Section 1.5—Significant Digits

Section 1.5 A

Counting significant digits

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

= 0.3333333333333333333 g/mL

If the actual measurements were only taken to 1 or 2 decimal places…

how can the answer be known to and infinite number of decimal places?

It can’t!

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

Significant Digit Rules

1 All measured numbers are significant

2 All non-zero numbers are significant

3 Middle zeros are always significant

4 Trailing zeros are significant if there’s a decimal place

5 Leading zeros are never significant

All the fuss about zeros

102.5 gMiddle zeros are important…we know that’s a zero (as opposed to being 112.5)…it was measured to be a zero

125.0 mLThe 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 is between 124.9 and 125.1

125 is between 124 and 126

0.0127 mThe leading zeros will dissapear if the units are changed without affecting the physical meaning or precision…therefore they are not significant

0.0127 m is the same as 127 mm

Sum it up into 2 Rules

1If there is no decimal point in the number, count from the first non-zero number to the last non-zero number

2If there is a decimal point (anywhere in the number), count from the first non-zero number to the very end

The 4 earlier rules can be summed up into 2 general rules

Examples of Summary Rule 1

Example:Count the number of significant figures in

each number

124

20570

200

150

1If there is no decimal point in the number, count from the first non-zero number to the last non-zero number

Examples of Summary Rule 1

Example:Count the number of significant figures in

each number

124

20570

200

150

1If there is no decimal point in the number, count from the first non-zero number to the last non-zero number

3 significant digits

4 significant digits

1 significant digit

2 significant digits

Examples of Summary Rule 2

Example:Count the number of significant figures in

each number

0.00240

240.

370.0

0.02020

2If there is a decimal point (anywhere in the number), count from the first non-zero number to the very end

Examples of Summary Rule 2

Example:Count the number of significant figures in

each number

0.00240

240.

370.0

0.02020

3 significant digits

3 significant digits

4 significant digits

4 significant digits

2If there is a decimal point (anywhere in the number), count from the first non-zero number to the very end

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

Let’s Practice

Example:Count the number of significant figures in

each number

1020 m

0.00205 g

100.0 m

10240 mL

10.320 g

Let’s Practice

Example:Count the number of significant figures in

each number

1020 m

0.00205 g

100.0 m

10240 mL

10.320 g

3 significant digits

3 significant digits

4 significant digits

4 significant digits

5 significant digits

Section 1.5 B

Calculations with significant digits

Performing Calculations with Sig Digs

1Addition & Subtraction: Answer has least number of decimal places as appears in the problem

2Multiplication & Division: Answer has least number of significant figures as appears in the problem

When recording a calculated answer, you can only be as precise as your least precise measurement

Always complete the calculations first, and then round at the end!

Addition & Subtraction Example #1

Example:Compute &

write the answer with the correct number of

sig digs

15.502 g+ 1.25 g

This answer assumes the missing digit in the problem is a zero…but we really don’t have any idea what it is

1Addition & Subtraction: Answer has least number of decimal places as appears in the problem

16.752 g

Addition & Subtraction Example #1

Example:Compute &

write the answer with the correct number of

sig digs

15.502 g+ 1.25 g

1Addition & Subtraction: Answer has least number of decimal places as appears in the problem

16.752 g

16.75 g

3 decimal places

2 decimal placesLowest is “2”

Answer is rounded to 2

decimal places

Addition & Subtraction Example #2

Example:Compute &

write the answer with the correct number of

sig digs

10.25 mL- 2.242 mL

This answer assumes the missing digit in the problem is a zero…but we really don’t have any idea what it is

1Addition & Subtraction: Answer has least number of decimal places as appears in the problem

8.008 mL

Addition & Subtraction Example #2

Example:Compute &

write the answer with the correct number of

sig digs

10.25 mL- 2.242 mL

1Addition & Subtraction: Answer has least number of decimal places as appears in the problem

8.01 mL

2 decimal places

3 decimal placesLowest is “2”

Answer is rounded to 2

decimal places

8.008 mL

Multiplication & Division Example #1

Example:Compute &

write the answer with the correct number of

sig digs

10.25 g2.7 mL

= 3.796296296 g/mL

2Multiplication & Division: Answer has least number of significant figures as appears in the problem

Multiplication & Division Example #1

Example:Compute &

write the answer with the correct number of

sig digs

3.8 g/mL

4 significant digits

2 significant digits

Lowest is “2”

Answer is rounded to 2

sig digs

10.25 g2.7 mL

= 3.796296296 g/mL

2Multiplication & Division: Answer has least number of significant figures as appears in the problem

Multiplication & Division Example #2

Example:Compute &

write the answer with the correct number of

sig digs

1.704 g/mL 2.75 mL

4.686 g

2Multiplication & Division: Answer has least number of significant figures as appears in the problem

Multiplication & Division Example #2

Example:Compute &

write the answer with the correct number of

sig digs

4.69 g

4 significant dig

3 significant digLowest is “3”

Answer is rounded to 3

significant digits

2Multiplication & Division: Answer has least number of significant figures as appears in the problem

1.704 g/mL 2.75 mL

4.686 g

Let’s Practice #1

Example:Compute &

write the answer with the correct number of

sig digs

0.045 g+ 1.2 g

Let’s Practice #1

Example:Compute &

write the answer with the correct number of

sig digs

1.2 g

3 decimal places

1 decimal placeLowest is “1”

Answer is rounded to 1

decimal place

1.245 g

Addition & Subtraction use number of decimal places!

0.045 g+ 1.2 g

Let’s Practice #2

Example:Compute &

write the answer with the correct number of

sig digs

2.5 g/mL 23.5 mL

Let’s Practice #2

Example:Compute &

write the answer with the correct number of

sig digs

59 g

2 significant dig

3 significant digLowest is “2”

Answer is rounded to 2

significant digits

2.5 g/mL 23.5 mL

58.75 g

Multiplication & Division use number of significant digits!

Let’s Practice #3

Example:Compute &

write the answer with the correct number of

sig digs

1.000 g2.34 mL

Let’s Practice #3

Example:Compute &

write the answer with the correct number of

sig digs

0.427 g/mL

4 significant digits

3 significant digits

Lowest is “3”

Answer is rounded to 3

sig digs

1.000 g2.34 mL

= 0.42735 g/mL

Multiplication & Division use number of significant digits!