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Add, subtract, multiply, divide negative numbers Introduction 3 What is a directed number? 4 The idea of opposites 4 What is an integer? 4 Adding and subtracting directed numbers Double negative 7 8 Multiplying and dividing directed numbers 11 Order of operations with directed numbers 13 Using a calculator Suggested responses to activities 4930CF: 2 Applying Everyday Mathematics DET NSW, 2005/008/012/06/2006 P0025996 15 18 1 2 4930CF: 2 Applying Everyday Mathematics DET NSW, 2005/008/012/06/2006 P0025996 Introduction In Topic 1 you learnt that the number system starts with counting numbers {1, 2, 3, 4, …….} then extends to include zero and negative numbers. In this topic you will learn how to add, subtract, multiply and divide negative numbers. You will also apply the rule of ‘order of operations’ to negative numbers. 4930CF: 2 Applying Everyday Mathematics DET NSW, 2005/008/012/06/2006 P0025996 3 What is a directed number? A directed number has both size and direction. For example, the number + 3 means ‘3 units to the right of zero’, while the number – 4 means ‘4 units to the left of zero’. All numbers to the right of zero are called positive numbers, and all numbers to the left of zero are called negative numbers. We usually leave the + sign off positive numbers. When you see the number 10, it means + 10. The idea of opposites Just as our English language contains pairs of opposite words, (hot-cold, updown, in-out), our Mathematical language uses + and – signs to signal opposites. So: + 23 is the opposite of – 23 – 5.9 is the opposite of + 5.9 Opposite numbers are an equal distance from 0. What is an integer? Integers are simply positive or negative whole numbers. The set of integers is the set {….– 4, – 3, – 2, – 1, 0, 1, 2, 3, 4, 5, …….}. A number line can be used to give a picture of the set of integers as shown in Figure 1. Figure 1: Integers on a number line Numbers to the right of zero are greater than numbers to the left of zero. For example 3 is greater than – 3. This is the same if you think of a temperature of 3C. 4 4930CF: 2 Applying Everyday Mathematics DET NSW, 2005/008/012/06/2006 P0025996 In general, when comparing 2 integers, the number on the right is bigger than the number on the left. For example, – 2 is bigger than – 3. Write ‘is greater than’ or ‘is less than’ in the space provided to make these statements true. 1 4______________________2 2 –1______________________+2 3 0______________________–2 4 –1______________________–3 1 4 is greater than 2 2 –1 is less than +2 3 0 is greater than –2 4 –1 is greater than –3 Activity 2A Write the answers to these problems in your workbook. 1 What is the opposite of? (a) 5 (b) 7.2 (c) –1 (d) 2 Write down all the integers between: (a) – 1 and + 3 (b) – 11 and – 5 4930CF: 2 Applying Everyday Mathematics DET NSW, 2005/008/012/06/2006 P0025996 5 You may use this number line to help answer the next two questions. 3 Write ‘is greater than’ or ‘is less than’ to make these statements true. (a) + 1_________________________– 2 (b) – 5_________________________– 2 (c) – 14________________________+ 12 (d) – 1_________________________– 6 (e) 4 0__________________________8 What number is? (a) 2 units to the right of + 3? (b) 5 units to the left of + 1? (c) 3 units to the left of –2? (d) 12 units to the right of –3? (e) 20 units to the left of +2? (f) 6 units to the left of +5? (g) 4 units to the right of –8? (h) 9 units to the right of –9? Check your answers with the suggested responses at the end of the topic. 6 4930CF: 2 Applying Everyday Mathematics DET NSW, 2005/008/012/06/2006 P0025996 Adding and subtracting directed numbers Consider the following examples. Example 1 If the temperature is 2C and falls by 5C, what is the temperature then? It is easy to solve this problem by using a number line. Using a number sentence, we can write: So the temperature is –3C. 4930CF: 2 Applying Everyday Mathematics DET NSW, 2005/008/012/06/2006 P0025996 7 Example 2 Start 2 floors below ground level and then go up one floor. What floor are you now on? Once again, a number line helps you to see the solution. Using a number sentence, we can write Example 3 Your bank account is overdrawn by $10 and you withdraw another $5. What is the balance of your account? Once again, a number line helps you to see the solution. Using a number sentence, we can write Directed numbers can be added or subtracted using a number line. to add, hop to the right to subtract, hop to the left Double negative has two minus, or negative, signs directly next to each other (ie there is no number separating the two minus signs). When this is the case, the ‘ ‘ becomes a ‘+’. That is: 8 4930CF: 2 Applying Everyday Mathematics DET NSW, 2005/008/012/06/2006 P0025996 Example 4 Consider the expression . Once again this question involves two minus signs next to each other. The ‘ ‘ becomes a ‘+’ so: Don’t forget you can still use a number line. (Start from –5, take 3 hops to the right). This rule might seem a little strange at first, however we also make use of the double negative in our everyday English language. For example, the prefix ‘un’ is used to make a ‘positive word’ into a ‘negative word’. Happy is a positive word. Unhappy is a negative word. ‘Not’ is a negative word used to change a positive expression into a negative one. What happens when we put two negative words together? ‘Not unhappy’ changes the meaning back to being positive. This is the double negative. To subtract a negative number, change the two minus signs into a plus sign. Activity 2B Don’t forget to write your answers in your workbook. 1 Use a number line to help answer these questions: (a) 15 – 20 (b) – 10 + 25 (c) – 1 – 4 (d) – 7 + 3 4930CF: 2 Applying Everyday Mathematics DET NSW, 2005/008/012/06/2006 P0025996 9 (e) 6 + 20 (f) 0 – 9 (g) – 12 + 12 (h) – 4 – 4 2 Rewrite each of these number sentences as additions, then use a number line to help answer the questions. The first one has been done for you. (a) (b) 4 – – 6 (c) 0 – – 2 (d) – 2 – – 5 (e) – 10 – – 6 3 Calculate the following. (a) 100 – 200 (b) 16 – – 7 (c) – 4 – 0 (d) – 7 – – 7 (e) 0.1 – 0.4 (f) – 5 – 6 Check your answers with the suggested responses at the end of the topic. 10 4930CF: 2 Applying Everyday Mathematics DET NSW, 2005/008/012/06/2006 P0025996 Multiplying and dividing directed numbers There are just a couple of simple rules to remember for multiplication and division questions: Multiplying and dividing two numbers with the same sign gives a positive result. Multiplying and dividing two numbers with different signs gives a negative result. Example 1 Solve: 4 × – 3 Step1: 4 × 3 = 12 Step2: signs are different (4 is positive, 3 is negative) so the answer will be negative. Therefore the answer is: 4 × – 3 = – 12 Example 2 Solve: – 5 × – 6 Step1: 5 × 6 = 30 Step2: signs are the same (– × – → +). Therefore the answer is: – 5 × – 6 = 30 Example 3 Solve: – 32 ÷ 8 Step1: 32 ÷ 8 = 4 Step2: signs are different (– ÷ + → –). Therefore the answer is: – 32 ÷ 8 = – 4 4930CF: 2 Applying Everyday Mathematics DET NSW, 2005/008/012/06/2006 P0025996 11 Example 4 Solve: – 6 ÷ – 4 Step1: 6 ÷ 4= 1.5 Step2: signs are the same (– ÷ – → +). Therefore the answer is: – 6 ÷ – 4 = 1.5 Activity 2C Solve the following problems. Don’t forget to write your answers in your workbook. 1 4 × __ 5 2 36 ÷ __ 3 3 – 7 × __ 6 4 –4×3 5 – 50 ÷ 10 6 (– 8)2 7 8 – 1.4 × 100 9 100 ÷ __ 5 10 45 ÷ 3 Check your answers with the suggested responses at the end of the topic. 12 4930CF: 2 Applying Everyday Mathematics DET NSW, 2005/008/012/06/2006 P0025996 Order of operations with directed numbers Do you remember learning about ‘order of operations’ in Recognising Everyday Mathematics? The principles used in order of operations also apply to directed numbers. Review: flowchart showing the order of operations in a number sentence. Example 1 Example 2 4930CF: 2 Applying Everyday Mathematics DET NSW, 2005/008/012/06/2006 P0025996 13 Example 3 Example 4 You do the top line first, because the fraction bar acts in the same way as brackets. Note: If you were to do this question on a calculator you would have to put brackets around the top line. Activity 2D Solve the following problems without using a calculator. Don’t forget to write your answers in your workbook. 1 2 – 10 – 4 – 8 3 (4 – 7) × – 8 4 16 ÷ – 4 × – 5 5 (– 5)3 6 (– 12 × 4) ÷ (– 4 + 10) 7 14 4930CF: 2 Applying Everyday Mathematics DET NSW, 2005/008/012/06/2006 P0025996 8 9 10 Check your answers with the suggested responses at the end of the topic. Using a calculator Depending on the model, your scientific calculator has a key or a key. This is called the sign change key. To enter – 6, put in the number 6 then press If your calculator has . as the sign change key, then you have to enter it first followed by the number 6. Let’s try some calculations now. Example 1 Find: Both methods will give the correct answer of – 32. 4930CF: 2 Applying Everyday Mathematics DET NSW, 2005/008/012/06/2006 P0025996 15 Example 2 Find: – 99 – 11.2 Your answer is – 110.2 Example 3 Find: Use a calculator to find the answer to the following problems. Try to do them without looking at the answers. 1 – 1.5 + 2.5 – 10 _______________ 2 3 16 _______________ (– 12 × 4) ÷ 20 _______________ 4 _______________ 5 _______________ 6 _______________ 4930CF: 2 Applying Everyday Mathematics DET NSW, 2005/008/012/06/2006 P0025996 1 –9 2 53 3 – 2.4 4 5 5 6 2 4930CF: 2 Applying Everyday Mathematics DET NSW, 2005/008/012/06/2006 P0025996 17 Suggested responses to activities Activity 2A 1 (a) – 5 (b) – 7.2 (c) 1 (d) 2 3 4 18 (a) 0, 1, 2 (b) (a) (b) (c) (d) – 10, – 9, – 8, – 7, – 6 +2 is greater than – 4 – 5 is less than – 2 – 14 is less than +12 – 1 is greater than – 6 (e) (a) (b) (c) (d) (e) (f) (g) (h) 0 is less than 8 +5 –4 –5 +9 – 18 –1 –4 0 4930CF: 2 Applying Everyday Mathematics DET NSW, 2005/008/012/06/2006 P0025996 Activity 2B 1 (a) (b) (c) (d) (e) (f) (g) (h) –5 +15 (or 15) –5 –4 26 –9 0 –8 2 (a) (b) (c) (d) (e) (a) (b) (c) (d) (e) 9 + 2 = 11 4 + 6 = 10 0+2=2 – 2 + 5 = + 3 (or 3) – 10 + 6 = – 4 – 100 16 + 7 = 23 –4 –7+7=0 – 0.3 3 (f) – 11 Activity 2C 1 – 20 2 – 12 3 42 4 – 12 5 –5 6 – 8 x – 8 = 64 7 5 8 – 140 9 – 20 10 15 4930CF: 2 Applying Everyday Mathematics DET NSW, 2005/008/012/06/2006 P0025996 19 Activity 2D 1 1 2 – 22 3 24 4 20 5 – 125 6 –8 7 1 8 –2 9 –3 10 4 20 4930CF: 2 Applying Everyday Mathematics DET NSW, 2005/008/012/06/2006 P0025996