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Natural Numbers The counting numbers 1, 2, 3, 4, 5, . . . Integers Include all the whole numbes and zero . . . -3, -2, -1, 0, 1, 2, 3, . . . Rational Numbers Include all the integers plus fractions Real Numbers Include all the Rational Numbers plus numbers that cannot be written as fractions N Z Q R
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Factor A factor of a number divides exactly into that number eg: Factors of 14 are: 1, 2, 7 and 14 Prime Number A number with exactly TWO factors: (1 and itself) 2, 3, 5, 7, 11, 13, 17, . . . Prime Factor A factor of a number which is also a prime number is called a prime factor
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Prime Factor A factor of a number which is also a prime number is called a prime factor
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eg 1: Write 24 as a product of Prime Factors Prime Factor A factor of a number which is also a prime number is called a prime factor 24 ÷ 2 12 ÷ 2 6 ÷ 2 3 ÷ 3 1 24 = 2 x 2 x 2 x 3 = 2 x 3 3 Keep dividing by prime numbers until you get to an answer of 1
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eg 2: Write 315 as a product of Prime Factors Prime Factor A factor of a number which is also a prime number is called a prime factor 315 ÷ 3 105 ÷ 3 35 ÷ 5 7 ÷ 7 1 315 = 3 x 3 x 5 x 7 = 3 x 5 x 7 2 Keep dividing by prime numbers until you get to an answer of 1
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eg 1: Write 357 as a product of Prime Factors Prime Factor A factor of a number which is also a prime number is called a prime factor 357 ÷ 3 119 ÷ 7 17 357 = 3 x 7 x 17 Keep dividing by prime numbers until you get to an answer of 1 ÷ 17 1
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Questions to Try Write each of these numbers as a product of prime factors. = 2 x 7 = 2 x 2 x 5 = 3 x 11 = 2 x 19 = 5 x 11 = 2 x 2 x 2 x 2 x 2 x 2 = 2 x 5 x 7 = 2 x 2 x 2 x 3 x 5 = 2 x 3 x 3 x 7 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 14 20 33 38 55 64 70 120 126 512 How well did you do?
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REAL NUMBERS Rational Numbers Irrational Numbers These include all the whole numbers and numbers which can be written as a fraction . ALL decimals which recur or terminate can be written as fractions. These are all the and numbers which can NOT be written as a fraction. Examples: pi Square roots of primes and multiples of primes.
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Writing a terminating decimal as a fraction (1) Write all the digits after the decimal point (2) Draw a line under these (3) Put a 0 under each digit (4) Put a 1 in front of the 0 s (5) Simplify if possible Examples 0.37 37 00 1 = 0.213 213 000 1 = 0.013 013 000 1 = 13 1000 =
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Writing a recurring decimal as a fraction If the recurring part starts straight after the decimal point, then it’s easy . . . If the fraction has 1 recurring digit, it’s that digit over 9 If the fraction has 2 recurring digits, it’s those digits over 99 If the fraction has 3 recurring digits, it’s those digits over 999, and so on. Examples 0.7 7 9 = 0.38 38 99 = 0.462 462 999 = 154 333 = . . . . .
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Writing a recurring decimal as a fraction If the recurring part doesn’t start straight after the decimal point, then we can express the decimal as a fraction by using methods as illustrated in the following examples: 0.73 x = 0.73 66 90 . 10 x = 7.3 90 x = 66 100 x = 73.3 x = . . . 11 15 = Scale up the above so that recurring part starts straight after decimal point Scale the above to line up recurring parts Subtract the two equations above
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Writing a recurring decimal as a fraction If the recurring part doesn’t start straight after the decimal point, then we can express the decimal as a fraction by using methods as illustrated in the following examples: 0.58 x = 0.58 53 90 . 10 x = 5.8 90 x = 53 100 x = 58.8 x = . . . Scale up the above so that recurring part starts straight after decimal point Scale the above to line up recurring parts Subtract the two equations above
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Writing a recurring decimal as a fraction If the recurring part doesn’t start straight after the decimal point, then we can express the decimal as a fraction by using methods as illustrated in the following examples: 0.658 x = 0.658 652 990 . . 10 x = 6.58 990 x = 652 1000 x = 658.58 x = . . . . . . 326 495 = Scale up the above so that recurring part starts straight after decimal point Scale the above to line up recurring parts Subtract the two equations above
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Writing a recurring decimal as a fraction If the recurring part doesn’t start straight after the decimal point, then we can express the decimal as a fraction by using methods as illustrated in the following examples: 0.174 x = 0.174 173 990 . . 10 x = 1.74 990 x = 173 1000 x = 174.74 x = . . . . . . Scale up the above so that recurring part starts straight after decimal point Scale the above to line up recurring parts Subtract the two equations above
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Writing a recurring decimal as a fraction If the recurring part doesn’t start straight after the decimal point, then we can express the decimal as a fraction by using methods as illustrated in the following examples: 0.369 x = 0.369 366 990 . . 10 x = 3.69 990 x = 366 1000 x = 369.69 x = . . . . . . Scale up the above so that recurring part starts straight after decimal point Scale the above to line up recurring parts Subtract the two equations above 61 165 =
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