SPH3UIB 1 ST DAY NOTES Significant digits, Uncertainties, Error Calculations

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.

RULES FOR SIGNIFICANT DIGITS/FIGURES  All non-zero digits are considered significant. For example, 91 has two significant figures (9 and 1), while has five significant figures (1, 2, 3, 4 and 5).  Zeros appearing anywhere between two non-zero digits are significant. Example: has five significant figures: 1, 0, 1, 1 and 2.  Leading zeros are not significant. For example, has two significant figures: 5 and 2.

RULES FOR SIGNIFICANT DIGITS/FIGURES  Trailing zeros in a number containing a decimal point are significant.  For example, has six significant figures: 1, 2, 2, 3, 0 and 0.  The number still has only six significant figures (the zeros before the 1 are not significant).  In addition, has five significant figures.

RULES FOR SIGNIFICANT DIGITS/FIGURES  This convention clarifies the precision of such numbers; for example, if a result accurate to four significant digits is given as then it might be understood that only two decimal places of accuracy are available.  Stating the result as makes clear that it is accurate to four decimal places.

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.

POSSIBLE METHODS FOR AMBIGUOUS CASES OF MEASURES (INFO ONLY!!)

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 " to 2 s.f." or " (2 sf)".  Alternatively, the uncertainty can be stated separately and explicitly, as in ± 1%, so that significant-figures rules do not apply. 

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).

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.

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 ± 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.

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.

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.

MATH RULES  When we add or subtract – we need to round our answers to the least number of decimals. Ex: = 3.4 (rounded to one decimal place) – 2.05 = (rounded to two decimal places)  When we multiply or divide – we need to round the answer to the least number of significant digits. Ex: 2.0 (12.5) = 25 (rounded to 2 sig digs) m/3.00 s = 4.02 ms -1 (rounded to 3 sig digs)

EXAMPLES TO SHOW ROUNDING RULES  <--the 4 is "fuzzy" in uncertainty (least significant) x 1.1 <- the 1 is also "fuzzy as it is least significant <- all these are "fuzzy" as used "fuzzy" 1 to find them <----- 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.

EXAMPLES TO SHOW ROUNDING RULES  x < the 4 is fuzzy < the last 1 is fuzzy < the 5 is the fuzzy digit As the 5 is the last number, no rounding is done.  Answer is 1.245

CALCULATIONS WITH ERROR  When adding/subtracting; you add the absolute errors  1) (1.3 ± 0.2) m + (1.1 ± 0.2) m = (2.4 ± 0.4) 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

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 to 2 sd)). 2) (5.0 ± 0.5) m / (1.0 ± 0.1) s = 5.0 m/s ± [0.5m/5.0m s/1.0s] (5.0 m/s) = (5.0 ± 1.0) m/s  Error worksheet