1. Significant Figures
When using calculators we must determine the correct answer. Calculators are ignorant boxes of switches and don’t know the correct answer.
There are 2 different types of numbers
Exact
Measured
Exact numbers are infinitely important
Measured number = they are measured with a measuring device (name all 4) so these numbers have ERROR.
When you use your calculator your answer can only be as accurate as your worst measurement…Doohoo 
Chapter Two

2. Uncertainty in Measurement
Depending on the apparatus used, the uncertainty in a measurement can vary.
Even digital devices are not infinitely precise!

3. Analog Devices
Some measuring tools will indicate the amount of uncertainty, however usually this is not the case
When using an analog device (one with lines) the uncertainty of the measurement is considered to be +- half of the smallest division.
Ex: on a graduated cylinder with 1 mL divisions:
+- .5 ml

4. Digital Devices
We consider digital devices (i.e. digital scales and thermometers) to be more precise.
Generally, the degree of uncertainty in a digital device is +- the smallest scale division (1 instead of half)
Ex: on a scale the reading is g and the uncertainty is g

5. Other sources of Uncertainty
In chemistry there are other sources besides the inherent uncertainty in a measurement.
In many reactions time measurements are taken to indicate when a particular reaction has completed
A researcher’s reaction time or judgements of temperature, color change, voltage are all sources of uncertainty
Note these even if they are non-quantifiable.

6. 2.4 Measurement and Significant Figures
Every experimental measurement has a degree of uncertainty.
The volume, V, at right is certain in the 10’s place, 10mL<V<20mL
The 1’s digit is also certain, 17mL<V<18mL
A best guess is needed for the tenths place.
Chapter Two

7. What is the Length? We can see the markings between 1.6-1.7cm
We can’t see the markings between the .6-.7
We must guess between .6 & .7
We record 1.67 cm as our measurement
The last digit an 7 was our guess...stop there

8. Learning Check What is the length of the wooden stick? 1) 4.5 cm

9. ?
8.00 cm or 3 (2.2/8)

10. Measured Numbers
Do you see why Measured Numbers have error…you have to make that Guess!
All but one of the significant figures are known with certainty. The last significant figure is only the best possible estimate.
To indicate the precision of a measurement, the value recorded should use all the digits known with certainty.

11. Below are two measurements of the mass of the same object
Below are two measurements of the mass of the same object. The same quantity is being described at two different levels of precision or certainty.
Chapter Two

12. Note the 4 rules
When reading a measured value, all nonzero digits should be counted as significant. There is a set of rules for determining if a zero in a measurement is significant or not.
RULE 1. Zeros in the middle of a number are like any other digit; they are always significant. Thus, g has five significant figures.
RULE 2. Zeros at the beginning of a number are not significant; they act only to locate the decimal point. Thus, cm has three significant figures, and mL has four.
Chapter Two

13. RULE 3. Zeros at the end of a number and after the decimal point are significant. It is assumed that these zeros would not be shown unless they were significant m has six significant figures. If the value were known to only four significant figures, we would write m.
RULE 4. Zeros at the end of a number and before an implied decimal point may or may not be significant. We cannot tell whether they are part of the measurement or whether they act only to locate the unwritten but implied decimal point.
Chapter Two

14. Practice Rule #1 Zeros 6 3 5 2 4
All digits count
Leading 0’s don’t
Trailing 0’s do
0’s count in decimal form
0’s don’t count w/o decimal
0’s between digits count as well as trailing in decimal form
48000.
48000
3.982106

15. 2.6 Rounding Off Numbers
Often when doing arithmetic on a pocket calculator, the answer is displayed with more significant figures than are really justified.
How do you decide how many digits to keep?
Simple rules exist to tell you how.
Chapter Two

16. Once you decide how many digits to retain, the rules for rounding off numbers are straightforward:
RULE 1. If the first digit you remove is 4 or less, drop it and all following digits becomes 2.4 when rounded off to two significant figures because the first dropped digit (a 2) is 4 or less.
RULE 2. If the first digit removed is 5 or greater, round up by adding 1 to the last digit kept is 4.6 when rounded off to 2 significant figures since the first dropped digit (an 8) is 5 or greater.
If a calculation has several steps, it is best to round off at the end.
Chapter Two

17. Practice Rule #2 Rounding
Make the following into a 3 Sig Fig number
Your Final number must be of the same value as the number you started with,
129,000 and not 129
1.5587
1367
128,522
106
1.56
.00374
1370
129,000
1.67 106

18. Examples of Rounding For example you want a 4 Sig Fig number
0 is dropped, it is <5
8 is dropped, it is >5; Note you must include the 0’s
5 is dropped it is = 5; note you need a 4 Sig Fig
780,582
1999.5
4965
780,600
2000.

19. RULE 1. In carrying out a multiplication or division, the answer cannot have more significant figures than either of the original numbers.
Chapter Two

20. RULE 2. In carrying out an addition or subtraction, the answer cannot have more digits after the decimal point than either of the original numbers.
Chapter Two

21. Multiplication and division
32.27  1.54 =
3.68  =
1.750  =
3.2650106  =  107
6.0221023  1.66110-24 =
49.7
46.4
.05985
1.586 107
1.000

22. Addition and Subtraction
Look for the last important digit
.71
82000 .1
= .713 = =
10 – =
__ ___ __

23. Mixed Order of Operation
 =
( )  ( ) =
= = =
239.6
2180.
=  = =

24. Experimental Error
Defined as the difference between the recorded value and the generally accepted or literature value.
Two types:
Random
systematic

25. Random Error
When approximating a reading, there is an equal chance that the reading was too high or too low.
Causes: readability of the device, slight variations in environmental conditions, insufficient data
To reduce random error: perform multiple trials and average the results.

26. Systematic Errors
These errors are a result of poor design or procedure.
Example: if the scale used to measure a measurement is not zeroed, all measurements will be off by the same amount. Also, measuring the top of the meniscus.
Reduce systematic error by having careful and well planned design.

27. Percentage Uncertainty
Sometimes it is helpful to express uncertainty as a percentage. (an uncertainty of 1 second is more significant for a measurement of 10s then it is 100)
Percentage uncertainty = (absolute uncertainty/measured value) X 100

28. Percentage Error Don’t confuse with percent uncertainty
Percent error is used to determine the closeness of an experimental result to the accepted or literature value.
Percentage error = (accepted value-experimental value/accepted value) x 100

29. Propagating Uncertainty
If two measurements with varying uncertainty are to be used to obtain a calculated result, the uncertainties must also be combined.
This is known as error propagation.
When adding and subtracting measurements, the uncertainty is the sum of the absolute uncertainties.
When multiplying or dividing measurements, the uncertainty is the sum of the PERCENT uncertainties.
Rule: If an uncertainty is greater than 2% of the answer use one sig fig and two sig figs if the uncertainty is less than 2%