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  1. TOPIC 1 : NUMBERS

  2. Topic SummaryNumbers 1.1 Types of Numbers 1.2 Factors and Multiples 1.3 Order of Operations 1.4 Indices 1.5 Standard Form 1.6 Rounding and Estimating 1.7 Proportions – Fractions, Decimals, Percentages 1.8 Direct and Inverse Proportions 1.9 Ratios 1.10 Interest Rates 1.11 Sets and Notation 1.12 Distance, Speed and Time

  3. 1.1 Types of Numbers

  4. 1.1 Types of Numbers Did you know that there are different types of numbers? Natural Numbers Rational Numbers Integers Irrational Numbers Square Numbers Real Numbers Cube Numbers Directed Numbers

  5. 1.1 Types of Numbers- Natural Numbers Whole numbers that are greater than zero 1, 2, 3, 4, 5, 6, ………

  6. 1.1 Types of Numbers- Integers Whole numbers, which include -ve numbers, zero and +ve numbers ……-4, -3, -2, -1, 0, 1, 2, 3, ….

  7. 1.1 Types of Numbers- Directed Numbers Directed Numbers These are numbers which has a + or – in front of it. Eg. Used on temperature scales: The winter temperature in Helsinki is -20oC. Adding and subtracting Directed Numbers: SUBTRACT ADD Multiplying and Dividing Directed Numbers: -4 -3 -2 -1 0 1 2 3 4 + + gives + - - gives + + - gives - - + gives -

  8. 1.1 Types of Numbers- Square Numbers Square of any integer Integer x Integer = Square Number P x P = P² 1, 4, 9, 16, 25, 36, 49, 64, ……

  9. 1.1 Types of Numbers- Cube Numbers Cube of any integer Integer x Integer x Integer = Cube Number P x P x P= P³ 1, 8, 27, 64, 125, 216, 343, 512, ……

  10. 1.1 Types of Numbers- Rational Number • Whole Numbers, eg. 4 (= 4/1), -5 (=-5/1) • Fractions, p/q where p and q are integers and q is NOT a zero • Terminating or recurring decimals, eg. 0.125 (= 1/8), 0.333333 (= 1/3), -0.143143143 (= -143/999)

  11. 1.1 Types of Numbers- Irrational Numbers • Messy – can’t be written in fractions • Never-ending, non-repeating decimals • Square roots and cube roots Eg. π is irrational

  12. 1.1 Types of Numbers- Real Numbers Rational Numbers + Irrational Numbers

  13. 1.1Types of Numbers- Prime Numbers Prime Numbers can’t be divided by anything other than itself and 1. 2, 3, 5, 7, 11, 13, …… • 1 is NOT a prime number • Prime numbers end in 1, 3, 7, or 9 • Not all prime numbers end in 1, 3, 7, or 9

  14. 1.2 Factors and MultiplesPrime Factors, HCF and LCM

  15. 1.2 Prime Factors Prime Factor is a factor that is a prime number. Some numbers can be written as the product of their prime factors 9 36 18 2 3 2 3 = 22 x 32 Prime Factors

  16. 1.2 Common Factors FACTORS of a number are all the numbers that DIVIDE INTO IT. Eg. Factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24 A COMMON FACTOR is a Shared Factor of Two Numbers. Eg. Factors of 12 are 1, 2, 3, 4, 6, 12 Factors of 15 are 1, 3, 5, 15 The common factor of 12 and 15 is 3

  17. 1.2 Common Multiples A Multiple is the Product of Two Numbers Eg. The multiples of 5 are 5, 10, 15, 20, 25, … A Common Multiple is a Multiple of two Different Numbers Eg. Multiples of 6 are 6, 12, 18, 24, 30, 36, … Multiples of 9 are 9, 18, 27, 36, 45, 54, … The common multiples of 6 and 9 are 18, 36, ….

  18. 1.2 HCF and LCM • Highest Common Factor (HCF) and Lowest Common Multiple (LCM) • HCF is the largest number that will divide the given numbers without a remainder. • Example: HCF of 12 and 36 is 12. • LCM is the smallest number that is a multiple of two given numbers. • Example: LCM of 10 and 12 is 60. You need the LCM when adding and subtracting fractions You need to know the HCF when factorising

  19. 1.2 HCF and LCM- How to find HCF Example: Find the HCF of 80 and 56 Write down the numbers as a product of their prime factors. . 80 = 2 x 2 x 2 x 2 x 5 56 = 2 x 2 x 2 x 7 20 14 7 80 40 2 2 2 10 56 28 2 2 2 2. Circle the common factors. These give the HCF as: 2 x 2 x 2 = 8 The HCF of 80 and 56 is 8 5 2

  20. 1.2 HCF and LCM- How to find LCM Example: Find the LCM of 80 and 56 Write down the numbers as a product of their prime factors. . 80 = 2 x 2 x 2 x 2 x 5 56 = 2 x 2 x 2 x 7 Line the numbers up carefully. 20 14 7 80 40 2 2 2 10 56 28 2 2 2 2. Take one number from each column. These give the LCM as: 2 x 2 x 2 x 2 x 5 x 7 = 560 The LCM of 80 and 56 is 560 5 2

  21. 1.3 Order of OperationsBODMAS

  22. 1.3 Order of Operations Brackets Order Division Multiplication Addition Subtraction Do Brackets First Do Subtraction Last

  23. 1.3 Order of Operations Do you know that BODMAS is sometimes known as BIDMAS or even PEMDAS? Regardless of what they are called, the ORDER of OPERATIONS are the same.

  24. 1.3 Order of Operations - Example EXAMPLE : T = (4 – 7)² + -2 x 3

  25. 1.4 Indices

  26. 1.4Indices means 5 x 5 x 5 x 5 x 5 x 5 x 5 Base Power of the index You say it as ‘five to the power seven’

  27. 1.4Indices- Multiplying When multiplying the powers of the same numbers, you add the indices. Eg.

  28. 1.4Indices- Dividing When dividing the powers of the same numbers, you subtract the indices. Eg.

  29. 1.4Indices- Negative Powers When the power is a negative power …… Eg.

  30. 1.4Indices- Zero Powers Any number (other than zero) to the power of zero equal 1…… Eg. 50 = 1 60 = 1 1000 = 1

  31. 1.4 Indices- Roots The power means Square Root The power means Cube Root The power means Fourth Root

  32. 1.4Indices- Powers of 10 The powers of 10 are ……

  33. 1.4Indices- Powers of 10 When the power of 10 is smaller than zero, we use a negative power …… Which means ……

  34. 1.4 Indices- Summary Rules of Indices

  35. 1.5 Standard Form

  36. 1.5 Standard Form • Standard Index Form • Used to represent very large or very small numbers. • A x 10n • For the power of 10, n: • If the number is > 1, n is positive • If the number is < 1, n is negative • If the number = 1, n must be 0 This numbers must always be 1  A < 10

  37. The planet Venus is 108,000,000 km from the sun. In Standard form, this is 1.08 x 108 km 1.5 Standard Form- Examples Large Numbers Small Numbers • A red corpuscle (blood cell) in a typical adult weights about 0.000 000 0001 g. • In Standard form, this is 1 x 10-8 g

  38. (3 x 107) x (5 x 106) = (3 x 5) x (107 x 106) = 15 x 1013 = 1.5 x 1014 (6 x 109)  (2 x 104) = (6  2) x (109 104) = 3 x 105 1.5 Standard Form- Calculations Multiplication Division SUBTRACT Powers ADD Powers • The Laws of Indices can be used when manually multiplying and dividing numbers written in standard form.

  39. 1.6 Rounding and Estimating

  40. 1.6 Rounding Numbers • Numbers are rounded frequently as this makes them easier to work with. • They can be rounded to their nearest 1, 10, 100, 1000 etc. • Decimals can be rounded to a particular number of decimal places (d.p.)

  41. 1.6 Rounding Numbers- Decimal Places (d.p.) “What is 7.45839 to 2 Decimal Places?” 7.45839 = 7.46 Last Digit to be written (2nd decimal place because we are rounding to 2 d.p.) The Last Digit rounds UP because the DECIDER is 5 or more DECIDER

  42. 1.6Rounding Numbers - Significant Figures (s.f.) • The significant figures(s.f.) of a number are those digits that carry meaning contributing to its precision. 2 s.f 4 s.f • Example: 2.3 cm and 2.300 cm • These two measurements may look the same. But the latter is more precise. It is to 4 significant figures.

  43. 1.6Significant Figures- How to find? • To find the 1st significant figure (s.f), look for the first non-zero digit. Then round up or down. • The 2nd, 3rd, 4th ... Significant figures may or may not be zero. • Example:

  44. 1.6Significant Figures- Rules • All non-zero digits are considered significant • Eg. 91 has two significant digits – 9 and 1. • Zeroes appearing anywhere between two non-zero digits are significant. • Eg. 102 has three significant figures – 1, 0 and 2 • Leading zeros are not significant. • Eg. 0.00052 has two significant digits – 5 and 2 • Trailing zeroes in a number containing a decimal point are significant. • Eg. 1.2300 has 5 significant digits – 1, 2, 3, 0 and 0 • Eg. 0.0012300 has 5 significant digits – 1, 2, 3, 0 and 0

  45. 1.6 Estimating Estimating can be used to check your answers. When estimating... • Round the numbers to easy numbers, 1 or 2 s.f. • Use 0.1 or 0.01 when multiplying or dividing. Do not use ZERO. • Use the symbol  as this means ‘approximately equal to’ • Small numbers can be approximated to zero when adding of subtracting.

  46. 1.6 Estimating- Example • 9.7 x 4.1  10 x 4 = 40 • 0.073 x 51  0.1 x 50 = 5

  47. 1.6 Limits of Accuracy- Upper & Lower Bounds • When a number has been rounded to a given number of significant figures, you might need to know the largest and smallest values the number may have been. • These are called the UPPER and LOWER BOUNDS of a number.

  48. 1.6 Limits of Accuracy- Upper & Lower Bounds Example 1: 62000 (2 s.f) people watched Liverpool’s football game this weekend. What are the lower and upper bounds of the numbers of supporters? Smallest number that rounds up to 62 000 Largest number that rounds down to 62 000 62 499 62 500 61 500 62 000 The lower bound = 61,500 supporters The upper bound = 62,499 supporters

  49. 1.6 Limits of Accuracy- Upper & Lower Bounds Example 2: A piece of string is 6.3 cm correct to the nearest millimeter. Write down the lower and upper bounds of the piece of string (1mm = 0.1cm) 6.2 6.3 6.4 6.25 6.35 The lower bound = 6.25 cm The upper bound = 6.35 cm (3 s.f) We can also say the length is up to 6.35 cm

  50. 1.7 ProportionsFractions, Decimals, Percentages

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