1. Significant digits, Uncertainties, Error Calculations
sph4U 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 – RECAP!
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.

4. 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.

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 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.

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 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.

7. What we do
A decimal point may be placed after the number; for example "100." indicates specifically that three significant figures are meant. (McMaster uses this sometimes.)
In a lab, there are not usually ambiguous numbers as the measuring devices have precision errors that dictate the number of sig digs.
On a test – I would accept answers of various sig digs, if the number is ambiguous.

8. Examples Determine the number of sig digs for each of the following.
1201
56.070
305.0
0.08
10.0
4.32 x 10-3

9. Math Rules
When adding or subtracting, round your answers to the least number of decimal places (not sig digs)
Ex) = = 2.02 (2 decimals)
When multiplying or dividing, round your answers to the least number of significant figures.
1.50 m/s * 0.50 s = m (2 s.d.)

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 ± 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. Digital devices
Note: Fr digital devices we’re limited to the last decimal, and we can NOT guess beyond that.
In this case we use that last decimal as the error.
Ex) Mass scale is g, so our error must be ± 0.01 g,
NOT ± g as the device can’t ever give us 3 decimals!

14. 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.

15. Calculations with Error
When adding/subtracting; you add the absolute errors
1) (1.3 ± 0.1) m + (1.1 ± 0.1) m = (2.4 ± 0.2) 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

16. Calculations with Error
When multiplying or dividing; you add the fractional errors and multiple this by your answer.
Ex: (2.00 ± 0.05)m/s (3.00 ± 0.10) s
= 6.00 m ± [0.05/ /3.00](6.00 m)
= 6.00 m ± 0.35 m
Note: errors are NOT used for sig digs, nor do you use sig digs to express them. You ALWAYS match the number of decimals!! (The sig digs of the answer determines this).

17. Optional next 2 slides

18. Examples to show rounding rules
<--the 4 is "fuzzy" in uncertainty (least significant)
x <- the 1 is also "fuzzy as it is least significant.
<- 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.

19. Examples to show rounding rules
< 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
Error worksheet