1. Copyright © Cengage Learning. All rights reserved. Real Numbers and Their Basic Properties 1

2. Copyright © Cengage Learning. All rights reserved. Section 1.4 Adding and Subtracting Real Numbers

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4. 4 Objectives 1. Add two or more real numbers with like signs. 2. Add two or more real numbers with unlike signs. 3. Subtract two real numbers. 4. Use signed numbers and one or more operations to model an application. 5. Use a calculator to add or subtract two real numbers. 1 1 2 2 3 3 4 4 5 5

5. 5 Add two or more real numbers with like signs 1.

6. 6 Adding Real Numbers with Like Signs 2 + 3 = ? Since the endpoint of the second arrow is the point with coordinate +5, we have (+2) + (+3) = +5

7. 7 Adding Real Numbers with Like Signs (-2) + (-3) = ? Since the endpoint of the final arrow is the point with coordinate –5, we have (–2) + (–3) = –5 Figure 1-20

8. 8 Adding Real Numbers with Like Signs 1. To add two positive numbers, add their absolute values and the answer is positive. 2. To add two negative numbers, add their absolute values and the answer is negative.

9. 9 Your Turn Add: a. (+4) + (+6) = +(4 + 6) = 10 b. (–4) + (–6) = –(4 + 6) = –10 c. +5 + (+10) = +(5 + 10) = 15

10. 10 Your Turn d. cont’d

11. 11 a(b + c) = ab + ac e.g., 3(4 + 5) = 3 ∙ 4 + 3 ∙ 5 = 12 + 15 = 27 -2(4 + 5) = (-2) ∙ 4 + (-2) ∙ 5 = -8 + (-10) = -18 Distributive Law of Algebra

12. 12 Add two or more real numbers with unlike signs 2.

13. 13 Adding Real Numbers with Unlike Signs (-6) + (+2) = ? The addition (–6) + (+2) can be represented by the arrows shown below.

14. 14 Adding Real Numbers with Unlike Signs Since the endpoint of the final arrow is the point with coordinate –4, we have (–6) + (+2) = –4

15. 15 Adding Real Numbers with Unlike Signs (+7) + (-4) = ? Since the endpoint of the final arrow is the point with coordinate +3, we have (+7) + (–4) = +3 Figure 1-22

16. 16 Add two or more real numbers with unlike signs Adding Real Numbers with Unlike Signs To add a positive and a negative number, subtract the smaller absolute value from the larger. 1.If the positive number has the larger absolute value, the answer is positive. 2.If the negative number has the larger absolute value, the answer is negative. 3.If a number and its opposite are added, the answer is zero.

17. 17 Your Turn Add: a. (+6) + (–5) = +(6 – 5) = 1 b. (–2) + (+3) = +(3 – 2) = 1 c. +6 + (–9) = –(9 – 6) = –3

18. 18 Example 2 d. cont’d

19. 19 Subtract two real numbers 3.

20. 20 Subtracting Real Numbers 5 – 3 = ? You can think of this as: 5 + (-3) 5 – 3 = 2

21. 21 Subtracting Real Numbers In general, to subtract one real number from another, we add the negative (or opposite) of the number that is being subtracted.

22. 22 Subtract two real numbers Subtracting Real Numbers If a and b are two real numbers, then a – b = a + (–b)

23. 23 Your Turn Evaluate: a. 12 – 4b. –13 – 5c. –14 – (–6) Solution: a. 12 – 4 = 12 + (–4) = 8 b. –13 – 5 = –13 + (–5) = –18 To subtract 4, add the opposite of 4. To subtract 5, add the opposite of 5.

24. 24 Example 5 – Solution c. –14 – (–6) = –14 + [–(–6)] = –14 + 6 = –8 To subtract –6, add the opposite of –6. The opposite of –6 is 6. cont’d

25. 25 Use signed numbers and one or more operations to model an application 4.

26. 26 Use signed numbers and one or more operations to model an application Words such as found, gain, credit, up, increase, forward, rises, in the future, and to the right indicate a positive direction. Words such as lost, loss, debit, down, decrease, backward, falls, in the past, and to the left indicate a negative direction.

27. 27 Your Turn– Account Balances The treasurer of a math club opens a checking account by depositing $350 in the bank. The bank debits the account $9 for check printing, and the treasurer writes a check for $22. Find the balance after these transactions. Solution: The deposit can be represented by +350. The debit of $9 can be represented by –9, and the check written for $22 can be represented by –22.

28. 28 Solution The balance in the account after these transactions is the sum of 350, –9, and –22. 350 + (–9) + (–22) = 341 + (–22) = 319 The balance is $319. Work from left to right. cont’d

29. 29 Use a calculator to add or subtract two real numbers 5.

30. 30 Use a calculator to add or subtract two real numbers A calculator can add positive and negative numbers. You do not have to do anything special to enter positive numbers. When you press 5, for example, a positive 5 is entered. To enter –5 into a calculator with a key, called the plus-minus or change-of-sign key, you must enter 5 and then press the key. To enter –5 into a calculator with a key, you must press the key and then press 5.

31. 31 Example 11 To evaluate –345.678 + (–527.339), we enter these numbers and press these keys: The display will read – 873.017. Using a calculator with a key Using a graphing calculator