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SIGNIFICANT FIGURES
UNDERSTANDING MEASUREMENTS AND
SIGNIFICANT FIGURES
The Academic Support Center @ Daytona State College (Science 118, Page 1 of 9)
MEASURED NUMBERS AND
SIGNIFICANT FIGURES
Learning Goal Identify a number as measured or
exact; determine the number of significant figures in a
measured number.
The Academic Support Center @ Daytona State College (Science 118, Page 2 of 9)
MEASURED NUMBERS
A measuring tool
• is used to determine a quantity such as the length
or the mass of an object
• provides numbers for a measurement called
measured numbers
The Academic Support Center @ Daytona State College (Science 118, Page 3 of 9)
REPORTING LENGTH
To report the length of an object
• observe the numerical values of the marked lines at
the end of the object
• estimate the last digit by visually dividing the space
between the smallest marked lines
This estimated number is the final digit that is reported
for a measured number.
The Academic Support Center @ Daytona State College (Science 118, Page 4 of 9)
REPORTING LENGTH – 4.5 CM
• The end of the object is between the 4-cm and 5cm marks.
• Estimate that the end is halfway between the
4-cm and 5-cm marks and report the value as 4.5
cm.
The Academic Support Center @ Daytona State College (Science 118, Page 5 of 9)
REPORTING LENGTH – 4.55 CM
• The metric ruler is marked at every 0.1 cm.
• You can now estimate that the length is halfway
between the 4.5-cm and 4.6-cm marks and report
the value as 4.55 cm.
The Academic Support Center @ Daytona State College (Science 118, Page 6 of 9)
LEARNING CHECK
. 8. . . . . . . . 9. . . . . . . .
10.
. cm
What is the length of the red line?
(A) 9.0 cm
(B) 9.03 cm
(C) 9.04 cm
The Academic Support Center @ Daytona State College (Science 118, Page 7 of 9)
SOLUTION
. 8. . . . . . . . 9. . . . . . . .
10.
. cm
The length of the line could be reported as:
(B) 9.03 cm
(C) 9.04 cm
The estimated digit may be slightly different. Both
readings are acceptable.
The Academic Support Center @ Daytona State College (Science 118, Page 8 of 9)
EXACT NUMBERS
Exact numbers are
• those numbers obtained by
counting items
• definitions that compare two
units in the same measuring
system
The Academic Support Center @ Daytona State College (Science 118, Page 9 of 9)
EXACT NUMBERS
Exact numbers are not measured, do not have a limited
number of significant figures, and do not affect the
number of significant figures in a calculation.
The Academic Support Center @ Daytona State College (Science 118, Page 10 of 9)
SIGNIFICANT FIGURES
Scientist use significant figures to determine how
precise a measurement is.
Significant digits in a measurement include all of the
known digits plus one estimated digit.
The Academic Support Center @ Daytona State College (Science 118, Page 11 of 9)
FOR EXAMPLE…
Look at the ruler below:
• Each line is 0.1cm
• You can read that the arrow is on 13.3 cm
• However, using significant figures, you must
estimate the next digit
• That would give you 13.30 cm
The Academic Support Center @ Daytona State College (Science 118, Page 12 of 9)
LET’S TRY THIS ONE
Look at the ruler below:
•
•
•
•
What can you read before you estimate?
12.8 cm
Now estimate the next digit…
12.85 cm
The Academic Support Center @ Daytona State College (Science 118, Page 13 of 9)
THE SAME RULES APPLY WITH ALL
INSTRUMENTS
• The same rules apply
• Read to the last digit that you know
• Estimate the final digit
The Academic Support Center @ Daytona State College (Science 118, Page 14 of 9)
LET’S TRY GRADUATED CYLINDERS
Look at the graduated cylinder below:
• What can you read with confidence?
• 56 ml
• Now estimate the last digit
• 56.0 ml
The Academic Support Center @ Daytona State College (Science 118, Page 15 of 9)
ONE MORE GRADUATED CYLINDER
Look at the cylinder below…
• What is the measurement?
• 53.5 ml
The Academic Support Center @ Daytona State College (Science 118, Page 16 of 9)
RULES FOR SIGNIFICANT FIGURES
RULE #1
• All non zero digits are ALWAYS significant
• How many significant digits are in the following
numbers?
274
3 Significant Figures
25.632
5 Significant Digits
8.987
4 Significant Figures
The Academic Support Center @ Daytona State College (Science 118, Page 17 of 9)
RULE #2
• All zeros between significant digits are ALWAYS
significant
• How many significant digits are in the following
numbers?
504
3 Significant Figures
60002
5 Significant Digits
9.077
4 Significant Figures
The Academic Support Center @ Daytona State College (Science 118, Page 18 of 9)
RULE #3
• All FINAL zeros to the right of the decimal ARE
significant
• How many significant digits are in the following
numbers?
32.0
3 Significant Figures
19.000
5 Significant Digits
105.0020
7 Significant Figures
The Academic Support Center @ Daytona State College (Science 118, Page 19 of 9)
RULE #4
• All zeros that act as place holders are NOT
significant
• Another way to say this is: zeros are only significant if
they are between significant digits OR are the very
final thing at the end of a decimal
The Academic Support Center @ Daytona State College (Science 118, Page 20 of 9)
FOR EXAMPLE
How many significant digits are in the following numbers?
0.0002
6.02 x 1023
100.000
150000
800
1 Significant Digit
3 Significant Digits
6 Significant Digits
2 Significant Digits
1 Significant Digit
The Academic Support Center @ Daytona State College (Science 118, Page 21 of 9)
RULE #5
• All counting numbers and constants have an infinite
number of significant digits
• For example:
1 hour = 60 minutes
12 inches = 1 foot
24 hours = 1 day
The Academic Support Center @ Daytona State College (Science 118, Page 22 of 9)
HOW MANY SIGNIFICANT DIGITS ARE
IN THE FOLLOWING NUMBERS?
0.0073
100.020
2500
7.90 x 10-3
670.0
0.00001
18.84
2 Significant Digits
6 Significant Digits
2 Significant Digits
3 Significant Digits
4 Significant Digits
1 Significant Digit
4 Significant Digits
The Academic Support Center @ Daytona State College (Science 118, Page 23 of 9)
RULES ROUNDING SIGNIFICANT DIGITS
RULE #1
If the digit to the immediate right of the last
significant digit is less that 5, do not round up
the last significant digit.
For example, let’s say you have the number
43.82 and you want 3 significant digits
The last number that you want is the 8 – 43.82
The number to the right of the 8 is a 2
Therefore, you would not round up & the
number would be 43.8
The Academic Support Center @ Daytona State College (Science 118, Page 24 of 9)
ROUNDING RULE #2
If the digit to the immediate right of the last
significant digit is greater that a 5, you round
up the last significant figure.
• Let’s say you have the number 234.87 and
you want 4 significant digits
• 234.87 – The last number you want is the 8
and the number to the right is a 7 Therefore,
you would round up & get 234.9
The Academic Support Center @ Daytona State College (Science 118, Page 25 of 9)
ROUNDING RULE #3
If the number to the immediate right of the
last significant is a 5, and that 5 is followed by
a non zero digit, round up.
78.657
(you want 3 significant digits)
• the number you want is the 6
• the 6 is followed by a 5 and the 5 is followed
by a non zero number Therefore, you round
up 78.7
The Academic Support Center @ Daytona State College (Science 118, Page 26 of 9)
ROUNDING RULE #4
If the number to the immediate right of the
last significant is a 5, and that 5 is followed by
a zero, you look at the last significant digit
and make it even.
• 2.5350
(want 3 significant digits)
• The number to the right of the digit you want
is a 5 followed by a 0
• Therefore you want the final digit to be even
2.54
The Academic Support Center @ Daytona State College (Science 118, Page 27 of 9)
SAY YOU HAVE THIS NUMBER
• 2.5250
(want 3 significant digits)
• The number to the right of the digit you want
is a 5 followed by a 0
• Therefore you want the final digit to be even
and it already is
• 2.52
The Academic Support Center @ Daytona State College (Science 118, Page 28 of 9)
LET’S TRY THESE EXAMPLES…
200.99
(want 3 SF)
201
18.22
(want 2 SF)
18
135.50
(want 3 SF)
136
0.00299
(want 1 SF)
0.003
98.59
(want 2 SF)
99
The Academic Support Center @ Daytona State College (Science 118, Page 29 of 9)
RULES FOR SIGNIFICANT FIGURES
The Academic Support Center @ Daytona State College (Science 118, Page 30 of 9)
LEARNING CHECK
1.
Which answer contains three significant
figures?
(a) 0.4760
2.
(c) 4.076 × 103
All the zeros are significant in
(a) 0.00307
3.
(b) 0.00476
(b) 25.300
(c) 2.050 × 103
The number of significant figures in 5.80 × 102
(a) one
(b) two
(c) three
The Academic Support Center @ Daytona State College (Science 118, Page 31 of 9)
CALCULATIONS
SIGNIFICANT FIGURES
The Academic Support Center @ Daytona State College (Science 118, Page 32 of 9)
SIGNIFICANT FIGURES IN CALCULATIONS
Learning Goal Adjust
calculated answers to
give the correct
number of significant
figures.
The Academic Support Center @ Daytona State College (Science 118, Page 33 of 9)
ROUNDING OFF
In calculations
• calculated answers are usually rounded off
• rounding rules are used to obtain the correct number
of significant figures
1.
If the first digit to be dropped is 4 or less, then it and
all following digits are simply dropped from the
number.
2.
If the first digit to be dropped is 5 or greater, then the
last retained digit is increased by 1.
The Academic Support Center @ Daytona State College (Science 118, Page 34 of 9)
LEARNING CHECK
Adjust the following calculated answers to give answers
with three significant figures:
A. 824.75 cm
B.
0.112486 g
C. 8.2 L
The Academic Support Center @ Daytona State College (Science 118, Page 35 of 9)
SOLUTION
Adjust the following calculated answers to give answers
with three significant figures:
A. 824.75 cm
B.
0.112486 g
C. 8.2 L
825 cm
0.112 g
8.20 L
The Academic Support Center @ Daytona State College (Science 118, Page 36 of 9)
ADDITION AND SUBTRACTION WITH
MEASURED NUMBERS
In addition or subtraction, the final answer is written so
that it has the same number of decimal places as the
measurement having the fewest decimal places.
For example,
2.367
+ 34.1
36.467
36.5
Thousandths place
Tenths place
Calculator display
Answer, rounded off to tenths place
The Academic Support Center @ Daytona State College (Science 118, Page 37 of 9)
LEARNING CHECK
For each calculation, round the answer to give the
correct number of digits.
1. 235.05 + 19.6 + 2 =
(a) 257
(b) 256.7
2. 58.925 – 18.2 =
(a) 40.725
(b) 40.73
(c) 256.65
(c) 40.7
The Academic Support Center @ Daytona State College (Science 118, Page 38 of 9)
SOLUTION
1.
2.
235.05
+19.6
+ 2
Hundredths place
Tenths place
Ones place
256.65
rounds to 257
58.925
–18.2
40.725
Thousandths place
Tenths place
rounds to 40.7
answer (c)
answer (a)
The Academic Support Center @ Daytona State College (Science 118, Page 39 of 9)
MULTIPLICATION AND DIVISION WITH
MEASURED NUMBERS
In multiplication and division, the final answer is
written to have the same number of significant
figures (SFs) as the measurement with the fewest
SFs.
For example,
24.65 × 0.67
4 SF
2 SF
= 16.5155

17
Calculator
Final answer (2 SF)
The Academic Support Center @ Daytona State College (Science 118, Page 40 of 9)
ADDING SIGNIFICANT ZEROS
The Academic Support Center @ Daytona State College (Science 118, Page 41 of 9)
LEARNING CHECK
The Academic Support Center @ Daytona State College (Science 118, Page 42 of 9)
SOLUTION
The Academic Support Center @ Daytona State College (Science 118, Page 43 of 9)
Questions
Prepared and Compiled from various sources by
A. Kandefer (Learning Specialist)
The Academic Support Center @ Daytona State College
http://www.daytonastate.edu/asc/ascsciencehandouts.html
The Academic Support Center @ Daytona State College (Science 118, Page 44 of 9)