Name Date TI-84+ GC 7 Avoiding Round-off Error in Multiple Calculations Objectives: Recall the meaning of exact and approximate Observe round-off error and learn to avoid it Perform calculations using the order of operations and extra parentheses Recall: An exact answer has no error. If we use an exact result to perform additional calculations, we ll continue to get exactly the right answer. If we perform the same calculation to different versions of an exact answer, we ll always get the same, exact final result. An approximate answer is close to the exact answer, but is a near miss. We usually find approximate answers by rounding or approximating. If we start with an approximate answer and perform additional calculations, we ll get approximate final results. CAUTION: You should always give an EXACT answer unless the instructions tell you to round. Round-off error is the absolute value of the difference between the exact answer and a rounded approximation of that answer, given by this formula: RoundoffError = exact approximate. Round-off error is the answer to the question How wrong is the rounded answer? 1 1 Example 1: = 0.15 exactly. Rounded to the nearest tenth,. 1 8 8 The round-off error, using the formula, in the answer 0.1 is 0.15 0.1 = 0. 05 In this example, the answer is wrong by 0.05. Round-off errors can become much bigger if a calculation is done from rounded partial results. 47.7908 Example : To illustrate the error of rounding partial results, calculate exactly and with 0.4678 rounded partial results, and then find the resulting round-off error. How wrong will the answer be? a) Calculate 47.7908 0.4678 exactly. Answer: 586 b) Round 47.7908 to the nearest tenth. Answer: 47.8 c) Round 0.4678 to the nearest tenth. Answer: 0.5 d) Divide your rounded answer for 47.7908 by your rounded answer for 0.4678 47.8 = 4945.6 Answer: 4945.6.5 e) Find the round-off error for this calculation. 586 4945.6 = 40.4 Answer: 40.4 How wrong is the answer? It s off by 40.4! That s a lot.
TI-84+ GC 7 Avoiding Round-off Error in Multiple Calculations, page IMPORTANT: Do not round parts of a calculation! Instead, do the calculation all at once (using extra parentheses for the correct order of operations) or use memory storage locations. If the instructions say to round, round only the final answer. Example : Calculate 41.78 (6.91) 9.188+ 76.4. Round to the nearest thousandth. Remember that the long fraction bar means that the entire numerator and entire denominator must be calculated before the results are divided. But the GC follows the order of operations and will not add or subtract before dividing unless we use extra parentheses, like this: ( 41.78 (6.91) ) ( 9.188+ 76.4) Then round the final answer to the nearest thousandth. Answer: -.96 Example 4: Calculate ( 1) 6. Round to the nearest ten-thousandth. 9.7 18.04 Notice that there is a subtraction in the exponent. The GC will not subtract before an exponent unless extra parentheses are added, like this: ( ) ( 6 1 ) 9.7 18.04 Also notice the long fraction bar, as before. Again, extra parentheses are needed so the GC will subtract the denominator before dividing, like this: ( ) ( 6 1 ) ( 9.7 18.04)
TI-84+ GC 7 Avoiding Round-off Error in Multiple Calculations, page Example 4, continued: Convert scientific notation to standard notation: 5 Multiply 6.948889 by 10, by moving the decimal point five places left to get 0.00006948889 Then round to the nearest ten thousandth (four decimal places): 0.0001 Answer: 0.0001 Example 5: Calculate 0.07 50000 1 + 65 1(65) Notice that there is a multiplication in the exponent. The GC will not multiply before an exponent unless extra parentheses are used: ( 1( 65) ) 0.07 50000 1 + 65 Round to nearest hundredth ( decimal places): 10,05.5 Answer: 10,05.5
TI-84+ GC 7 Avoiding Round-off Error in Multiple Calculations, page 4 Example 6: Calculate.6 7.1 4.5+ 9.. Round to the nearest thousandth. Notice that the parentheses supplied surround both the numerator and the denominator. These do not ensure that the numerator will be subtracted first! To get the correct answer, use additional parenthesis inside the given parentheses, like this: (.6 7.1) ( 4.5+ 9.) Answer: -0.017 Example 7: Calculate 10 1 7 Notice the square roots are different. First, the square root in the numerator is the square root of a difference use parentheses around the difference. Second, the denominator is a difference of a square root and, so close the parentheses for the square root, and use another set around the entire denominator, like this: ( 10 1) ( 7) ( ) Answer: 14.61
TI-84+ GC 7 Avoiding Round-off Error in Multiple Calculations, page 5 Practice: 1) Find the round-off error if 16 5 is rounded to the nearest tenth. ) Find the round-off error if 189 is rounded to the nearest hundred. ) Calculate 4) Calculate 5) Calculate 6) Calculate 7) Calculate 8) Calculate 9) Calculate ( 4.9) 7. 97.4 6.08 9.71+ 14 77 7 ( 1) + 718 654 (.64) 8 79 0.055 75000 1 + 1 0.08 90 1 4 7 1 0.0941 716 1 11 8 + 5 ( 5) 147 6098 ( ) ( 4). Round to the nearest tenth. 0(1). Round to the nearest whole. 11 16 76 6 4(). Round to the nearest ten. 8. Round to the nearest thousandth. 0.0088 0.001907.7 6. 10) Calculate 5.4+ 8.1 11) Calculate 1) Calculate 1) Calculate 14) Calculate. Round to the nearest thousandth. 14 1 6 7. 1. 6. 1.9 + 6 14 1 1 4 17 4 7
TI-84+ GC 7 Avoiding Round-off Error in Multiple Calculations, solutions, page 6 1) 5 = 0. 15 rounds to 0.. Round-off error = 0.15 0. = 0. 015 Answer: 0.015 16 ) 189 rounded to the nearest hundred is 100. Round-off error 189 100 = 11 = 11 Answer: 11 ) Answer: 0486. 4) Answer: 4.56 5) Answer: 1,,898.14
TI-84+ GC 7 Avoiding Round-off Error in Multiple Calculations, solutions, page 7 6) Answer: 95 7) Answer: 770. 8) Answer: -517.55 9) Answer: -0.006
TI-84+ GC 7 Avoiding Round-off Error in Multiple Calculations, solutions, page 8 10) 11) Answer: 0.071 1) Answer: -17.5 1) Answer: 0.8 14) Answer: -0.06 Answer: 1.0