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Integers INTEGERS Negative numbers are used often in every day life. For example, in Oregon, temperatures can reach 3 degrees below 0, or -3° F. Or, if your banking account is overdrafted, you receive a statement showing a balance of $-12.50 (in debt). Another example is a loss in value of a certain stock is shown as $-3.4 (down 3.40 per share). Integers are the set of positive and negative whole numbers. {…-5,-4,-3,-2,-1,0,1,2,3,4,5…} Integers are often shown on a number line such as the one below. This helps many students perform all the operations with integers, such as ordering, adding, subtracting, multiplying, and dividing. ● COMPARING INTEGERS Recall the meaning of the following symbols: < less than > greater than A number line can help determine the order of two integers. Notice that the numbers get larger the further right you go. When comparing two integers, the number on the right is always larger than the number on the left. Student Practice: Fill in the blank with either a < , >. 1. 2 4. -2 7. -50 9 0 -60 2. -8 3 3. -5 5. -7 -3 6. 5 8. 0 9. -1 4 -12 -5 -4 Integers ● ABSOLUTE VALUE The absolute value of a number is it’s distance from 0 on the number line. Two vertical lines around a number are used to denote absolute value a . Sample Problem: Solution: Simplify. b. 4 a. 6 c. 0 a. 6 6 since 6 is 6 units from 0 on the number line. b. 4 4 since -4 is 4 units away from 0 on the number line. c. 0 0 since 0 is 0 units away from itself. Tip: Many students like to think of the absolute value as simply making any number POSITIVE always. Student Practice: Simplify. 11. 6 10. 7 12. 3 13. 11 ADDITION OF INTEGERS • ADDING INTEGERS USING A NUMBER LINE To add two integers a + b using a number line, 1. Start at a. 2. Then, if b is positive, move to the right. If b left. 3. The answer is where you end up. Sample Problem: Solution: Add. a. 4 (2) is negative, move to the c. 3 (4) b. 5 8 a. -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 b. c. Integers Tip: It may help to think of positive numbers as a GAIN and negative numbers as a LOSS when adding integers. When you add two positive integers or gains, you have a bigger gain so the result is still positive. When you add two negative integers or losses, you have a bigger loss so the result remains negative. When you add a positive integer and a negative integer, or a gain and a loss, your result is the DIFFERENCE of these values and the sign will be that of which was larger, the gain or loss. • ADDING INTEGERS WITHOUT A NUMBER LINE To add to integers a + b , use the following rules: • If the signs are the same (both positive or both negative), add the absolute values and keep the common sign. • If the signs are different (one positive and one negative), subtract the absolute values keep the sign of the larger number. Sample Problem: Solution: Add. a. 11 (4) a. 11 (4) 15 b. 10 3 7 c. 20 (5) 15 b. 10 3 c. 20 (5) add and keep the sign. subtract and keep the sign of the larger. subtract and keep the sign of the larger. Student Practice: Add. 14. –15 + 5 15. 25 + (–15) 16. –20 + (–12) 17. –4 + 4 18. 1 + (–4) 19. –2 + (–4) 20. 6 5 21. 1 (9) 22. 10 (20) 23. 8 12 24. 3 (15) 25. 9 (9) Integers • ADDING A SERIES OF INTEGERS When adding a series of integers, you can either • Add integers from left to right, OR • Add all positive numbers together and all the negative numbers together, then add the results. Student Practice: Add. 26. 15 21 (8) (12) 3 8 27. 10 (4) (20) 5 5 SUBTRACTION OF INTEGERS Before we determine the rule for subtracting, consider the following examples. b. 12 4 = a. 10 5 = 10 (5) = c. 2 10 = 12 (4) = d. 3 5 = 2 (10) = 3 (5) = • SUBTRACTING INTEGERS One way to subtract two integers is to add the opposite. In other words, change the subtraction to addition and change the sign of the second integer. Then use the addition rules. If a and b are integers, then a b a (b) . Sample Problem: Subtract. a. 20 15 Solution: a. b. c. d. b. 5 15 20 15 20 (15) 5 5 15 5 (15) 10 5 9 5 (9) 14 6 (10) 6 (10) 6 10 4 c. 5 9 d. 6 (10) Integers Tip: It may help to think of subtraction ALSO AS A LOSS, just like a negative. In this case, 3 – 6 means a gain of 3 but a loss of 6, so it’s -3. And, –5 – 10 means a loss of 5, then a loss of 10 more, so it’s –15. Student Practice: Subtract. 28. 5 20 29. 10 8 30. 8 7 31. 4 (3) 32. 12 (10) 33. 1 15 34. 7 12 35. 3 (9) 36. 12 3 37. 7 (7) 38. 10 10 39. 20 (20) 40. 4 24 41. 43. 10 5 8 3 20 2 (20) 44. 4 (6) 12 17 2 42. 2 3 45. 6 4 20 2 24 Integers MULTIPLYING INTEGERS Recall that multiplication is simply repeated addition. In other words, 2 times 4 really means add 2 four times. 24 2 2 2 2 8 In the same way, -2 times 4 really means add -2 four times. 2 4 (2) (2) (2) (2) 8 Additionally, 6 times -3 means add -3 six times which results in -18. This suggests that the product of a negative integer and a positive integer is always negative. Now consider the following pattern: 3 3 9 3 2 6 3 1 3 3 0 0 3 1 3 2 This suggests that the product of two negative integers is always positive. • MULTIPLYING INTEGERS When multipling two integers a b , remember this: • If the signs are different, one positive and one negative, the result is NEGATIVE, regardless which is bigger. ()() () ()() () • If the signs are the same, either both positive or both negative, the result is POSITIVE. ()() ( ) Sample Problem: Solution: Multiply. a. 3(9) 27 b. 4(3) 12 c. 1(100) 100 a. 3(9) b. 4(3) c. 1(100) (-) (+) = (-) (-) (-) = (+) (-) (+) = (-) + Tip: Some students like to use the following triangle. Cover any two and the result is always the third. _ multiplication division _ Integers Student Practice: Multiply. 1. 3 6 2. 5 (3) 3. (2)(10) 4. 0(8) 5. (1)(8) 6. 4(20) 7. 9(1) 8. 500(2) 9. 6(6) 10. (2)(3)(5)(2) 11. (4)(1)(3)(2) EXPONENTS If the base of an exponential expression is negative, then: • If the exponent is even, the result is positive. • If the exponent is odd, the result is negative. Sample Problem: Evaluate: Solution: a. (8) 2 b. (2) 3 a. (8) 2 (8)(8) 64 b. (2) 3 (2)(2)(2) 8 c. 4 2 (4 4) 16 c. 4 2 Note: Only the 4 is squared. Student Practice: Evaluate. 12. (5) 2 13. (3) 3 14. (2) 4 Integers 15. 62 18. (9) 2 16. (1) 88 17. (4) 3 19. (2) 3 20. (10) 2 DIVIDING INTEGERS • DIVIDING INTEGERS a , remember this: b • If the signs are different, one positive and one negative, the result is NEGATIVE, regardless which is bigger. When dividing two integers ( ) ( ) () () ( ) ( ) • If the signs are the same, either both positive or both negative, the result is POSITIVE. ( ) ( ) ( ) Sample Problem: Solution: Divide. 9 3 3 12 6 b. 2 15 5 c. 3 a. 9 3 b. 12 2 c. 15 3 a. + Tip: Some students like to use the following triangle. Cover any two and the result is always the third. _ multiplication division _ Integers Student Practice: Divide. 21. 8 2 24. 81 9 27. 8 8 22. 2 2 25. 23. 25 5 100 10 26. 18 2 29. 12 2 28. 21 (7) SPECIAL CASES 30. 0 3 31. 5 0 a is undefined for any value of a 0 Now that we have seen ALL operations and rules with integers, let’s practice simplifying some expressions that consist of mixed operations to help us remember the rules of signs. Integers ORDER OF OPERATIONS Sample Problem: 12 (3 5) 3 2 11 Simplify: 12 (3 5) 3 2 11 Solution: 12 ( 2) 3 2 11 12 ( 8) 2 11 12 ( 16 ) 11 12 16 11 4 11 15 Student Practice: Simplify each expression using the proper order of operations. Be careful with your signs. 1. 2(3) 5 2. 8 (6) 3 3. 7 4(5) 4. 20 10(3) 5. 20 10(3) 6. 7(2) (10) 7. (30 5) 5 10 8. 10 (3) 2 9. 14 8 (2) 10 11. 4 2 3 (1 3) 2 12. 35 33 10. 40 (5)(2) Integers EVALUATING EXPRESSIONS 3x 3 4 x 3 Sample Problem: Evaluate Solution: 3x3 4 x 3 if x = -3 3( 3) 3 4( 3) 3 3( 27) 4( 3) 3 81 4( 3) 3 81 ( 12) 3 81 12 3 96 Student Practice: Evaluate each expression if x = -3, y = 4, and z = -2 . 13. x y z 14. x y z 15. x 2 2 16. x 2 2 17. 3x 2 2 z 18. 2 z 3 y 5 x Integers JUST FOR FUN Imagine if you were asked to find the sum of the first n whole numbers. For example, find the sum of the first 3 whole numbers…. the first 4 whole numbers… the first 5 whole numbers… 1+2+3 1+2+3+4 1+2+3+4 +5 How about the first 50 whole numbers? Can you generalize a rule that will help us sum up the first n whole numbers without actually having to add up the numbers? Could this somehow be useful? Problem 1: Imagine a room with 60 people. If each person shakes hands ONCE with every other person in the room, how many handshakes would there be? Problem 2: A volleyball league has just started at STC on Tuesday evenings. A team called Full Throttle has won the last 3 consecutive volleyball tournaments. Ever heard of them? Your instructor is the captain of that team. What does that have to do with this? NOTHING. I’m just gloating. Now back to the problem. Suppose there are 16 teams in the league and that each team must play each other ONE TIME. The team with the best record wins the league. How many games will there be total?