Assumed Knowledge (Number) Contents: A Number types [1.1] B Operations and brackets [1.2] C HCF and LCM [1.3] D Fractions E Powers and roots [1.4] F Ratio and proportion [1.5] G Number equivalents [1.7] H Rounding numbers [1.11] I Time [1.12] A NUMBER TYPES [1.1] The set of natural or counting numbers is N = f0, 1, 2, 3, 4, 5, 6, ......g. The set of natural numbers is endless, so we call it an infinite set. The set of integers or whole numbers is Z = f:::: , ¡4, ¡3, ¡2, ¡1, 0, 1, 2, 3, 4, ......g: The set Z includes N as well as all negative whole numbers. For example: 0, 127, ¡15, 10 000 and ¡34 618 are all members of Z : a Q a b The set of rational numbers, denoted , is the set of all numbers of the form b where and are integers and b 6=0. 1 15 ¡5 1 1 ¡9 For example: ² 3 , 32 , 4 are all rational ²¡2 4 is rational as ¡2 4 = 4 5 20 431 ² 5 is rational as 5= 1 or 4 ² 0:431 is rational as 0:431 = 1000 8 ² 0:8 is rational as 0:8= 9 All decimal numbers that terminate or recur are rational numbers. The set of real numbers, denoted R , includes all numbers which can be located on a number line. 2 p 0 and ¡5 cannot be placed on a number, and so are not real. 0 0 0 0 5 5 5 5 50 50 50 50 95 95 75 75 25 25 25 75 95 25 75 95 IGCSE01 100 100 100 100 cyan magenta yellow black Y:\HAESE\IGCSE01\IG01_AS\001IGCSE01_AS.CDR Wednesday, 12 November 2008 1:55:02 PM PETER 2 Assumed Knowledge (Number) The set of irrational numbers includes all real numbers which cannot be written in the form a a b b =0 b where and are integers and 6 . p p For example: ¼, 2 and 3 are all irrational. p p p 3 p 11 9 and 1:21 are rational since 9=3= 1 and 1:21 = 1:1= 10 : PRIMES AND COMPOSITES The factors of a positive integer are the positive integers which divide exactly into it, leaving no remainder. For example, the factors of 10 are: 1, 2, 5 and 10. A positive integer is a prime number if it has exactly two factors, 1 and itself. A positive integer is a composite number if it has more than two factors. For example: 3 is prime as it has two factors: 1 and 3. 6 is composite as it has four factors: 1, 2, 3 and 6. 1 is neither prime nor composite. If we are given a positive integer, we can use the following procedure to see if it is prime: Step 1: Find the square root of the number. Step 2: Divide the whole number in turn by all known primes less than its square root. If the division is never exact, the number is a prime. The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, ...... Example 1 Self Tutor Is 131 a prime number? p 131 = 11:445......, so we divide 131 by 2, 3, 5, 7 and 11. 131 ¥ 2=65:5 131 ¥ 3=43:66::::: 131 ¥ 5=26:2 131 ¥ 7=18:7142::::: 131 ¥ 11 = 11:9090::::: None of the divisions is exact, so 131 is a prime number. OTHER CLASSIFICATIONS A perfect square or square number is an integer which can be written as the square of another integer. For example, 4 and 25 are perfect squares since 4=22 and 25 = 52. A perfect cube is an integer which can be written as the cube of another integer. For example, 8 and ¡125 are perfect cubes since 8=23 and ¡125 = (¡5)3. 0 0 0 0 5 5 5 5 50 50 50 50 95 95 75 75 25 25 25 75 95 25 75 95 IGCSE01 100 100 100 100 cyan magenta yellow black Y:\HAESE\IGCSE01\IG01_AS\002IGCSE01_AS.CDR Monday, 27 October 2008 9:52:29 AM PETER Assumed Knowledge (Number) 3 EXERCISE A 1 Copy and complete the table given. Number N Z Q R 19 X X X X 2 ¡ 3 p 7 5:6389 p 16 2¼ ¡11 6 0 p ¡2 2 Show that: a 16, 529 and 20 802 721 are square numbers b 27, 343 and 13 824 are cube numbers c 11, 17 and 97 are prime numbers d 64 and 729 are both squares and cubes. 3 List all prime numbers between 50 and 70. 4 Explain why 1 is not a prime number. 5 Is 263 a prime number? Show all working. B OPERATIONS AND BRACKETS [1.2] RULES FOR THE ORDER OF OPERATIONS The word BEDMAS may help you remember this order. ² Perform the operations within brackets first. Brackets are grouping symbols ² Calculate any part involving exponents. that tell us what to evaluate first. ² Starting from the left, perform all divisions and multiplications as you come to them. ² Starting from the left, perform all additions and subtractions as you come to them. RULES FOR THE USE OF BRACKETS ² If an expression contains one set of brackets or group symbols, work that part first. ² If an expression contains two or more sets of grouping symbols one inside the other, work the innermost first. ² The division line of fractions also behaves as a grouping symbol. This means that the numerator and the denominator must be found separately before doing the division. 0 0 0 0 5 5 5 5 50 50 50 50 95 95 75 75 25 25 25 75 95 25 75 95 IGCSE01 100 100 100 100 cyan magenta yellow black Y:\HAESE\IGCSE01\IG01_AS\003IGCSE01_AS.CDR Thursday, 11 September 2008 9:30:06 AM PETER 4 Assumed Knowledge (Number) Example 2 Self Tutor Simplify: a 3+7¡ 5 b 6 £ 3 ¥ 2 a 3+7¡ 5 fWork left to right as only + and ¡ are involved.g =10¡ 5 =5 b 6 £ 3 ¥ 2 fWork left to right as only £ and ¥ are involved.g =18¥ 2 =9 Example 3 Self Tutor Simplify: a 23 ¡ 10 ¥ 2 b 3 £ 8 ¡ 6 £ 5 a 23 ¡ 10 ¥ 2 b 3 £ 8 ¡ 6 £ 5 =23¡ 5 f¥ before ¡g =24¡ 30 f£ before ¡g =18 = ¡6 EXERCISE B 1 Simplify: a 6+9¡ 5 b 56 ¡ 9+ c 6 ¡ 9 ¡ 5 d 3 £ 12 ¥ 6 e 12 ¥ 6 £ 3 f 6 £ 12 ¥ 3 2 Simplify: a 5+8£ 4 b 79 £ 4+ c 17 ¡ 7 £ 2 d 6 £ 7 ¡ 18 e 36 ¡ 6 £ 5 f 19 ¡ 7 £ 0 g 3 £ 6 ¡ 6 h 70 ¡ 5 £ 4 £ 3 i 45 ¥ 3 ¡ 9 j 8 £ 5 ¡ 6 £ 4 k 7+3+5£ 2 l 17 ¡ 6 £ 4+9 Example 4 Self Tutor Simplify: 3 + (11 ¡ 7) £ 2 3 + (11 ¡ 7) £ 2 =3+4£ 2 fevaluating the brackets firstg =3+8 f£ before + g =11 3 Simplify: a 14 + (8 ¡ 5) b (19 + 7) ¡ 13 c (18 ¥ 6) ¡ 2 d 18 ¥ (6 ¡ 4) e 72 ¡ (18 ¥ 6) f (72 ¡ 18) ¥ 6 g 36 + (14 ¥ 2) h 36 ¡ (7 + 13) ¡ 9 i (22 ¡ 5) + (15 ¡ 11) j (18 ¥ 3) ¥ 2 k 32 ¥ (4 ¥ 2) l 28 ¡ (7 £ 3) ¡ 9 0 0 0 0 5 5 5 5 50 50 50 50 95 95 75 75 25 25 25 75 95 25 75 95 IGCSE01 100 100 100 100 cyan magenta yellow black Y:\HAESE\IGCSE01\IG01_AS\004IGCSE01_AS.CDR Wednesday, 10 September 2008 2:10:30 PM PETER Assumed Knowledge (Number) 5 Example 5 Self Tutor Simplify: [12 + (9 ¥ 3)] ¡ 11 [12 + (9 ¥ 3)] ¡ 11 = [12 + 3] ¡ 11 fevaluating the inner brackets firstg =15¡ 11 fouter brackets nextg =4 4 Simplify: a 8 ¡ [(4 ¡ 6) + 3 £ 2] b [22 ¡ (11 + 4)] £ 3 c 25 ¡ [(11 ¡ 7) + 8] d [28 ¡ (15 ¥ 3)] £ 4 e 300 ¥ [6 £ (15 ¥ 3)] f [(14 £ 5) ¥ (28 ¥ 2)] £ 3 g [24 ¥ (8 ¡ 6) £ 9] h 18 ¡ [(1+6) £ 2] i [(14 ¥ 2) £ (14 ¥ 7)] ¥ 2 Example 6 Self Tutor 12 + (5 ¡ 7) Simplify: 18 ¥ (6 + 3) 12 + (5 ¡ 7) 18 ¥ (6 + 3) 12 + (¡2) = fevaluating the brackets firstg 18 ¥ 9 10 = 2 fsimplifying the numerator and denominatorg =5 5 Simplify: 240 27 39 ¥ 3 a b c 8 £ 6 17 ¡ 8 14 + 12 18 + 7 58 ¡ 16 6 £ 7+7 d e f 7 ¡ 2 11 ¡ 5 7 54 (6 + 9) ¡ 5 2 ¡ 2 £ 2 g h i 11 ¡ (2 £ 4) 7+(9¡ 6) 2 £ (2 + 2) 6 Find: a 7 ¡¡8 b 10 ¡ (2 + 5) c 6 £¡7 ¡4 d e ( 2) ( 2) f 4 ( 3) 8 ¡ £ ¡ ¡ ¡ ¡24 g (¡3) £ (¡7) h i 2 ¡ 5 £ (¡3) ¡6 ¡2 £ 3 2 £ (¡3) j 2 £ 5 ¡ (¡3) k l ¡4 ¡4 £ 6 0 0 0 0 5 5 5 5 50 50 50 50 95 95 75 75 25 25 25 75 95 25 75 95 IGCSE01 100 100 100 100 cyan magenta yellow black Y:\HAESE\IGCSE01\IG01_AS\005IGCSE01_AS.CDR Wednesday, 10 September 2008 2:10:37 PM PETER 6 Assumed Knowledge (Number) Example 7 Self Tutor 75 a 12 + 32 (8 6) b Use your calculator to evaluate: ¥ ¡ 7+8 a Key in 12 + 32 ¥ ( 8 ¡ 6 ) ENTER Answer: 28 b Key in 75 ¥ ( 7 + 8 ) ENTER Answer: 5 For help using your calculator, refer to the graphics calculator instructionson page 11 . Example 8 Self Tutor Calculate: a 41 £¡7 b ¡18 £ 23 a Key in 41 £ (¡) 7 ENTER Answer: ¡287 b Key in (¡) 18 £ 23 ENTER Answer: ¡414 7 Evaluate each of the following using your calculator: a 87 + 27 £ 13 b (29 + 17) £ 19 c 6136 ¥ 8+1 d 136 ¥ (8 + 9) e 39 £¡27 f ¡128 ¥¡32 97 + ¡7 ¡25 ¡ 15 g h ¡ 67 + 64 ¥¡4 i ¡5 £ 3 9 ¡ (16 ¥ 4) j (¡3)4 k ¡34 l (¡1:4)3 £¡2:72 8 Use a calculator to answer the following questions: a Kevin can throw a tennis ball 47:55 m, and Dean can throw a tennis ball 42:8 m.