Significant Figures in Operations © R. A. Hill

Odd Even Rounding Rule:   If the digit after the digit to be retained is: 1) < 5, then round down. 2.34 = 2.3 2) > 5, “ “ up. 2.36 = 2.4 3) = 5 & the digit to be retained is odd, round up. 2.35 = 2.4 4) = 5 “ “ “ “ “ “ “ even, “ down. 2.45 = 2.4

2 key characteristics of measurements:   # of S.F.  # of digits coming directly from the measuring device. accuracy – how close a real measurement is to the immeasurable measurement containing an infinite number of digits. 3.1 3.12 3.123 3.123409345007613952013997700041359019802910… Which measurement is most accurate as compared to

3.123 is more accurate than the other 2 measurements 2 key characteristics of measurements:   # of S.F.  # of digits coming directly from the measuring device. accuracy – how close a real measurement is to the immeasurable measurement containing an infinite number of digits. 3.1 3.12 3.123 3.123409345007613952013997700041359019802910… 3.123 is more accurate than the other 2 measurements

3.123 is more accurate than the other 2 measurements 2 key characteristics of measurements:   # of S.F.  # of digits coming directly from the measuring device. accuracy – how close a real measurement is to the immeasurable measurement containing an infinite number of digits. (…“how far to the right” a measurement has been made) 3.1 3.12 3.123 3.123409345007613952013997700041359019802910… 3.123 is more accurate than the other 2 measurements

2 key characteristics of measurements:   # of S.F.  # of digits coming directly from the measuring device. accuracy – how close a real measurement is to the immeasurable measurement containing an infinite number of digits. (…“how far to the right” a measurement has been made) Ex: 2.25 or 564.3 is more accurate has more S.F.

5 . 0 4 3 + 1 . 2 6 . 2 4 3 Addition & Subtraction   The answer cannot be any more or less accurate than the # used during the calculation which was least accurate. 5 . 0 4 3 + 1 . 2 6 . 2 4 3 Recall

5 . 0 4 3 + 1 . 2 6 . 2 4 3 Addition & Subtraction   The answer cannot be any more or less accurate than the # used during the calculation which was least accurate. 5 . 0 4 3 + 1 . 2 6 . 2 4 3 Recall

5 . 0 4 3 + 1 . 2 6 . 2 4 3 Addition & Subtraction   The answer cannot be any more or less accurate than the # used during the calculation which was least accurate. 5 . 0 4 3 + 1 . 2 6 . 2 4 3 Recall

5 . 0 4 3 + 1 . 2 6 . 2 Addition & Subtraction   The answer cannot be any more or less accurate than the # used during the calculation which was least accurate. 5 . 0 4 3 + 1 . 2 6 . 2 Recall Final Answer

Assume that the following numbers are measurements and determine the answer with the appropriate number of significant figures. (1) 31.1 + 1.02 32.12

Assume that the following numbers are measurements and determine the answer with the appropriate number of significant figures. (1) 31.1 + 1.02 32.12

Assume that the following numbers are measurements and determine the answer with the appropriate number of significant figures. (1) 31.1 + 1.02 32.12

Assume that the following numbers are measurements and determine the answer with the appropriate number of significant figures. (1) 31.1 + 1.02 32.12

Assume that the following numbers are measurements and determine the answer with the appropriate number of significant figures. (1) 31.1 + 1.02 32.1

Assume that the following numbers are measurements and determine the answer with the appropriate number of significant figures. (1) 31.1 + 1.02 32.1

Assume that the following numbers are measurements and determine the answer with the appropriate number of significant figures. (1) 31.1 (2) 123.3 + 1.02 41.05 32.1 + 12.123 176.473

Assume that the following numbers are measurements and determine the answer with the appropriate number of significant figures. (1) 31.1 (2) 123.3 + 1.02 41.05 32.1 + 12.123 176.473

Assume that the following numbers are measurements and determine the answer with the appropriate number of significant figures. (1) 31.1 (2) 123.3 + 1.02 41.05 32.1 + 12.123 176.473

Assume that the following numbers are measurements and determine the answer with the appropriate number of significant figures. (1) 31.1 (2) 123.3 + 1.02 41.05 32.1 + 12.123 176.473

Assume that the following numbers are measurements and determine the answer with the appropriate number of significant figures. (1) 31.1 (2) 123.3 + 1.02 41.05 32.1 + 12.123 176.473

Assume that the following numbers are measurements and determine the answer with the appropriate number of significant figures. (1) 31.1 (2) 123.3 + 1.02 41.05 32.1 + 12.123 176.5

Assume that the following numbers are measurements and determine the answer with the appropriate number of significant figures. (1) 31.1 (2) 123.3 + 1.02 41.05 32.1 + 12.123 176.5

Assume that the following numbers are measurements and determine the answer with the appropriate number of significant figures. (1) 31.1 (2) 123.3 (3) 1300 + 1.02 41.05  10.2 32.1 + 12.123 1289.8 176.5

Assume that the following numbers are measurements and determine the answer with the appropriate number of significant figures. How many S.F.? What is the accuracy? (1) 31.1 (2) 123.3 (3) 1300 + 1.02 41.05  10.2 32.1 + 12.123 1289.8 176.5

Assume that the following numbers are measurements and determine the answer with the appropriate number of significant figures. (1) 31.1 (2) 123.3 (3) 1300 + 1.02 41.05  10.2 32.1 + 12.123 1289.8 176.5

Assume that the following numbers are measurements and determine the answer with the appropriate number of significant figures. (1) 31.1 (2) 123.3 (3) 1300 + 1.02 41.05  10.2 32.1 + 12.123 1289.8 176.5

Assume that the following numbers are measurements and determine the answer with the appropriate number of significant figures. (1) 31.1 (2) 123.3 (3) 1300 + 1.02 41.05  10.2 32.1 + 12.123 1289.8 176.5

Assume that the following numbers are measurements and determine the answer with the appropriate number of significant figures. (1) 31.1 (2) 123.3 (3) 1300 + 1.02 41.05  10.2 32.1 + 12.123 1289.8 176.5 When in doubt as the # of S.F. in a measurement, assume the least # of S.F. are present.

Assume that the following numbers are measurements and determine the answer with the appropriate number of significant figures. (1) 31.1 (2) 123.3 (3) 1300 + 1.02 41.05  10.2 32.1 + 12.123 1289.8 176.5 When in doubt as the # of S.F. in a measurement, assume the least # of S.F. are present.

Assume that the following numbers are measurements and determine the answer with the appropriate number of significant figures. (1) 31.1 (2) 123.3 (3) 1300 + 1.02 41.05  10.2 32.1 + 12.123 1289.8 176.5 When in doubt as the # of S.F. in a measurement, assume the least # of S.F. are present.

Assume that the following numbers are measurements and determine the answer with the appropriate number of significant figures. (1) 31.1 (2) 123.3 (3) 1300 + 1.02 41.05  10.2 32.1 + 12.123 1300 176.5 When in doubt as the # of S.F. in a measurement, assume the least # of S.F. are present.

Assume that the following numbers are measurements and determine the answer with the appropriate number of significant figures. (1) 31.1 (2) 123.3 (3) 1300 + 1.02 41.05  10.2 32.1 + 12.123 1300 176.5 When in doubt as the # of S.F. in a measurement, assume the least # of S.F. are present.

Assume that the following numbers are measurements and determine the answer with the appropriate number of significant figures. (1) 31.1 (2) 123.3 (3) 1300 (4) 32.4 + 1.02 41.05  10.2 32 32.1 + 12.123 1300 + 30 176.5 94.4

Assume that the following numbers are measurements and determine the answer with the appropriate number of significant figures. (1) 31.1 (2) 123.3 (3) 1300 (4) 32.4 + 1.02 41.05  10.2 32 32.1 + 12.123 1300 + 30 176.5 94.4

Assume that the following numbers are measurements and determine the answer with the appropriate number of significant figures. (1) 31.1 (2) 123.3 (3) 1300 (4) 32.4 + 1.02 41.05  10.2 32 32.1 + 12.123 1300 + 30 176.5 94.4

Assume that the following numbers are measurements and determine the answer with the appropriate number of significant figures. (1) 31.1 (2) 123.3 (3) 1300 (4) 32.4 + 1.02 41.05  10.2 32 32.1 + 12.123 1300 + 30 176.5 94.4

Assume that the following numbers are measurements and determine the answer with the appropriate number of significant figures. (1) 31.1 (2) 123.3 (3) 1300 (4) 32.4 + 1.02 41.05  10.2 32 32.1 + 12.123 1300 + 30 176.5 94.4

Assume that the following numbers are measurements and determine the answer with the appropriate number of significant figures. (1) 31.1 (2) 123.3 (3) 1300 (4) 32.4 + 1.02 41.05  10.2 32 32.1 + 12.123 1300 + 30 176.5 90

Assume that the following numbers are measurements and determine the answer with the appropriate number of significant figures. (1) 31.1 (2) 123.3 (3) 1300 (4) 32.4 + 1.02 41.05  10.2 32 32.1 + 12.123 1300 + 30 176.5 90

Multiplication & Division   The answer cannot have any more or less S.F. than the # used in the calculation which has the least # of S.F.

Multiplication & Division   The answer cannot have any more or less S.F. than the # used in the calculation which has the least # of S.F. (1) 272 x 12 = 3264

Multiplication & Division   The answer cannot have any more or less S.F. than the # used in the calculation which has the least # of S.F. (1) 272 x 12 = 3264

Multiplication & Division   The answer cannot have any more or less S.F. than the # used in the calculation which has the least # of S.F. (1) 272 x 12 = 3300

Multiplication & Division   The answer cannot have any more or less S.F. than the # used in the calculation which has the least # of S.F. (1) 272 x 12 = 3300

Multiplication & Division   The answer cannot have any more or less S.F. than the # used in the calculation which has the least # of S.F. (1) 272 x 12 = 3300 (2) 523 / 12.56 = 41.6401…

Multiplication & Division   The answer cannot have any more or less S.F. than the # used in the calculation which has the least # of S.F. (1) 272 x 12 = 3300 (2) 523 / 12.56 = 41.6401…

Multiplication & Division   The answer cannot have any more or less S.F. than the # used in the calculation which has the least # of S.F. (1) 272 x 12 = 3300 (2) 523 / 12.56 = 41.6

Multiplication & Division   The answer cannot have any more or less S.F. than the # used in the calculation which has the least # of S.F. (1) 272 x 12 = 3300 (2) 523 / 12.56 = 41.6

Multiplication & Division   The answer cannot have any more or less S.F. than the # used in the calculation which has the least # of S.F. (1) 272 x 12 = 3300 (2) 523 / 12.56 = 41.6 (3) 100 x 3500 = 350 000

Multiplication & Division   The answer cannot have any more or less S.F. than the # used in the calculation which has the least # of S.F. (1) 272 x 12 = 3300 (2) 523 / 12.56 = 41.6 (3) 100 x 3500 = 350 000

Multiplication & Division   The answer cannot have any more or less S.F. than the # used in the calculation which has the least # of S.F. (1) 272 x 12 = 3300 (2) 523 / 12.56 = 41.6 (3) 100 x 3500 = 350 000

Multiplication & Division   The answer cannot have any more or less S.F. than the # used in the calculation which has the least # of S.F. (1) 272 x 12 = 3300 (2) 523 / 12.56 = 41.6 (3) 100 x 3500 = 350 000

Multiplication & Division   The answer cannot have any more or less S.F. than the # used in the calculation which has the least # of S.F. (1) 272 x 12 = 3300 (2) 523 / 12.56 = 41.6 (3) 100 x 3500 = 400 000

Multiplication & Division   The answer cannot have any more or less S.F. than the # used in the calculation which has the least # of S.F. (1) 272 x 12 = 3300 (2) 523 / 12.56 = 41.6 (3) 100 x 3500 = 400 000

Multiplication & Division   The answer cannot have any more or less S.F. than the # used in the calculation which has the least # of S.F. (1) 272 x 12 = 3300 (2) 523 / 12.56 = 41.6 (3) 100 x 3500 = 400 000 (4) 6.000 / 3.00 = 2

Multiplication & Division   The answer cannot have any more or less S.F. than the # used in the calculation which has the least # of S.F. (1) 272 x 12 = 3300 (2) 523 / 12.56 = 41.6 (3) 100 x 3500 = 400 000 (4) 6.000 / 3.00 = 2

Multiplication & Division   The answer cannot have any more or less S.F. than the # used in the calculation which has the least # of S.F. (1) 272 x 12 = 3300 (2) 523 / 12.56 = 41.6 (3) 100 x 3500 = 400 000 (4) 6.000 / 3.00 = 2

Multiplication & Division   The answer cannot have any more or less S.F. than the # used in the calculation which has the least # of S.F. (1) 272 x 12 = 3300 (2) 523 / 12.56 = 41.6 (3) 100 x 3500 = 400 000 (4) 6.000 / 3.00 = 2.00

Multiplication & Division   The answer cannot have any more or less S.F. than the # used in the calculation which has the least # of S.F. (1) 272 x 12 = 3300 (2) 523 / 12.56 = 41.6 (3) 100 x 3500 = 400 000 (4) 6.000 / 3.00 = 2.00

3.45 x 0.45 = 0.5175 3 Sequential Operations   When multiple operations are done in sequence, rounding to the proper # of S.F. is done only in the last step. When all the operations belong to the same category, i.e. all + and or  or all x and or , then use the appropriate pattern to round in the last step. 3.45 x 0.45 = 0.5175 3

3.45 x 0.45 = 0.5175 3 Sequential Operations   When multiple operations are done in sequence, rounding to the proper # of S.F. is done only in the last step. When all the operations belong to the same category, i.e. all + and or  or all x and or , then use the appropriate pattern to round in the last step. 3.45 x 0.45 = 0.5175 3

3.45 x 0.45 = 0.5 3 Sequential Operations   When multiple operations are done in sequence, rounding to the proper # of S.F. is done only in the last step. When all the operations belong to the same category, i.e. all + and or  or all x and or , then use the appropriate pattern to round in the last step. 3.45 x 0.45 = 0.5 3

3.45 x 0.45 = 0.5 3 Sequential Operations   When multiple operations are done in sequence, rounding to the proper # of S.F. is done only in the last step. When all the operations belong to the same category, i.e. all + and or  or all x and or , then use the appropriate pattern to round in the last step. 3.45 x 0.45 = 0.5 3

Sequential Operations   When multiple operations are done in sequence, rounding to the proper # of S.F. is done only in the last step. When all the operations belong to the same category, i.e. all + and or  or all x and or , then use the appropriate pattern to round in the last step. When mixed operations are involved, some + and or  and some x and or , then the estimate must be tracked (underlined) in order to round in the last step.

Sequential Operations   When multiple operations are done in sequence, rounding to the proper # of S.F. is done only in the last step. When mixed operations are involved, some + and or  and some x and or , then the estimate must be tracked (underlined) in order to round in the last step.

3.45 x 0.45 + 1.01 = 1.5275 3 o.5175 Sequential Operations   When multiple operations are done in sequence, rounding to the proper # of S.F. is done only in the last step. When mixed operations are involved, some + and or  and some x and or , then the estimate must be tracked (underlined) in order to round in the last step. 3.45 x 0.45 + 1.01 = 1.5275 3 o.5175

3.45 x 0.45 + 1.01 = 1.5 3 o.5175 Sequential Operations   When multiple operations are done in sequence, rounding to the proper # of S.F. is done only in the last step. When mixed operations are involved, some + and or  and some x and or , then the estimate must be tracked (underlined) in order to round in the last step. 3.45 x 0.45 + 1.01 = 1.5 3 o.5175

56.2 x 0.081 = 7.00338… 2.15 − 1.5 Sequential Operations   When multiple operations are done in sequence, rounding to the proper # of S.F. is done only in the last step. When mixed operations are involved, some + and or  and some x and or , then the estimate must be tracked (underlined) in order to round in the last step. 56.2 x 0.081 = 7.00338… 2.15 − 1.5

56.2 x 0.081 4.5522 = = 7.00338… 2.15 − 1.5 0.65 Sequential Operations   When multiple operations are done in sequence, rounding to the proper # of S.F. is done only in the last step. When mixed operations are involved, some + and or  and some x and or , then the estimate must be tracked (underlined) in order to round in the last step. 56.2 x 0.081 4.5522 = = 7.00338… 2.15 − 1.5 0.65

56.2 x 0.081 4.5522 = = 7 2.15 − 1.5 0.65 Sequential Operations   When multiple operations are done in sequence, rounding to the proper # of S.F. is done only in the last step. When mixed operations are involved, some + and or  and some x and or , then the estimate must be tracked (underlined) in order to round in the last step. 56.2 x 0.081 4.5522 = = 7 2.15 − 1.5 0.65

3.4 + 0.71 = 0.101631…. 36.4 x 1.111 Sequential Operations   When multiple operations are done in sequence, rounding to the proper # of S.F. is done only in the last step. When mixed operations are involved, some + and or  and some x and or , then the estimate must be tracked (underlined) in order to round in the last step. 3.4 + 0.71 = 0.101631…. 36.4 x 1.111

Sequential Operations   When multiple operations are done in sequence, rounding to the proper # of S.F. is done only in the last step. When mixed operations are involved, some + and or  and some x and or , then the estimate must be tracked (underlined) in order to round in the last step. 3.4 + 0.71 4.11 = = 0.101631…. 36.4 x 1.111 40.4404

3.4 + 0.71 4.11 = = 0.10 36.4 x 1.111 40.4404 Sequential Operations   When multiple operations are done in sequence, rounding to the proper # of S.F. is done only in the last step. When mixed operations are involved, some + and or  and some x and or , then the estimate must be tracked (underlined) in order to round in the last step. 3.4 + 0.71 4.11 = = 0.10 36.4 x 1.111 40.4404

Significant figures will be used in every calculation in this course from this point on… It is used in college physical science courses for all calculations…