Significant Figures (digits)

1. Significant Figures (digits) How to determine the least significant figure. How to determine the least significant figure after mathematical manipulations.

2. Rules for writing significant figures All non-zero and digits to the right of any non- zero digit are significant. Zeros in a string of zeros to the left of a non-zero digit are not significant. 0.01120 - has 4 sig figs 320 - has 3 sig figs. (SIO 1989 convention) Zeros to the right have meaning (i.e. indicate the precision) and are obtained during device reading or in mathematical manipulations 2

3. Rounding Rules If the digit past the least significant digit is a: 0 - 4 then “round down” i.e. truncate 5 - 0 then “round up” i.e. add 1 to the least significant digit and truncate 3

4. Generally there are two types of devices from which numbers are read - analogue and digital Digital devices are simple to determine the least significant figure. It is the last digit (to the right) that is read out. For analogue devices determine the value that each mark on the device signifies. One should be able to estimate one more digit beyond that designated by the marks.

5. Rule for writing significant figures All non-zero and digits to the right of any non- zero digit are significant. Zeros in a string of zeros to the left of a non-zero digit are not significant. • 0.01120 - has 4 sig figs • 320 - has 3 sig figs. (SIO 1989 convention) Zeros to the right have meaning (i.e. indicate the precision) and are obtained during device reading or in mathematical manipulations

6. As an example of an digital device:a digital balance 125.7283 g The least significant figure is the "3" (i.e. read all the digits!)

7. For an analogue device:1) Determine what each line value corresponds to.2) Estimate one more digit. 0 mL The marks correspond to 0.1 mL Therefore, one can estimate to ~0.01 mL Here read 2.25 mL 1 mL 2 mL 2 mL 3 mL 4 mL 5 mL 3 mL 6 mL 7 mL 50 mL

8. If the estimate indicates that the measurement is on the line, then the trailing zero must be present In this example it appears that the level in on the line. One must indicate the estimate to ~0.01 mL Therefore this is 2.50 mL 2 mL 3 mL

9. Rule for multiplication and division Count the number of significant digits in each number being multiplied of divided Example: 524.2 4 significant figures H 345.725 6 significant figures 181 229.045 Should be to 4 sig figs. Thus: 1.812 H 105 is the correct answer*. *note: sometime one must use scientific notation to express the answer correctly.

10. Rule for addition and subtraction 1) Determine the uncertainty in each of the numbers. 2) The uncertainty in the answer is the same as the highest uncertainty determined in step 1) Example: 34.5 The uncertainty is 0.1 digit + 53.25 The uncertainty is 0.01 digit 77.75 But the uncertainty should be with 0.1 digit therefore the answer should be: 77.8

11. More examples of addition/subtraction 0.1210 The uncertainty is in the 0.0001 digit -0.01310 The uncertainty is in the 0.00001 digit 0.10790 But the uncertainty should be with the 0.0001 digit, therefore the answer is: 0.1079 1.42 H10-5 The uncertainty is in 0.01H10-5 + 2 H 10-6 The uncertainty is in 1 H 10-6 digit 1.62H10-5 But the uncertainty should be with the 1H10-6 digit, therefore the answer is: 1.6H10-5