Adding and Subtracting Real Numbers Section 1-5

Goals Goal Rubric To find sums and differences of real numbers. Level 1 – Know the goals. Level 2 – Fully understand the goals. Level 3 – Use the goals to solve simple problems. Level 4 – Use the goals to solve more advanced problems. Level 5 – Adapts and applies the goals to different and more complex problems.

Vocabulary Absolute value Opposite Additive inverses

Real Numbers 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. 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.

Example: Adding & Subtracting 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 11 10 9 8 7 6 5 4 3 2 1 –4 + (–7) = –11

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

Your Turn: 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 -2 -1 1 2 3 4 5 6 7 8 9 –3 + 7 = 4

Your Turn: 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 9 8 7 6 5 4 3 2 1 –3 – 7 = –10

Your Turn: 8 7 6 5 4 3 2 1 1 2 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 4 3 2 1 1 2 –5 – (–6.5) = 1.5

Definition Absolute Value – The distance between a number and zero on the number line. Absolute value is always nonnegative since distance is always nonnegative. The symbol used for absolute value is | |. Example: The |-2| is 2 and the |2| is 2.

Absolute Value on the Number Line 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 5 units - 6 - 5 - 4 - 3 - 2 - 1 1 2 3 4 5 6 |–5| = 5 |5| = 5

Rules For Adding

Example: Adding Real Numbers 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.

Your Turn: Add. a. –5 + (–7) (5 + 7 = 12) –12 b. –13.5 + (–22.3) 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

Your Turn: Add. c. 52 + (–68) (68 – 52 = 16) –16 Different signs: subtract the absolute values. Use the sign of the number with the greater absolute value. –16

Definition Additive Inverse – The negative of a designated quantity. The additive inverse is created by multiplying the quantity by -1. Example: The additive inverse of 4 is -1 ∙ 4 = -4.

Opposites 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.

Additive Inverse Property

Subtracting Real Numbers To subtract signed numbers, you can use additive inverses. Subtracting a number is the same as adding the opposite of the number. Example: The expressions 3 – 5 and 3 + (-5) are equivalent.

Subtracting Real Numbers A number and its opposite are additive inverses. To subtract signed numbers, you can use additive inverses. Subtracting 6 is the same as adding the inverse of 6. Additive inverses 11 – 6 = 5 11 + (–6) = 5 Subtracting a number is the same as adding the opposite of the number.

Rules For Subtracting Subtracting Real Numbers

Example: Subtracting Real Numbers –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. –10.8 Both numbers are negative, so the sum is negative.

Example: Subtracting Real Numbers 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.

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

Your Turn: Subtract. 13 – 21 (21 – 13 = 8) –8 To subtract 21, add –21. = 13 + (–21) To subtract 21, add –21. Different signs: subtract absolute values. (21 – 13 = 8) Use the sign of the number with the greater absolute value. –8

Your Turn: Subtract. –14 – (–12) (14 – 12 = 2) –2 –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.

elevation at top of iceberg Example: 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. elevation at bottom of iceberg elevation at top of iceberg minus 75 – –247 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.

elevation at top of iceberg Your Turn: What if…? 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 elevation of 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. Distance from the top of the iceberg to the Titanic is 13,018 feet.

Joke Time What’s brown and sticky? A stick. What happened when the wheel was invented? It caused a revolution. Why was the calendar depressed? Because it’s days were numbered.

Assignment 1.5 Exercises Pg. 41 – 43: #10 – 76 even