sph3Uib 1 st Day notes

1. sph3Uib 1st Day notes Significant digits, Uncertainties, Error Calculations

2. Significant digits/figures • The concept of significant figures is often used in connection with rounding. • A practical calculation that uses any irrational number necessitates rounding the number, and hence the answer, to a finite number of significant figures. • The significant digits/figures of a number are those digits that carry meaning contributing to its precision.

3. Rules for Significant digits/figures • All non-zero digits are considered significant. For example, 91 has two significant figures (9 and 1), while 123.45 has five significant figures (1, 2, 3, 4 and 5). • Zeros appearing anywhere between two non-zero digits are significant. Example: 101.12 has five significant figures: 1, 0, 1, 1 and 2. • Leading zeros are not significant. For example, 0.00052 has two significant figures: 5 and 2.

4. Rules for Significant digits/figures • Trailing zeros in a number containing a decimal point are significant. • For example, 12.2300 has six significant figures: 1, 2, 2, 3, 0 and 0. • The number 0.000122300 still has only six significant figures (the zeros before the 1 are not significant). • In addition, 120.00 has five significant figures.

5. Rules for Significant digits/figures • This convention clarifies the precision of such numbers; for example, if a result accurate to four decimal places is given as 12.23 then it might be understood that only two decimal places of accuracy are available. • Stating the result as 12.2300 makes clear that it is accurate to four decimal places.

6. Rules for Significant digits/figures • The significance of trailing zeros in a number not containing a decimal point can be ambiguous. • For example, it may not always be clear if a number like 1300 is accurate to the nearest unit (and just happens coincidentally to be an exact multiple of a hundred) or if it is only shown to the nearest hundred due to rounding or uncertainty. • Various conventions exist to address this issue, but none that are Universal. In IB, 1200 is considered as 2 sig digs, unless more info is provided.

7. Possible methods for ambiguous cases of measures (info only!!) • A bar may be placed over the last significant digit; any trailing zeros following this are insignificant. For example, , has three significant figures (and hence indicates that the number is accurate to the nearest ten). • The last significant figure of a number may be underlined; for example, "20000" has two significant figures. • A decimal point may be placed after the number; for example "100." indicates specifically that three significant figures are meant. (McMaster uses this in some cases.)

8. Rules for Significant digits/figures • If all else fails, the level of rounding can be specified explicitly. • The abbreviation s.f. is sometimes used, for example "20 000 to 2 s.f." or "20 000 (2 sf)". • Alternatively, the uncertainty can be stated separately and explicitly, as in 20 000 ± 1%, so that significant-figures rules do not apply. • http://en.wikipedia.org/wiki/Significant_figures

9. Significant digits/figures • The issues of trailing zeroes with no decimals will not affect labs, as they will have errors determined by measuring devices. • This will also be avoided on tests by not using numbers with ambiguous significant digits, or a decimal will be used (100. cm is 3 sig digs).

10. Uncertainties with labs • Uncertainties affect all sciences. • Experimental errors and human errors in reading measuring apparatus cause errors in experimental data. • A system of rules is required to indicate errors and to plot graphs indicating error. • It is important to include errors in your labs and analysis of data problems. • Significant digits are one way in which scientists deal with uncertainties.

11. Uncertainties with labs • Sig dig rules are shortcuts to looking at uncertainties. • Sig digs are not perfect rules. • The error must match the number of decimals of the measurement. (4.55 ± 0.002 is not possible). • In experiments, a series of measurements may be done and repeated carefully (precisely) many times but still have differences due to error.

12. Error types • Errors are random uncertainties that may include the observer (momentary lapse) or the environment (temperature, material variations, imperfections.....). • Any built in errors with devices are called systematic errors. • We usually use half the smallest division to indicate this. • Random uncertainties can be reduced by repeating measurements and by using graphs. • Errors show the level of confidence we have in a measure.

13. Error types • A measure is written as, for example; 2.08 m ± 0.05 m • The ± is the absolute error. • This can be converted to a percent of the measure into a relative error: 2.08 ± (0.05/2.08)x100 = 2.08 m ± 2.4% • Graphs will be plotted with absolute or relative errors. (Excel handles this easily). • See Excel graphing practice (website) for more info on this.

14. Calculations with Error • When adding/subtracting; you add the absolute errors • 1) (1.3 ± 0.1) m + (1.1 ± 0.2) m = (2.4 ± 0.3) m • 2) (6.6 ± 0.5) m - (1.6 ± 0.5) m = (5.0 ± 1.0) m • This method yields a worst case scenario in the errors!! • Limitation: a small difference between large numbers give large uncertainties: • (400 ± 5) s - (350 ± 5) s = (50 ± 10) s

15. Calculations with Error • When multiplying or dividing; you add the relative errors and express your final answers as absolute errors. • 1) (20 m/s ± 2.4%) (4.2 s ± 3.6 %) = 84 m ± 6.0 % = (84 ± 5) m (Note: error is rounded to match decimals of answer calculated (which was rounded by sd)). • 2) (5.0 ± 0.5) m / (1.0 ± 0.1) s = 5.0 m/s ± [0.5m/5.0m + 0.1 s/1.0s] (5.0 m/s) = (5.0 ± 1.0) m/s

16. Examples to show rounding rules • 1.234 <--the 4 is "fuzzy" in uncertainty (least significant) x 1.1 <- the 1 is also "fuzzy as it is least significant. 0.1234 <- all these are "fuzzy" as used "fuzzy" 1 to find them. 1.234 1.3574<----- the last 4 digits are "fuzzy" so we round off as 1.4 This is the basis for why we round off to 2 sig digs for that example.

17. Examples to show rounding rules • 1.234 + 1.1 x 10-2 1.234 <---------the 4 is fuzzy + 0.011 <---------- the last 1 is fuzzy 1.245 <---------the 5 is the fuzzy digit As the 5 is the last number, no rounding is done. • Answer is 1.245 Error worksheet