0 TAKE OA ? UNIT AND DRAW AB OA = SUCH THAT AB ? UNIT JOIN OB THEN, OB OA AB ? ? ? ? ? UNITS WITH O AS CENTRE AND OB AS RADIUS, DRAW AN ARC, MEETING OX AT P THEN, OP OB ? ? UNITS THUS, THE POINT P REPRESENTS 2 ON THE REAL LINE NOW, DRAW BC OB = SUCH THAT BC ? UNIT JOIN OC THEN, ( ) OC OB BC ? ? ? ? ? UNITS WITH O AS CENTRE AND OC AS RADIUS, DRAW AN ARC, MEETING OX AT Q THEN, OQ OC ? ? UNITS THUS, THE POINT Q REPRESENTS 3 ON THE REAL LINE NOW, DRAW CD OC = SUCH THAT CD ? UNIT JOIN OD THEN, ( ) OD OC CD ? ? ? ? ? ? UNITS NOW, DRAW DE OD = SUCH THAT DE ? UNIT JOIN OE THEN, OE OD DE ? ? ? ? ? UNITS WITH O AS CENTRE AND OE AS RADIUS, DRAW AN ARC, MEETING OX AT R THEN, OR OE ? ? UNITS THUS, THE POINT R REPRESENTS 5 ON THE REAL LINE HENCE, THE POINTS P, Q, R REPRESENT THE NUMBERS , 3 AND 5 RESPECTIVELY REMARK IN THE SAME WAY, WE CAN LOCATE N FOR ANY POSITIVE INTEGER N, AFTER N ? HAS BEEN LOCATED EXAMPLE 6 AND 7 ON THE REAL LINE SOLUTION DRAW A HORIZONTAL LINE , X OX L TAKEN AS THE X-AXIS TAKE O AS THE ORIGIN TO REPRESENT 0 LET OA ? UNITS AND LET AB OA = SUCH THAT AB = 1 UNIT JOIN OB THEN, OB OA AB ? ? ? ? ? WITH O AS CENTRE AND OB AS RADIUS, DRAW AN ARC, MEETING OX AT P THEN, OP OB ? ? THUS, P REPRESENTS 5 OR THE REAL LINE NOW, DRAW BC OB = AND SET OFF BC ? ? ? ? ? WITH O AS CENTRE AND OC AS RADIUS, DRAW AN ARC, MEETING OX AT Q THEN, OQ OC ? ? SCHOOL MATHEMATICS HIGH NUMBER SYSTEMS THUS, Q REPRESENTS 6 ON THE REAL LINE NOW, DRAW CD OC = AND SET OFF CD ? UNIT JOIN OD THEN, ( ) OD OC CD ? ? ? ? ? WITH O AS CENTRE AND OD AS RADIUS, DRAW AN ARC, MEETING OX AT R THEN, OR OD ? ? THUS, THE POINTS P, Q, R REPRESENT THE REAL NUMBERS , 6 AND 7 RESPECTIVELY EXISTENCE OF X FOR A GIVEN POSITIVE REAL NUMBER X LET X BE A GIVEN POSITIVE REAL NUMBER WE SHALL PROVE THE EXISTENCE OF X IN THE FOLLOWING THREE STEPS STEP FINDING X GEOMETRICALLY DRAW A LINE SEGMENT AB X ? UNITS AND EXTEND IT TO C SUCH THAT BC ? UNIT FIND THE MIDPOINT O OF AC WITH O AS CENTRE AND OA AS RADIUS, DRAW A SEMICIRCLE NOW, DRAW , BD AC = INTERSECTING THE SEMICIRCL