1. Notes 1.2: Significant Figures
Physics Honors I
Lecture 1.2

2. Objectives: Differentiate between Accuracy and Precision.
Learn the rules for significant figures and apply them.

3. Further Learning:
Physics: Principles and Problems (Red Book in class):
- Chapter 1, Section 1, Page 7 (Significant Figures)
- Chapter 1, Section 2, Page 11 – (Accuracy, Precision, and Measurement)
Accuracy versus Precision:
Significant Figures:

4. Accuracy vs Precision:

5. Accuracy vs Precision:
Accuracy – describes how well the results of a measurement agree with the “real” value; that is, the accepted value as measured by competent experimenters.
Accuracy – how close a measurement is to the object’s true or correct value.
Precision – the degree of exactness of a measurement.
Precision – how closely repeated measurements of a quantity come to each other and to the average.
Measurement – a comparison between an unknown quantity and a standard.

6. RULES FOR SIGNIFICANT FIGURES:
Zeroes to the right of a decimal point are always counted as significant.
Zeroes to the left of the first nonzero digit are never counted as significant.
Zeros on the end of a number without a decimal point are assumed not to be significant.
For multiplication and division, the number of significant figures in the answer should not be greater than the number of significant figures in the least precise measurement.

7. RULES FOR SIGNIFICANT FIGURES:
For addition and subtraction, the answer should have the same number of decimal places as the quantity with the fewest number of decimal places.
Exact values have an unlimited / infinite number of significant figures.
When a problem has both addition/subtraction and multiplication/division, technically you wait until the very end to do your rounding but that can make things difficult.

8. Rule 1: Zeroes to the right of a decimal point are always counted as significant
4.500 m tells us that the researcher could give us a measurement that is to the ten thousandth of a meter precise e.g. the researcher could give us a measurement that is precise as low as a millimeter.

9. Rule 2: Zeroes to the left of the first nonzero digit are never counted as significant
means that all the zeroes in front of the 6 are not significant. They just act as place holders to so you how precise the measurement is. Therefore, according to this rule, only the 6 is significant.
HELPFUL HINT: Take for example. The zero in front of the decimal IS NOT significant. Personally, I like to make sure I write it because it is very very easy to lose a decimal when you are doing long problems that require you to carry the down many lines. Suddenly, you find you are multiplying or dividing by 857 instead of 0.857, which will cause you to be many orders of magnitude off. However, if you do forget to write the decimal, you will have Well, you know you do not start a number with 0, so you must have forgot the decimal. It is a way to keep yourself from making simple mistakes.

10. Rule 3: Zeros on the end of a number without a decimal point are assumed not to be significant
What this rule means is that a number such as 85,700 only has three significant figures. The two zeroes on the end are not significant.
Moreover, if you have many zeroes before a number, all of those zeroes are not significant, nor is the number at the end. For instance, if I tell you that I have 4 billion, 500 million, and 5 dollars e.g. $4,500,000,005; you will probably be wondering why I included the 5 dollars. There are only two significant figures in it which is the 4.5.

11. Comprehension check I:
Give an example of being accurate.
Give an example of being precise.
How many significant figures does have?
How many significant figures does have?
How many significant figures does $15, have?

12. Rule 4: Multiplication and Division
For multiplication and division, the number of significant figures in the answer should not be greater than the number of significant figures in the least precise measurement.
3.14 ∙
3.14 has three significant figures.
2.751 has four significant figures.
0.64 has two significant figures.
This rule states that for multiplication and division, it is the least number of significant figures. Thus, the 0.64 (with only 2 significant figures) tells us how many significant figures should be in the final answer.

13. Rule 4: Multiplication and Division
When we multiply and divide, we get with the calculator: =
But, we know our least precise measurement has two significant figures, so we round up.
= 13.4xxxxxxxx

14. Rule 4: Multiplication and Division
All the numbers after the four does not matter. You don’t round the four up to five because of the nine and then round up to two significant figures. You look at the Since the rule states that your round up if the number is 5 or greater, we do not round up in the case. So our final answer is:
= 13.
However, if our number was 13.5, we would have rounded up and our final answer would have been 14.

15. Rule 5: Addition and Subtraction
For addition and subtraction, the answer should have the same number of decimal places as the quantity with the fewest number of decimal places.
That is, the number with the less digits to the right of the decimal place determines how many decimal places should be in the final answer.
3.247 has three decimal places (the .247).
41.36 has two decimal places (the .36).
125.2 has one decimal place (the .2)
Therefore, the determines how many decimal places the final answer should have. When you add up the numbers, you get

16. Rule 5: Addition and Subtraction
The rule states that it is the number with the least amount of decimal places which is the with only one decimal places. Therefore, our final answer should just have one decimal place.
= 169.8
NOTE: Multiplication and division is concerned with significant figures, but adding and subtracting is not. Adding and subtracting is only concerned with the least number of decimal place.

17. Comprehension check 2: How many significant figures? 478 0.0021 ==
25.34 – 4.058
(3247)(0.123) 3

18. Rule 6: Exact values have an unlimited / infinite number of significant figures
Conversions, such as 1 ft = 12 in, have an infinite number of significant figures. Exact counts, such as the number of people in a room, have an infinite number of significant figures.
Let’s do an example. Convert 42.3 inches to feet. (Conversion factor: 1 ft = 12 in)
42.3 𝑖𝑛 𝑓𝑡 12 𝑖𝑛

19. Rule 6: Exact values have an unlimited / infinite number of significant figures
42.3 𝑖𝑛 𝑓𝑡 12 𝑖𝑛
So the conversion factor 1 𝑓𝑡 12 𝑖𝑛 essentially means that it is:
𝑓𝑡 𝑖𝑛
with all of the zeroes being significant.

20. Rule 6: Exact values have an unlimited / infinite number of significant figures
Our calculator gives us the number Our final answer should have only three significant figures, thus:
= 3.53 in   

21. Rule 7: wait until the very end to do your rounding
When a problem has both addition/subtraction and multiplication/division, technically you wait until the very end to do your rounding but that can make things difficult. Take the problem below:
1.07−
Well first we look at the numerator because by the laws of the order of operations, we have to subtract before we divide since technically this would be written as ( )/0.762.
1.07 – =
(In our calculator)

22. Rule 7: wait until the very end to do your rounding
By the rules of addition and subtraction, it is the least number of decimal places, so we see that 1.07 has only two decimal places and has four decimal places. Therefore, after we subtract, we should have only two significant figures but we are told to not round until all of the calculation is finish (meaning we still have to divide. So what we do is put an underscore under the number of digits after the subtraction or highlight it, but we should not round yet. That is,
Now divide:

23. Rule 7: wait until the very end to do your rounding
We see that the 8 is only at the hundredth decimal place and no other numbers are in front of the decimal places meaning that in the end, it will be two significant figures. Whereas the has three significant figures.
Thus, our final answer should have two significant figures. Divide and we get:
(In the calculator)

24. Rule 7: wait until the very end to do your rounding
We have determined that my final answer should have 2 significant figures. Thus, we look at the Since the last uncertainly number is 5, we round up. Therefore, our final answer is:
= 0.25

25. Rule 7: wait until the very end to do your rounding
The reason we are supposed to solve combination problems the way as seen above is because when you round, you introduce a specific amount of uncertainty in the calculation.
Let’s look at the problem if we had solved it by rounding after each order of operation.
1.07−

26. Rule 7: wait until the very end to do your rounding
We do the subtraction.
1.07 – =
(In our calculator)

27. Rule 7: wait until the very end to do your rounding
We round to the least number of decimal places per the rules of significant figures for addition and subtraction.
= 0.19

28. Rule 7: wait until the very end to do your rounding
Now we prepare to divide. =
(In our calculator)
The 0.19 has the least amount of significant figures so our final answer should only have two significant figures.
= 0.249

29. Rule 7: wait until the very end to do your rounding
Remember to round and we get our final answer of.
=0.25
Which is the same answer we got in the previous answer. However, if you compare in our first method with , you can see how it may could have caused a rounding error.
Overall, you will often times be okay after rounding when you finish adding or subtraction and when you finish multiplying and dividing. Yet, technically it introduces more uncertainty into the problem and you shouldn’t round until the end.

30. Comprehension check 3:
Convert 254 meters to centimeters. (Conversion factor: 1 m = 100 cm)
How many significant figures? Make sure not to round until the end.
(14.25 𝑐𝑚)( 𝑐𝑚) (2.223 𝑐𝑚−1.04 𝑐𝑚)

31. Quick Recap:

32. Exit Ticket: Solve to the appropriate significant figures:
g g
g 11.7 mL
10.0 g g g
43.4 𝑖𝑛 1 𝑓𝑡 12 𝑖𝑛