1. Significant digits, Uncertainties, Error Calculations
sph3U/3UB 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 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 will 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.)
All labs will use the precision of measuring devices, so there are never any ambiguous cases in labs.
On a test: ask if you are unsure. Otherwise, I have to accept a range of answers to various amounts of sig digs.

8. Scientific Notation
Used to represent very large and very small numbers
Always written as a value between 1 and 10 with a x10 to a power
Sig Digs are maintained (the x10 part is not used for sig digs)
Ex) = x 104
= 1.25 x 10-5

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. Sci Notn and sig digs ws
Hand out

11. Metric System Metric prefix word Metric prefix symbol Power of ten
nano n
10-9
micro
10-6
milli m
10-3
centi c
10-2
kilo k
103
mega M
106
giga G
109

12. Metric system
The standard units in Physics are kilograms (kg), seconds (s) and metres (m).
The Newton (N) is 1 kgm/s2 and
The Joule (J) is 1 kgm2/s2.
To use standard units, students must be able to convert units to standard units for effective communication of data in labs and to compare to given values.

13. Examples
12.0 cm is converted into metres by shifting the decimal place left two spaces.
12.0 cm = m
12.0 g is converted into kg by dividing by 1000 or shifting the decimal left three times.
12.0 g = kg
If a mass is given as 12.0 mg, then the decimal shifts left 3 times to grams and then 3 more to kg. Once we get to really small/large numbers, scientific notation is needed.
12.0 mg = kg = 1.20 x 10-5 kg.

14. Metric practice 12.0 µm 33.45 mm 12.0 µg 12.67 ns 123.4 Gm 7654 Mg
Convert to standard units: kg, m, s:
12.0 µm
33.45 mm
12.0 µg
12.67 ns
123.4 Gm
7654 Mg Ms km cs

15. Metric practice 12.0 µm  1.20 x 10-5 m 33.45 mm  0.03345 m
Convert to standard units: kg, m, s:
12.0 µm  1.20 x 10-5 m
33.45 mm  m
12.0 µg  1.20 x 10-8 kg
12.67 ns  x 10-8 s
123.4 Gm 1.234 x 1011 m
7654 Mg  x 106 kg
Ms  x 107 s
km  m
cs  x 10-4 s

16. Unit analysis Useful to catch simple algebra mistakes!
Carry units all the way through your calculations.
Multipliers used when it’s not just a simple decimal shift, when two units are changed at the same time.
Ex) Convert 232 km/h into standard units.
232 km (1000 m) (1h) =
h km s
64.4 m/s

17. ws
Unit analysis ws

18. 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.

19. 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.

20. 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!

21. 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.

22. 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.

23. Measures with errors ws
Reading rulers, thermometers and triple beam balances…oh my! WS

24. Errors on Labs All lab reports will require a list of 5 errors.
One of these errors must be a list of +/- errors (precision errors) for each device used. (These all count as ONE error).
You need to come up with 4 more errors – systematic or random.
For each error, you will need to explain how they affect the lab and how you might improve this.
Ex: There may be some dirt on the mass pan of the mass scale, which adds extra mass. To fix this, I would clean the mass pan of dust and debris.

25. Rounding Rules
For addition or subtraction, you round the answer to least number of decimal places.
Ex) = 2.9
For multiplication and division, you round the answer to the least number of sig digs.
Ex) 2.40 x 0.50 = 1.2
Ex) 1.20 x 10-3 / 1.0 x 10-5 = 1.2 x 102

26. Math Ws’s
Practice and Hand In sheets

27. 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

28. 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 s/1.0s] (5.0 m/s)
= (5.0 ± 1.0) m/s

29. Error calculation WS

30. Examples to show rounding rules - Optional
<--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.

31. 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