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Warm Up Simplify. 1.

14. 55. 3. 7. 2. 2. 2. 2. 5. 3. Warm Up Simplify. 1. –4. – |4|. 2. |–3|. 3. Write an improper fraction to represent each mixed number. 6. 2. 3. 4. 4. 7. 3. 7. Write a mixed number to represent each improper fraction. 12. 24. 5. 6. 5. 9. California

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Warm Up Simplify. 1.

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  1. 14 55 3 7 2 2 2 2 5 3 Warm Up Simplify. 1. –4 –|4| 2. |–3| 3 Write an improper fraction to represent each mixed number. 6 2 3. 4 4. 7 3 7 Write a mixed number to represent each improper fraction. 12 24 5. 6. 5 9

  2. California Standards 2.0 Students understand and use such operations as taking the opposite, finding the reciprocal, taking a root, and raising to a fractional power. They understand and use the rules of exponents.

  3. Vocabulary real numbers absolute value opposites additive inverse

  4. The set of all numbers that can be represented on a number line are called real numbers. You can use a number line to model addition and subtraction of real numbers.

  5. Addition To model addition of a positive number, move right. To model addition of a negative number, move left. Subtraction To model subtraction of a positive number, move left. To model subtraction of a negative number, move right.

  6. 0 1 2 3 4 5 -4 -3 -2 -1 1/2 Venn Diagram Rational Numbers 1 2 4 9 • The Set of Real Numbers • All numbers fall on the Number Line Irrational Numbers π =3.1415 √2 Whole Numbers 0, Natural Numbers 1, 2, 3, 4, 5, . . . Integers …, -3, -2, -1

  7. 11 10 7 6 5 4 3 2 1 9 8 0 Adding and Subtracting Numbers on a Number Line Add or subtract using a number line. –4 + (–7) Start at 0. Move left to –4. To add –7, move left 7 units. + (–7) –4 –4 + (–7) = –11

  8. -3 -2 -1 0 1 2 3 4 5 6 8 9 7 Adding and Subtracting Numbers on a Number Line Add or subtract using a number line. 3 – (–6) Start at 0. Move right to 3. To subtract –6, move right6units. –(–6) + 3 3 – (–6) = 9

  9. -3 -1 0 1 2 3 4 6 8 9 -2 7 5 TEACH! Example 1 Add or subtract using a number line. –3 + 7 Start at 0. Move left to –3. To add 7, move right 7 units. +7 –3 –3 + 7 = 4

  10. TEACH! Example 1b Add or subtract using a number line. –3 – 7 Start at 0. Move left to –3. To subtract 7, move left 7 units. –7 –3 11 10 6 9 8 7 5 4 3 2 1 0 –3 – 7 = –10

  11. TEACH! Example 1c Add or subtract using a number line. Start at 0. Move left to –5. –5 – (–6.5) To subtract –6.5, move right 6.5 units. – (–6.5) –5 8 7 6 5 2 1 0 1 2 4 3 –5 – (–6.5) = 1.5

  12. The absolute value of a number is the distance from zero on a number line. The absolute value of 5 is written as |5|. 5 units 5units - - - - - 6 5 - 1 0 1 2 3 4 5 6 4 3 2 |–5| = 5 |5| = 5

  13. Example 2: Adding Real Numbers Add. A. Different signs: subtract the absolute values. Use the sign of the number with the greater absolute value. B. –6+ (–2) (6 + 2 = 8) Same signs: add the absolute values. –8 Both numbers are negative, so the sum is negative.

  14. TEACH! Example 2 Add. a. –5 + (–7) Same signs: add the absolute values. (5 + 7 = 12) –12 Both numbers are negative, so the sum is negative. b. –13.5 + (–22.3) Same signs: add the absolute values. (13.5 + 22.3 = 35.8) Both numbers are negative, so the sum is negative. –35.8

  15. TEACH! Example 2b Add. c.52 + (–68) (68 – 52 = 16) Different signs: subtract the absolute values. Use the sign of the number with the greater absolute value. –16

  16. Two numbers are opposites if their sum is 0. A number and its opposite are additive inverses and are the same distance from zero. They have the same absolute value.

  17. To subtract signed numbers, you can use additive inverses. Subtracting a number is the same as adding the opposite of the number.

  18. Subtracting RealNumbers

  19. Additional Example 3A:Subtracting Real Numbers Subtract. –6.7 – 4.1 –6.7 – 4.1 = –6.7 + (–4.1) To subtract 4.1, add –4.1. (6.7 + 4.1 = 10.8) Same signs: add absolute values. Both numbers are negative, so the sum is negative. –10.8

  20. TEACH! Example 3a Subtract. 13 – 21 13 – 21 To subtract 21, add –21. = 13 + (–21) Different signs: subtract absolute values. (21 – 13 = 8) Use the sign of the number with the greater absolute value. –8

  21. Additional Example 3B:Subtracting Real Numbers Subtract. 5 – (–4) 5 − (–4) = 5 + 4 To subtract –4, add 4. (5 + 4 = 9) Same signs: add absolute values. 9 Both numbers are positive, so the sum is positive.

  22. 1 2 –3 To subtract , add . 1 2 3 TEACH! Example 3b Subtract. Same signs: add absolute values. Both numbers are positive, so the sum is positive. 4

  23. , , add . To subtract Rewrite with a denominator of 10. Subtracting Real Numbers (Example 3c) Subtract. Same signs: add absolute values . Both numbers are negative, so the sum is negative. –5.3

  24. TEACH! Example 3c Subtract. –14– (–12) –14– (–12) = –14 + 12 To subtract –12, add 12. (14 – 12 = 2) Different signs: subtract absolute values. –2 Use the sign of the number with the greater absolute value.

  25. Helpful Hint On many scientific and graphing calculators, there is one button to express the opposite of a number and a different button to express subtraction.

  26. Example 4: Oceanography Application An iceberg extends 75 feet above the sea. The bottom of the iceberg is at an elevation of –247 feet. What is the height of the iceberg? Find the difference in the elevations of the top of the iceberg and the bottom of the iceberg. elevationat bottom of iceberg elevation at top of iceberg minus – –247 75 75 – (–247) 75 – (–247) = 75 + 247 To subtract –247, add 247. Same signs: add the absolute values. = 322 The height of the iceberg is 322 feet.

  27. TEACH! Example 4 The tallest known iceberg in the North Atlantic rose 550 feet above the ocean's surface. How many feet would it be from the top of the tallest iceberg to the wreckage of the Titanic, which is at an elevation of –12,468 feet? elevation at top of iceberg elevationof the Titanic minus – 550 –12,468 550 – (–12,468) To subtract –12,468, add 12,468. 550 – (–12,468) = 550 + 12,468 = 13,018 Same signs: add the absolute values.

  28. TEACH! Example 4 Continued The tallest known iceberg in the North Atlantic rose 550 feet above the ocean's surface. How many feet would it be from the top of the tallest iceberg to the wreckage of the Titanic, which is at an elevation of –12,468 feet? Distance from the top of the iceberg to the Titanic is 13,018 feet.

  29. Lesson Quiz Add or subtract using a number line. 2. –5 – (–3) –2 1. –2 + 9 7 Add or subtract. 3. –23 + 42 19 4. 4.5 – (–3.7) 8.2 5. 6. The temperature at 6:00 A.M. was –23°F. At 3:00 P.M., it was 18°F. Find the difference in the temperatures. 41°F

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