SPH3UIB 1ST DAY NOTESSignificant 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 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.

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.

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 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.

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!!)

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.)

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

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 ± 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.

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: 2.00 + 1.4 = 3.4 (rounded to one decimal place)12.05 – 2.05 = 10.00 (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)12.06 m/3.00 s = 4.02 ms-1 (rounded to 3 sig digs)

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.

EXAMPLES TO SHOW ROUNDING RULES

1.234 + 1.1 x 10-2

1.234 <---------the 4 is fuzzy+ 0.011 <---------- the last 1 is fuzzy1.245 <---------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 ERRORWhen 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 + 0.1 s/1.0s] (5.0 m/s) = (5.0 ± 1.0) m/s Error worksheet