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KS4 Mathematics

KS4 Mathematics. N1 Integers. N1 Integers. Contents. A. N1.2 Calculating with integers. A. N1.1 Classifying numbers. N1.3 Multiples, factors and primes. A. N1.4 Prime factor decomposition. A. N1.5 LCM and HCF. A. Classifying numbers. Rational numbers

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KS4 Mathematics

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  1. KS4 Mathematics N1 Integers

  2. N1 Integers Contents • A N1.2 Calculating with integers • A N1.1 Classifying numbers N1.3 Multiples, factors and primes • A N1.4 Prime factor decomposition • A N1.5 LCM and HCF • A

  3. Classifying numbers Rational numbers Numbers that can be expressed in the form n/m, where n and m are integers. All fractions and all terminating and recurring decimals are rational numbers, for example, ¾, –0.63, 0.2. . Natural numbers Positive whole numbers 0, 1, 2, 3, 4 … Integers Positive and negative whole numbers … –3, –2, 1, 0, 1, 2, 3, … Irrational numbers Numbers that cannot be expressed in the form n/m, where n and m are integers. Examples of irrational numbers are  and 2.

  4. Even numbers E(1) = 2 E(2) = 4 E(3) = 6 E(4) = 8 E(5) = 10 Even numbers are numbers that are exactly divisible by 2. For example, 48 is aneven number. It can be written as48 = 2 × 24. All even numbers end in 0, 2, 4, 6 or 8. Even numbers can be illustrated using dots or counters arranged as follows: The nth even number can be written as E(n) = 2n.

  5. Odd numbers U(1) = 1 U(2) = 3 U(3) = 5 U(4) = 7 U(5) = 9 Odd numbersleave a remainder of 1 when divided by 2. For example, 17 is anodd number. It can be written as17 = 2 × 8 + 1 All odd numbers end in 1, 3, 5, 7 or 9. Odd numbers can be illustrated using dots or counters arranged as follows: The nth odd number can be written as U(n) = 2n – 1.

  6. Triangular numbers T(1) = 1 T(2) = 3 T(3) = 6 T(4) = 10 T(5) = 15 Triangular numbers are numbers that can be written as the sum of consecutive whole numbers starting with 1. For example, 15 is a triangular number. It can be written as15 = 1 + 2 + 3 + 4 + 5 Triangular numbers can be illustrated using dots or counters arranged in triangles:

  7. Triangular numbers T(1) = 1 T(2) = 3 T(3) = 6 T(4) = 10 T(5) = 15 2T(4) = 20 2T(5) = 30 2T(1) = 2 2T(2) = 6 2T(3) = 12 Suppose we want to know the value of T(50), the 50th triangular number. We could either add together all the numbers from 1 to 50 or we could find a rule for the nth term, T(n). If we double the number of counters in each triangular arrangement we can make rectangular arrangements:

  8. Triangular numbers T(1) = 1 T(2) = 3 T(3) = 6 T(4) = 10 T(5) = 15 2T(1) = 2 2T(2) = 6 2T(3) = 12 2T(4) = 20 2T(5) = 30 n(n + 1) T(n) = 2 Any rectangular arrangement of counters can be written as the product of two whole numbers: 2T(1) = 1 × 2 2T(2) = 2 × 3 2T(3) = 3 × 4 2T(4) = 4 × 5 2T(5) = 5 × 6 From these arrangements we can see that 2T(n) = n(n + 1) So, for any triangular number T(n)

  9. Triangular numbers 50(50 + 1) T(50) = 2 50 × 51 T(50) = 2 2550 T(50) = 2 n(n + 1) T(n) = 2 We can now use this rule to find the value of the 50th triangular number. T(50) =1275

  10. Gauss’ method for adding consecutive numbers There is a story that when the famous mathematician Karl Friedrich Gauss was a young boy at school, his teacher asked the class to add up the numbers from one to a hundred. The teacher expected this activity to keep the class quiet for some time and so he was amazed when Gauss put up his hand and gave the answer, 5050, almost immediately!

  11. Gauss’ method for adding consecutive numbers 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 11 + 11 + 11 + 11 + 11 + 11 + 11 + 11 + 11 + n(n + 1) T(n) = 2 Gauss worked the answer out by noticing that you can quickly add together consecutive numbers by writing the numbers once in order and once in reverse order and adding them together. For example, to add the numbers from 1 to 10: 11 = 110 Sum of the numbers from 1 to 10 = 110 ÷ 2 = 55 Use this method to show that the nth triangular number is:

  12. Square numbers S(1) = 1 S(2) = 4 S(3) = 9 S(4) = 16 S(5) = 25 Square numbers are obtained when a whole number is multiplied by itself. They are sometimes called perfect squares. For example, 49 is a square number. It can be written as49 = 7 × 7 or 49 = 72. Square numbers can be illustrated using dots or counters arranged in squares:

  13. Making square numbers Thenth square number S(n) can be written as S(n) = n2. There are several ways to generate a sequence of square numbers. • We can multiply a whole number by itself. • For example, 25 = 5 × 5 or 25 = 52. • We can add consecutive odd numbers starting from 1. • For example, 25 = 1 + 3 + 5 + 7 + 9. • We can add together two consecutive triangular numbers. • For example, 25 = 10 + 15 • We can find the product of two consecutive even or odd • numbers and add 1. • For example, 25 = 4 × 6 + 1.

  14. Difference between consecutive squares Show that the difference between two consecutive square numbers is always an odd number. If we use the general form for a square number n2, where n is a whole number, we can write the square number following it as (n + 1)2. The difference between two consecutive square numbers can therefore be written as (n + 1)2 – n2 = (n + 1)(n + 1) – n2 = n2 + n + n + 1 – n2 = 2n + 1 2n + 1 is always an odd number for any whole number n.

  15. Cube numbers C(1) = 1 Cube numbers are obtained when a whole number is multiplied by itself and then by itself again. For example, 64 is a cube number. It can be written as64 = 4 × 4 × 4 or 64 = 43. Cube numbers can be illustrated using spheres arranged in cubes: C(2) = 8 C(3) = 27 C(4) = 64 C(5) = 125 Thenth cube number C(n) can be written as C(n) = n3.

  16. Squares, triangles and primes

  17. N1 Integers Contents N1.1 Classifying numbers • A • A N1.2 Calculating with integers N1.3 Multiples, factors and primes • A N1.4 Prime factor decomposition • A N1.5 LCM and HCF • A

  18. Negative numbers A positive or negative whole number, including zero, is called an integer. For example, –3 is an integer. –3 is read as ‘negative three’. This can also be written as –3. It is 3 less than 0. 0 – 3 = –3 Here the ‘–’ sign means minus 3 or subtract 3. Here the ‘–’ sign means negative 3. We say, ‘zero minus three equals negative three’.

  19. Integers on a number line –3 –8 Negative integers Positive integers Positive and negativeintegers can be shown on a number line. We can use the number line to compare integers. For example, –3> –8 –3 ‘is greater than’ –8

  20. Adding integers We can use a number line to help us add positive and negative integers. –2 + 5 = = 3 -2 3 To add a positive integer we move forwards up the number line.

  21. Adding integers We can use a number line to help us add positive and negative integers. –3 + –4 = = –7 -7 -3 To add a negative integer we move backwards down the number line. –3 + –4 is the same as –3 – 4

  22. Subtracting integers We can use a number line to help us subtract positive and negative integers. 5 – 8 = = –3 -3 5 To subtract a positive integer we move backwards down the number line. 5 – 8 is the same as 5 – +8

  23. Subtracting integers We can use a number line to help us subtract positive and negative integers. 3 ––6 = = 9 3 9 To subtract a negative integer we move forwards up the number line. 3 ––6 is the same as 3 + 6

  24. Subtracting integers We can use a number line to help us subtract positive and negative integers. –4 ––7 = = 3 -4 3 To subtract a negative integer we move forwards up the number line. –4 ––7 is the same as –4 + 7

  25. Adding and subtracting integers To add a positive integer we move forwards up the number line. To add a negative integer we move backwards down the number line. a + –b is the same as a – b. To subtract a positive integer we move backwards down the number line. To subtract a negative integer we move forwards up the number line. a – –b is the same as a + b.

  26. Integer circle sums

  27. Rules for multiplying and dividing × ÷ = = – – – – – – – – – – – – + + + + + + + + + + + + × ÷ = = × ÷ = = ÷ × = = When multiplying negative numbers remember: Dividing is the inverse operation to multiplying. When we are dividing negative numbers similar rules apply:

  28. Multiplying and dividing integers –3× 8 = –36÷ = –4 42 ÷ = –6 × –8 = 96 ÷ –90 = –6 –7× = 175 –4×–5 × –8 = 47 × = –141 –72÷–6 = 3 × –8÷ = 1.5 Complete the following: –24 9 –7 540 –12 –25 –160 –3 –16 12

  29. Using a calculator (–) (–) 5 4 6 (–) = ÷ 6 We can enter negative numbers into a calculator by using the sign change key: For example: –456÷ –6 can be entered as: The answer will be displayed as 76. Always make sure that answers given by a calculator are sensible.

  30. Sums and products What two integers have a sum of 2 and a product of –8? Start by writing down all of the pairs of numbers that multiply together to make –8. Since –8 is negative, one of the numbers must be positive and one of the numbers must be negative. We can have: –1 × 8 = –8 1 × –8 = –8 –2 × 4 = –8 or 2 × –4 = –8 –1 + 8 = 7 1 + –8 = –7 –2 + 4 = 2 2 + –4 = –2 The two integers are –2 and 4.

  31. Sums and products

  32. N1 Integers Contents N1.1 Classifying numbers • A N1.2 Calculating with integers • A N1.3 Multiples, factors and primes • A N1.4 Prime factor decomposition • A N1.5 LCM and HCF • A

  33. Multiples What are the first six multiples of 7? A multiple of a number is found by multiplying the number by any whole number. To find the first six multiples of 7 multiply 7 by 1, 2, 3, 4, 5 and 6 in turn to get: 7, 14, 21, 28, 35 and 42. Any given number has infinitely many multiples.

  34. Factors A factor (or divisor) of a number is a whole number that divides into it exactly. Factors come in pairs. For example, What are the factors of 30? 1 and 30, 2 and 15, 3 and 10, 5 and 6. So, in order, the factors of 30 are: 1, 2, 3, 5, 6, 10, 15 and 30.

  35. Prime numbers If a whole number has two, and only two, factors it is called a prime number. For example, the number 17 has only two factors, 1 and 17. Therefore, 17 is a prime number. The number 1 has only one factor, 1. Therefore, 1 is not a prime number. There is only one even prime number. What is it? 2 is the only even prime number.

  36. Prime numbers There are 25 prime numbers less than 100. These are: 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 What if we go above 100? Around 400 BC the Greek mathematician, Euclid, proved that there are infinitely many prime numbers.

  37. N1 Integers Contents N1.1 Classifying numbers • A N1.2 Calculating with integers • A N1.4 Prime factor decomposition N1.3 Multiples, factors and primes • A • A N1.5 LCM and HCF • A

  38. Prime factors A prime factor is a factor that is a prime number. For example, What are the prime factors of 70? The factors of 70 are: 1 2 5 7 10 14 35 70 The prime factors of 70 are 2, 5, and 7.

  39. Products of prime factors 70 = 2 × 5 × 7 56 = 2 × 2 × 2 × 7 This can be written as 56 = 23× 7 99 = 3 × 3 × 11 This can be written as 99 = 32× 11 Every whole number greater than 1 is either a prime number or can be written as a product of two or more prime numbers.

  40. The prime factor decomposition When we write a number as a product of prime factors it is called the prime factor decomposition or prime factor form. For example, The prime factor decomposition of 100 is: 100 = 2 × 2 × 5 × 5 = 22× 52 There are two methods of finding the prime factor decomposition of a number.

  41. Factor trees 36 4 9 2 2 3 3 36 = 2 × 2 × 3 × 3 = 22× 32

  42. Factor trees 36 3 12 4 3 2 2 36 = 2 × 2 × 3 × 3 = 22× 32

  43. Factor trees 2100 30 70 6 5 10 7 2 3 2 5 2100 = 2 × 2 × 3 × 5 × 5 × 7 = 22× 3 × 52 × 7

  44. Factor trees 780 78 10 2 39 5 2 3 13 780 = 2 × 2 × 3 × 5 × 13 = 22× 3 × 5× 13

  45. Dividing by prime numbers 2 2 2 2 2 3 2 96 2 48 96 = 2 × 2 × 2 × 2 × 2 × 3 2 24 2 12 = 25 × 3 2 6 3 3 1

  46. Dividing by prime numbers 3 3 5 7 3 315 315 = 3 × 3 × 5 × 7 3 105 5 35 = 32 × 5 × 7 7 7 1

  47. Dividing by prime numbers 2 3 3 3 13 2 702 3 351 702 = 2 × 3 × 3 × 3 × 13 3 117 3 39 = 2 × 33× 13 13 13 1

  48. Using the prime factor decomposition 2 2 3 3 3 3 Use the prime factor form of 324 to show that it is a square number. 2 324 324 = 2 × 2 × 3 × 3 × 3 × 3 2 162 = 22 × 34 This can be written as: 3 81 (2 × 32) × (2 × 32) 3 27 or (2 × 32)2 3 9 If all the indices in the prime factor decomposition of a number are even, then the number is a square number. 3 3 1

  49. Using the prime factor decomposition 3 3 3 5 5 5 Use the prime factor form of 3375 to show that it is a cube number. 3 3375 3375 = 3 × 3 × 3 × 5 × 5 × 5 = 33 × 53 3 1125 This can be written as: 3 375 (3 × 5) × (3 × 5) × (3 × 5) 5 125 or (3 × 5)3 5 25 If all the indices in the prime factor decomposition of a number are multiples of 3, then the number is a cube number. 5 5 1

  50. Using the prime factor decomposition 4116 = 22 × 3 × 73 168 = 23 × 3 × 7 294 = 2 × 3 × 72 Use the prime factor decompositions of the numbers given above to answer the following questions. 1) What is 168 × 294 as a product of prime factors? 168 × 294 = (23 × 3 × 7) × (2 × 3 × 72 ) = 23 × 2 × 3 × 3 × 7 × 72 = 24 × 32 × 73

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