1. Significant Figures
Significant figures are important in science as they convey uncertainty in measurements and calculations involving measurement. In science, there are no repeating decimal values. Every value has a specific number of digits which are used to express it due to uncertainty in measurement and measuring instruments.

2. There are 2 different types of numbers Exact Measured
Significant Figures
There are 2 different types of numbers
Exact
Measured

3. Infinite Significance
Exact numbers are infinitely significant! These values are obtained by definition or counting.

4. Counting
For example; the number of students in a class is 21 – there aren’t nor You have a student or you do not! There are 7 pencils in your book bag. This results from counting; no measuring tool was used! Therefore, you either have 7 pencils or you DO NOT.

5. By Definition
Likewise, a centimeter is defined as 0.01 of a meter. 1 cm = 10-2 m No uncertainty! A yard is defined as 36 inches. 1 yard = 36 inch No uncertainty! No measuring tool was employed in these determinations.

6. Significant Figures
Measured numbers - they are determined with a measuring device so these values have uncertainty. Every measuring tool has its own limitation. Each also has a degree of uncertainty built into its manufacture. And, the technician or operator also adds a degree of uncertainty

7. Expressing Measurement
Every measurement has a limit and an error. The volume, V, at right is certain in the 10’s place, 10 mL < V < 20 mL. The 1’s digit is also certain, 17 mL < V < 18 mL. An estimate is needed for the tenths place.

8. Composition of a Significant Figure
All but one of the significant figures are known with certainty. The last significant figure is only the best possible estimate. Recognize, an estimated value still conveys meaning. However, the “implied uncertainty” is always ± 1 of the last significant digit in the value.

9. Precision
Look at the masses below. They represent the same object massed on different balances. Obviously one of the balances is more precise than the other.

10. Conveying Uncertainty
In order to communicate reliability of a measurement or data, the uncertainty in the measurement must be expressed. It provides additional meaning to the piece of data.

11. Significant Figures
Significant figures are a simple means for conveying uncertainty associated with a measurement.
They are frequently employed when only one or two measurements are made instead of a series of measurements from which an average (mean) and an actual uncertainty could be determined.

12. What is a Significant Figure?
A significant figure or significant digit is one known with “reasonable reliability.”

13. What Digits are Significant Figures?
All digits in a measurement are considered significant except for place holding zeros whose function is to provide the value with its magnitude.
Therefore, magnitude of a number has nothing to do with its significance.

14. Rules for Determining Significant Digits
Rules for determining significant digits in a measured value.
1. All nonzero digits (1 – 9) are significant.
2. All zeros between significant digits are significant.
3. Zeros ending a number to the right of the decimal are significant.

15. Rules for Determining Significant Digits cont’d
4. Zeros ending a number to the left of the decimal point are not significant unless so indicated.
Indications may be expression of the decimal point or a repitand bar over the last significant zero.
5. Zeros to the left of significant digits are not significant.

16. Calculators
When using calculators a correct answer must be determined; recognize calculators only do what they are told; and don’t know the correct answer. When you use your calculator your answer can only be as precise as your worst measurement…………..

17. MAKE A NOTE!!
NOTE: All problems given on tests and quizzes in this class will be expressed to the desired significance. Treat them accordingly!!!

18. Operations Using Significant Figures
Addition and subtraction
When adding or subtracting:
1) Identify the least significant value on a units/decimal place basis.
Value with largest uncertainty (±)
2) Perform the operation (addition or subtraction).
3) Round the result to the same units/decimal place as the least significant value.

19. Operations Using Significant Figures
Multiplication and division
When multiplying or dividing:
1) Identify the least significant value based upon the number of significant digits.
2) Perform the operation (multiplying or dividing).
3) Round the result to have the same number of significant digits as the least significant figure.

20. Mixed Operations
When working complex problems that involve mixed operations (+, -, x, and ÷), significant figures must be accounted for every time the “type” of operation changes; meaning (+ or -) to (x or ÷) or (x or ÷) to (+ or -). This happens because our basis for significance changes. Units/decimal place for (+ or -) vs. number of significant digits for (x or ÷).

21. Rules for Rounding
1. If the first digit dropped is less than 5, leave the preceding number unchanged.  (3.133 becomes 3.13)
2. If the first digit dropped is greater than 5, increase the preceding digit by 1.  (3.127 becomes 3.13)
3. If the first digit dropped is exactly 5, round off to make the preceding digit to an even number.  (3.125 becomes 3.12, but becomes 3.14)
Exactly 5, (or 1/2), means there could be any number of zeros after the five, however, no nonzero digits.

22. Significance and Dimensional Analysis
When performing dimensional analysis problems involving only conversion factors obtained from definition, the answer will have the same number of significant figures as the “given” in the problem.

23. Significance and Dimensional Analysis
When performing dimensional analysis problems which involve at least one other conversion factor obtained from “measurement,” the final answer will have the same number of significant digits as the “given” or the other “measured” factor(s) pending upon which has the fewest and limits.

24. Example
What is the total amount of water to pass over a dam in hours if the rate of flow is 21.6 gallons per minute? hrs x 60 min. X 21.6 gal. = 1,940 gal. 1 hr. 1 min. In this problem, the value 21.6 gal = 1 min. had to result from a measurement, therefore, since it has only three sig figs, the answer would have only three sig figs.

25. Significance and Scientific Notation
When working with scientific (exponential) notation, any significant digit must appear in the digit or coefficient term of the scientific notation form. And, any digit in the digit or coefficient term must be a significant digit when in scientific notation form.