1. Section 1.5—Significant Digits

2. Counting significant digits
Section 1.5 A
Counting significant digits

3. Taking & Using Measurements
You learned in Section 1.3 how to take careful measurements
Most of the time, you will need to complete calculations with those measurements to understand your results
1.00 g
3.0 mL
= g/mL
how can the answer be known to and infinite number of decimal places?
If the actual measurements were only taken to 1 or 2 decimal places…
It can’t!

4. Significant Digits
A significant digit is anything that you measured in the lab—it has physical meaning
The real purpose of “significant digits” is to know how many places to record in an answer from a calculation
But before we can do this, we need to learn how to count significant digits in a measurement

5. Significant Digit Rules1
All measured numbers are significant-if it is not measured and is instead counted or
known then we say that they have infinite significant figures. 2
All non-zero numbers are significant 3
Middle zeros are always significant
Trailing zeros are significant if there’s a decimal place 4 5
Leading zeros are never significant

6. All the fuss about zeros
Middle zeros are important…we know that’s a zero (as opposed to being 112.5)…it was measured to be a zero
102.5 g
The convention is that if there are ending zeros with a decimal place, the zeros were measured and it’s indicating how precise the measurement was.
125.0 mL
125.0 is between and 125.1
125 is between 124 and 126
The leading zeros will dissapear if the units are changed without affecting the physical meaning or precision…therefore they are not significant m
m is the same as 12.7 mm

7. Sum it up into 2 Rules
The 4 earlier rules can be summed up into 2 general rules
If there is no decimal point in the number, count from the first non-zero number to the last non-zero number 1
If there is a decimal point (anywhere in the number), count from the first non-zero number to the very end 2

8. Examples of Summary Rule 1
If there is no decimal point in the number, count from the first non-zero number to the last non-zero number 1
124
20570
200
150
Example:
Count the number of significant figures in each number

9. Examples of Summary Rule 1
If there is no decimal point in the number, count from the first non-zero number to the last non-zero number 1
124
20570
200
150
3 significant digits
Example:
Count the number of significant figures in each number
4 significant digits
1 significant digit
2 significant digits

10. Examples of Summary Rule 2
If there is a decimal point (anywhere in the number), count from the first non-zero number to the very end 2
240.
370.0
Example:
Count the number of significant figures in each number

11. Examples of Summary Rule 2
If there is a decimal point (anywhere in the number), count from the first non-zero number to the very end 2
240.
370.0
3 significant digits
Example:
Count the number of significant figures in each number
3 significant digits
4 significant digits
4 significant digits

12. Importance of Trailing Zeros
Just because the zero isn’t “significant” doesn’t mean it’s not important and you don’t have to write it!
“250 m” is not the same thing as “25 m” just because the zero isn’t significant
The zero not being significant just tells us that it’s a broader range…the real value of “250 m” is between 240 m & 260 m.
“250. m” with the zero being significant tells us the range is from 249 m to 251 m

13. Count the number of significant figures in each number
Let’s Practice
1020 m g
100.0 m
10240 mL g
Example:
Count the number of significant figures in each number

14. Count the number of significant figures in each number
Let’s Practice
1020 m g
100.0 m
10240 mL g
3 significant digits
Example:
Count the number of significant figures in each number
3 significant digits
4 significant digits
4 significant digits
5 significant digits

15. Calculations with significant digits
Section 1.5 B
Calculations with significant digits

16. Performing Calculations with Sig Digs
When recording a calculated answer, you can only be as precise as your least precise measurement
Addition & Subtraction: Answer has least number of decimal places as appears in the problem 1
Multiplication & Division: Answer has least number of significant figures as appears in the problem 2
Always complete the calculations first, and then round at the end!

17. Addition & Subtraction Example #1
Addition & Subtraction: Answer has least number of decimal places as appears in the problem 1
Example:
Compute & write the answer with the correct number of sig digs g g g
This answer assumes the missing digit in the problem is a zero…but we really don’t have any idea what it is

18. Addition & Subtraction Example #1
Addition & Subtraction: Answer has least number of decimal places as appears in the problem 1
Example:
Compute & write the answer with the correct number of sig digs g g
3 decimal places
Lowest is “2”
2 decimal places g
Answer is rounded to 2 decimal places
16.75 g

19. Addition & Subtraction Example #2
Addition & Subtraction: Answer has least number of decimal places as appears in the problem 1
Example:
Compute & write the answer with the correct number of sig digs mL mL
8.008 mL
This answer assumes the missing digit in the problem is a zero…but we really don’t have any idea what it is

20. Addition & Subtraction Example #2
Addition & Subtraction: Answer has least number of decimal places as appears in the problem 1
Example:
Compute & write the answer with the correct number of sig digs mL mL
2 decimal places
Lowest is “2”
3 decimal places
8.008 mL
Answer is rounded to 2 decimal places
8.01 mL

21. Multiplication & Division Example #1
Multiplication & Division: Answer has least number of significant figures as appears in the problem 2
Example:
Compute & write the answer with the correct number of sig digs
10.25 g
2.7 mL
= g/mL

22. Multiplication & Division Example #1
Multiplication & Division: Answer has least number of significant figures as appears in the problem 2
Example:
Compute & write the answer with the correct number of sig digs
4 significant digits
Lowest is “2”
10.25 g
2.7 mL
= g/mL
Answer is rounded to 2 sig digs
2 significant digits
3.8 g/mL

23. Multiplication & Division Example #2
Multiplication & Division: Answer has least number of significant figures as appears in the problem 2
Example:
Compute & write the answer with the correct number of sig digs
1.704 g/mL
 2.75 mL
4.686 g

24. Multiplication & Division Example #2
Multiplication & Division: Answer has least number of significant figures as appears in the problem 2
Example:
Compute & write the answer with the correct number of sig digs
1.704 g/mL
 2.75 mL
4 significant dig
Lowest is “3”
3 significant dig
4.686 g
Answer is rounded to 3 significant digits
4.69 g

25. Compute & write the answer with the correct number of sig digs
Let’s Practice #1
Example:
Compute & write the answer with the correct number of sig digs
0.045 g g

26. Compute & write the answer with the correct number of sig digs
Let’s Practice #1
Example:
Compute & write the answer with the correct number of sig digs
0.045 g g
3 decimal places
Lowest is “1”
1 decimal place
1.245 g
Answer is rounded to 1 decimal place
1.2 g
Addition & Subtraction use number of decimal places!

27. Compute & write the answer with the correct number of sig digs
Let’s Practice #2
Example:
Compute & write the answer with the correct number of sig digs
2.5 g/mL
 23.5 mL

28. Compute & write the answer with the correct number of sig digs
Let’s Practice #2
Example:
Compute & write the answer with the correct number of sig digs
2.5 g/mL
 23.5 mL
2 significant dig
Lowest is “2”
3 significant dig
58.75 g
Answer is rounded to 2 significant digits
59 g
Multiplication & Division use number of significant digits!

29. Compute & write the answer with the correct number of sig digs
Let’s Practice #3
Example:
Compute & write the answer with the correct number of sig digs
1.000 g
2.34 mL

30. Compute & write the answer with the correct number of sig digs
Let’s Practice #3
Example:
Compute & write the answer with the correct number of sig digs
4 significant digits
Lowest is “3”
1.000 g
2.34 mL
= g/mL
Answer is rounded to 3 sig digs
3 significant digits
0.427 g/mL
Multiplication & Division use number of significant digits!