1. Significant Figures – Measurements
AS – Measurements and Sig Fig

2. Significant Figures
When using our calculators we must determine the correct answer; our calculators are mindless drones 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 
AS Level Measurements and Sig Fig

3. Exact Numbers
An exact number is obtained when you count objects or use a defined relationship.
Counting objects are always exact 2 soccer balls 4 pizzas
Exact relationships, predefined values, not measured 1 foot = 12 inches 1 meter = 100 cm
For instance is 1 foot = inches? No
1 ft is EXACTLY 12 inches.
AS Level Measurements and Sig Fig

4. Learning Check A. Exact numbers are obtained by
1. using a measuring tool
2. counting
3. definition
B. Measured numbers are obtained by
AS Level Measurements and Sig Fig

5. Solution
A. Exact numbers are obtained by 2. counting 3. definition B. Measured numbers are obtained by 1. using a measuring tool
AS Level Measurements and Sig Fig

6. Learning Check
Classify each of the following as an exact or a measured number.
1 yard = 3 feet
The diameter of a red blood cell is 6 x 10-4 cm.
There are 6 hats on the shelf.
Gold melts at 1064°C.
AS Level Measurements and Sig Fig

7. Solution
Classify each of the following as an exact (1) or a measured(2) number. This is a defined relationship. A measuring tool is used to determine length. The number of hats is obtained by counting. A measuring tool is required.
AS Level Measurements and Sig Fig

8. How to express a measurement including the uncertainty.
The steps to follow are the same regardless the instrument you are using:
1.Observe what is the value for the smallest division in your instrument represents.
2.Divide this number by 2.
3.This number represents the error you will show in your measurements.
4.Do not forget to include units in these numbers
AS Level Measurements and Sig Fig

9. What is the Length? Many students would measure this as 1.7 cm.
This is not correct.
There is a rule (specially for Cambridge Students)
If the object ends just on the line of the instrument, the last digit MUST BE ZERO.
If the object ends in between two lines, the last digit, MUST BE SHOWN AS FIVE.
We record 1.70 cm as our measurement
The last digit a ZERO which means that the rule measures to the hundredths of cm.
AS Level Measurements and Sig Fig

10. Learning Check What is the length of the wooden stick? 1) 4.5 cm
AS Level Measurements and Sig Fig

11. What is the length of the pencil in cm? 1) 8.0 cm 2) 3.25 cm
AS Level Measurements and Sig Fig

12. 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.
AS Level Measurements and Sig Fig

13. 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.
V=17.5 mL
AS Level Measurements and Sig Fig

14. Measuring with burettes
Which is the correct volume in all burettes?
AS Level Measurements and Sig Fig

15. Measuring with burettes
Which is the correct volume in both burettes?
The smallest division represent 0.1 mL, divided by two, is 0.05, so we need to show two decimals in every measurement.
The burette on the left reads 4.45 mL
The burette on the right reads mL
The burette below reads mL
AS Level Measurements and Sig Fig

16. Significant digits 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.
1) all NON ZERO digits are significant
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.
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.
AS Level Measurements and Sig Fig

17. 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
AS Level Measurements and Sig Fig

18. 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
AS Level Measurements and Sig Fig

19. Coefficient X 10 Exponent
Scientific Notation
Scientific notation, or exponential notation, is a convenient way to write down a very large or a very small number.
In scientific notation each number is written as a product of two numbers:
Coefficient X 10 Exponent
Coefficients are usually expressed with one digit to the left of the decimal point.
An exponent gives the position of the decimal point in the number and is either:
-positive (generally for numbers greater than or equal to 10)
-zero (generally for numbers between 0 and 10)
-negative (generally for numbers less than 0)
AS Level Measurements and Sig Fig

20. Converting a Number to Scientific Notation
Write in scientific (exponential) notation. 
First, write the coefficient:
1.5
Second, count the places between the current decimal place and its position in the coefficient: 2
Third, determine the sign of the exponent.
Moving to the right gives a negative sign (the number is less than 0):
-Finally write the number in scientific notation:
1.5 x 10-2
AS Level Measurements and Sig Fig

21. Converting Scientific Notation to a Decimal System Number
Write 1.23 x 103 as a decimal system number.
First, decide which way the decimal point will move based on whether the exponent is positive (move to right) or negative (move to left) :
+ therefore moves to right (number is greater than 10)
Second, decide how many places the decimal point will move based on the size of the exponent:
3 places
Finally write the number using zeroes to fill in the places between the decimal point in the coefficient and in the new number:
1230
AS Level Measurements and Sig Fig

22. Sig Fig in Exponential Notation
Scientific notation is helpful for indicating how many significant figures are present in a number that has zeros at the end but to the left of a decimal point.
The distance from the Earth to the Sun is 150,000,000 km. Written in standard notation this number could have anywhere from 2 to 9 significant figures.
Scientific notation can indicate how many digits are significant. Writing 150,000,000 as 1.5 x 108 indicates 2 and writing it as x 108 indicates 4.
AS Level Measurements and Sig Fig

23. Rounding to the correct amount of sig fig
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.
AS Level Measurements and Sig Fig

24. Make the following into a 3 Sig Fig number
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
AS Level Measurements and Sig Fig

25. Examples of Rounding 0 is dropped, it is <5
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.
AS Level Measurements and Sig Fig

26. Rounding in calculations
MULTIPLICATIONS OR DIVISIONS
In carrying out a multiplication or division, the answer cannot have more significant figures than either of the original numbers.
AS Level Measurements and Sig Fig

27. Rounding in calculations
ADDITIONS OR SUBTRACTIONS:
In carrying out an addition or subtraction, the answer cannot have more digits after the decimal point than either of the original numbers.
AS Level Measurements and Sig Fig

28. 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
AS Level Measurements and Sig Fig

29. Addition/Subtraction‑
AS Level Measurements and Sig Fig

30. Addition and Subtraction
Look for the last important digit
= .713 = =
10 – =
__ ___ __
.71
82000 .1
AS Level Measurements and Sig Fig