Measurements
Measured and Exact Numbers
Measured Numbers
When you use a measuring tool to
determine a quantity such as your
height or weight, the numbers you
obtain are called measured numbers.
This type of observation is quantitative because it involves
numbers. Words such as “tall” or “heavy” are qualitative
observations, and don’t involve measured numbers
Reading a Meterstick
. l2. . . . I . . . . I3 . . . .I . . . . I4. .
cm
First digit (known)
=2
2.?? cm
Second digit (known)
= 0.7
2.7? cm
Third digit (estimated) between 0.05- 0.07
Length reported
=
2.75 cm
or
2.76 cm
or
2.77 cm
(All three of the reported lengths are valid !)
Known + Estimated Digits
 Known digits 2 and 7 are 100% certain
 The third digit 6 is estimated (uncertain)
 In the reported length, all three digits
(2.76 cm) are significant including the
estimated one
Learning Check MN1
. l8. . . . I . . . . I9. . . .I . . . . I10. .
cm
What is the length of the line?
1) 9.2 cm
2) 9.22 cm
3) 9.23 cm
How does your answer compare with your
neighbor’s answer? Who is correct?
Solution MN1
. l8. . . . I . . . . I9. . . .I . . . . I10. .
cm
Estimate to the hundreth’s place (0.01 cm)
2) 9.22 cm
3) 9.23 cm
The estimated digit may be slightly
different for different people. Both
readings are acceptable. (They have the same
amount of certainty and the same degree of estimate)
Learning Check MN2
l5. . . . I . . . . I6. . . .I . . . . I7. .
What is the length of the line?
1) 6.0 cm
2) 6.06 cm
3) 6.60 cm
cm
Solution MN2
l5. . . . I . . . . I6. . . .I . . . . I7. .
cm
Estimate to the hundreth’s place (0.01 cm).
The reading is between 6.0 and 6.1.
Another reasonable answer is 6.07 cm.
2) 6.06 cm
Zero as a Measured Number
. l3. . . . I . . . . I4 . . . . I . . . . I5. .
cm
What is the length of the line?
First digit
4.?? cm
Second digit
4.5? cm
Last (estimated) digit is
4.50 cm
(not noticeably to the left or right of .5)
Exact Numbers
Obtained when you count objects
2 soccer balls
1 watch
4 pizzas
Obtained from a defined relationship
1 foot = 12 inches
1 meters = 100 cm
Not obtained with measuring tools
Learning Check MN3
A. Exact numbers are obtained by
1. measuring
2. counting
3. definition
B. Measured numbers are obtained by
1. measuring
2. counting
3. definition
Solution MN3
A. Exact numbers are obtained by
2. counting
3. definition
B. Measured numbers are obtained by
1. Using a measuring tool
Learning Check MN4
Classify each of the following as an exact (1)
or a measured (2) number.
A.___Gold melts at 1947°F
B.___1 yard = 3 feet
C.___A red blood cell’s diameter is 6 x 10-4 cm
D.___There were 6 hats on the shelf
E.___A can of soda contains 355 mL of soda
Solution MN4
Classify each of the following as an exact (1) or
a measured(2) number. Give reason.
A. 2 Requires a thermometer(measuring tool)
B. 1 From a definition in U.S. system
C. 2_ Need measuring tool to determine
D. 1_ Counted the hats
E. 2 Measured
Learning Check MN5
True(1) or False(2)
1. ___All measuring tools have limitations
2. ___Some measurements can be exact
3. ___Every measurement has some error
4. ___Every measurement has uncertainty
Solution MN5
True(1) or False(2)
1. _1_All measuring tools have limitations
2. _2_Some measurements can be exact
3. _1_Every measurement has some error
4. _1_Every measurement has uncertainty