Transcript Measurement and Significant Figures

The Importance of
measurement
Scientific Notation
Scientific notation
36,000 is 3.6 x 104
The exponent indicates how many
times the coefficient 3.6 must be
multiplied by 10 to equal the number
36,000.
Large numbers have positive
exponents
Small numbers (less than one) have
negative exponents.
0.0081 is 8.1 x 10-3, the exponent -3
indicates that the coefficient 8.1
must be divided by 10 three times to
equal 0.0081
Multiplication and Division
To multiply numbers written in
scientific notation, multiply the
coefficients and add the exponents
(3.0 x 104) x (2.0 x 102)= 6.0 x 106
To divide numbers written in scientific
notation, first divide the coefficient,
then subtract the exponent in the
dominator from the exponent in the
numerator
3.0 x 104 / 2.0 x 102 = 1.5 x 102
Sample Problems
Write each measurement in scientific
notation
a. 222 meters
2.22 x 102 meters
b. 728500 meters
7.285 x 105 meters
c. 0.00054 meters
5.4 x 10-4 meters
Multiplication and Division
a. (3.0 x 105) x (2.6 x 103)=
7.8 x 108
b. (2.8 x 103) / (1.4 x 10-2)=
2.0 x 105
Uncertainty in Measurements
Accuracy, Precision and Error
- You must be able to make reliable
measurements in the lab. Ideally,
measurements are both correct and
reproducible.
Accuracy- is a measure of how close a
measurement comes to the actual or
true value of the object measured
Precision- is a measure of how close a
series of measurements are to one
another
Example- Which set of measurements
is more precise?
a. 2g, 3g, 4g
b. 2.1g, 2.2g, 2.3g
b. is correct because the
measurements are closer together
Error in Measurements
Percent error can be used to evaluate the
accuracy of a measurement, it must be
compared to the correct value.
% error =
experimental value - accepted value
accepted value
x100
Significant Figures in a measurement
include all of the digits that are known,
plus a last digit that is estimated.
Measurements must always be recorded to
the correct number of significant digits.
Calculated answers depend upon the
number of significant figures in the values
used in the calculation
Making measurements with sig figs
Making measurements with sig figs
Determining Significant Digits
Nonzero digits: all are considered to be
significant
Example: 3279g has 4 sig figs
Leading zeros: none are significant. They
are considered to be place holders and not
part of the measurement
Example: 0.0045 has 2 sig figs (only the 4
and the 5)
Captive zeros: zeros between two
nonzero digits
They are considered to be
significant.
Example: 5.007 has 4 sig figs
Trailing zeros: zeros at the end of a
measurement
They are counted only if the
number contains a decimal point
Examples:
100 has 1 sig fig
100. has 3 sig figs
100.0 has 4 sig figs
0.0100 has 3 sig figs
Scientific notation: all numbers listed
in the coefficient are considered to
be significant.
Examples:
1.7 x 10-4 has 2 sig figs
1.30 x 10-2 has 3 sig figs
Exact numbers: have unlimited
significant digits
Examples:
4 chairs (determined by counting)
1 inch = 2.54 cm (determined by
definition)
How many sig figs are in each
measurement?
1.
2.
3.
4.
5.
123 meters
3
0.123 meters
3
40,506 meters
5
9.8000 x 103 meters
5
30.0 meters
3
More practice
6.
7.
8.
22 meter sticks
unlimited sig figs (count sig figs
for measured values only)
0.07080 meters
4 sig figs
98,000 meters
2 sig figs
Sig figs in calculations
Rounding: In general, a calculated
answer cannot be more precise than
the least precise measurement from
which it was calculated.
Once you know the number of significant
digits your answer should have, round to
that many digits, counting from the left.
Rounding
Round each measurement to the number of
sig figs shown in parentheses.
1.
2.
3.
314.721 meters (round to 4 sig figs)
315.0 meters
0.001775 meters (round to 2 sig figs)
0.0018 meters
8792 meters (round to 2 sig figs)
8800 meters
Rules for multiplication and
division:
When multiplying or dividing with
measurements, round the answer to
the same number of sig figs as the
measurement with the least
number of sig figs.
Rules for multiplication and
division:
Example: 7.55 meters x 0.34 meters
= 2.567 meters.
Round the answer to 2.6 meters (2 sig figs)
(The position of the decimal point has
nothing to do with the rounding
process when multiplying and
dividing measurements.)
Rules for addition and subtraction:
The answer to an addition or
subtraction calculation should be
rounded to the same number of
decimal places (not digits) as the
measurement with the least
number of decimal places.
Rules for addition and subtraction
Example: 7.055 meters + 0.35 meters
= 7.405 meters.
Round the answer to 7.41 meters (2 places
to the right of the decimal)
Sample Problems
1.
2.
3.
4.
7.55 meters x 0.34 meters =
2.6 m2
2.10 meters x 0.70 meters =
1.5 m2
2.4526 meters / 8.4 seconds=
.29 m/s
0.365 meters / 18.25 seconds =
0.0200 m/s
More problems
5.
6.
7.
12.52 meters + 349.0 meters+ 8.24
meters =
369.8 m
74.626 meters- 28.34 meters=
46.29 m
80.0 meters + 0.0002 meters =
80.0 m