Transcript of Significant Figures

Measurements and Significant Digits
How big?
How small?
How accurate?
Agenda
• MORE PRACTICE ON SIGNIFICANT DIGITS
• HW: complete scientific notation, rounding,
sig. digits worksheets
Using Scientific Measurements
Precision and Accuracy
1. Precision – the closeness of a set of measurements
of the same quantities made in the same way (how
well repeated measurements of a value agree with
one another).
2. Accuracy – is determined by the agreement
between the measured quantity and the correct
value.
ACCURATE = CORRECT
Ex: Throwing Darts
PRECISE = CONSISTENT
Accuracy vs. Precision
Good accuracy
Good precision
Poor accuracy
Good precision
Poor accuracy
Poor precision
Systematic errors:
reduce accuracy
(instrument)
Random errors:
reduce precision
(person)
Precision
Accuracy
 reproducibility
 correctness
 check by
repeating
measurements
 check by using a
different method
 poor precision
results from poor
technique
 poor accuracy
results from
procedural or
equipment flaws.
Percent Error
is calculated by subtracting the experimental value
from the accepted value, then dividing the difference
by the accepted value. Multiply this number by 100.
Accuracy can be compared quantitatively with the
accepted value using percent error.
Measurement
• Exact number
- results from counting items that cannot be
subdivided
- has an infinite number of significant digits.
• Approximate number
- results from measuring
- does not express absolute accuracy
- has a defined number of significant digits that
depends on the accuracy of the measuring device
What time is it?
• Someone might say “1:30” or “1:28” or “1:27:55”
• Each is appropriate for a different situation
• In science we describe a value as having a certain number
of “significant digits”
• The # of significant digits in a value includes all digits that
are certain and one that is uncertain
• “1:30” likely has 2, 1:28 has 3, 1:27:55 has 5
Reporting Measurements
• Using significant figures
• Report what is known with
certainty
• Add ONE digit of
uncertainty (estimation)
Counting Significant Figures
• When you report a measured value it is assumed
that all the numbers are certain except for the last
one, where there is an uncertainty of ±1.
• Example of nail: the nail is 6.36cm long. The 6.3 are
certain values and the final 6 is uncertain! There are
3 significant figures in the value 6.36cm (2 certain
and 1 uncertain). All measured values will have one
(and one only) uncertain number (the last one) and
all others will be certain. The reader can see that the
6.3 are certain values because they appear on the
ruler, but the reader has to estimate the final 6.
Significant Figures
• Indicate precision of a measurement.
• Recording Significant Figures (SF)
– Sig figs in a measurement include the known digits
plus a final estimated digit
2.35 cm
Practice Measuring
0
cm
1
2
3
4
5
4.5 cm
0
cm
1
2
3
4
5
4.54 cm
0
cm
1
2
3
4
5
3.0 cm
20
15 mL ?
1.50 x 101 mL
15.0 mL?
10
There are rules that dictate the
number of significant digits in a value.
1. Read the handout up to A.
2. Try A
3. Bored? There are more:
a. 38.4703 mL
b. 0.00052 g
c. 0.05700 s
d. 500 g
a. 6
b. 2
c. 4
d. 1
The rules for counting the number of
significant figures in a value are:
1. All numbers other then zero will always be counted as
significant figures.
2. Captive zeros always count. All zeros between two nonzero numbers are significant.
3. Leading zeros do not count. Zeros before a non-zero
number after a decimal point are not significant.
4. Trailing zeros count only if there is a decimal.
- All zeros after a non-zero number, after a decimal point
are significant.
- Zeros after a non zero number with no decimal point
are not significant.
Answers to question A
1.
2.83
3
2.
36.77
4
3.
14.0
3
4.
0.0033
2
5.
0.02
1
6.
0.2410
4
–2
7.
4 2.350 x 10
8.
1.00009
6
9. infinite
3
10.
0.0056040
5
Rounding
Rounding using the statistical approach:
When a number ends in 5 and only 5 when you need to round:
• If the preceding number is even –leave it, don’t round up
Ex. The number 21.45 rounded off to 3 significant figures
becomes 21.4
• If the preceding number is odd – round up
Ex. The number 21.350 rounded off to 3 significant figures
becomes 21.4
BUT
If any nonzero digits follow the 5, raise the preceding digit by 1.
Ex. The number 21.4501 rounded off to 3 significant figures
becomes 21.5
Scientific notation
• All significant digits must be maintained
• Only one number is written before the decimal point
and express the decimal points as a power of ten.
Decimal notation
Scientific notation
127g
1.27 x 102 g
0.0907m
9.07 x 10 –2 m
0.000506cg
5.06 x 10 –4 cg
2 300 000 000 000m
2.3 x 1012 m
Scientific notation
•
•
If your value is expressed in proper scientific
notation, all of the figures in the pre-exponential
value are significant, with the last digit being the
least significant figure.
“7.143 x 10-3 grams” contains 4 significant figures
If that value is expressed as 0.007143, it still has 4
significant figures. Zeros, in this case, are
placeholders. If you are ever in doubt about the
number of significant figures in a value, write it in
scientific notation.
Give the number of significant
figures in the following values:
a. 6.19 x 101 years
c. 3.80 x 10-19 J
•
•
b. 7 400 000 years
Helpful Hint :Convert to scientific notation if you
are not certain as to the proper number of
significant figures.
When solving multiple step problems DO NOT
ROUND OFF THE ANSWER UNTIL THE VERY END OF
THE PROBLEM.
ANS: a. 3 b. 3
c. 3
Significant Digits
• It is better to represent 100 as 1.00 x 102
• Alternatively you can underline the position of the last
significant digit. E.g. 100.
• This is especially useful when doing a long calculation or for
recording experimental results
• Don’t round your answer until the last step in a calculation.
• Note that a line overtop of a number indicates that it
repeats indefinitely. E.g. 9.6 = 9.6666…
• Similarly, 6.54 = 6.545454…
Fill in the table
Ordinary Notation ( g)
0.0012
0.00102
0.00120
1.200
12.00
1200
1200
Scientific Notation (g)
# of Significant Figures
1.2 x 10 -3
1.02 x 10 -3
1.20 x 10 -3
1.200 x 10 0
2
3
3
4
1.200 x 10 1
1.2 x 10 3
1.20 x 10 3
4
2
3
Fill in the table considering the
number of significant figures.
Previous Number (mL) Number ( mL) Following Number (mL)
110
120
130
120.0
119.9
120.1
120
119
121
Significant Figures in Calculations
1. In addition and subtraction, your answer should have
the same number of decimal places as the measurement
with the least number of decimal places.
Example: 12.734mL - 3.0mL = __________
Solution: 12.734mL has 3 figures past the decimal point.
3.0mL has only 1 figure past the decimal point.
Therefore your final answer should be rounded off to
one figure past the decimal point.
12.734mL
- 3.0mL
9.734 -------- 9.7mL
Adding with Significant Digits
• How far is it from Warsaw to room C40? To B12?
• Adding a value that is much smaller than the last sig.
digit of another value is irrelevant
• E.g. a) 13.64 + 0.075 + 67 b) 267.8 – 9.36
13.64
+ 0.075
+ 67.
81
80.715
267.8
– 9.36
258.44
• Try question B on the handout
B) Answers
i)
83.25
– 0.1075
83.14
ii)
4.02
+ 0.001
4.02
iii)
0.2983
+ 1.52
1.82
Multiplying with Significant Digits
2. In multiplication and division, your answer should have
the same number of significant figures as the least
precise measurement (or the measurement with the
fewest number of SF).
Examples:
a. 61cm x 0.00745cm = 0.45445 = 0.45cm2 = 4.5 x 10-1 cm2
2SF
3SF
2SF
b. 608.3m x 3.45m = 2098.635 = 2.10 x 103 m2
4SF
3SF
3SF
c. 4.8 g  392g = 0.012245 = 0.012 or 1.2 x 10 – 2
2SF
3SF
2SF
• Try question C and D on the handout (recall: for long
questions, don’t round until the end)
C), D) Answers
i) 7.255  81.334 = 0.08920
ii) 1.142 x 0.002 = 0.002
iii) 31.22 x 9.8 = 3.1 x 102 (or 310 or 305.956)
6.12 x 3.734 + 16.1  2.3
22.85208
+
7.0
= 29.9
ii) 0.0030 + 0.02 = 0.02
iii) 1700
+ 134000
iv) 33.4
Note: 146.1  6.487
+ 112.7
135700
= 22.522 = 22.52
+
0.032
=1.36 x105
146.132  6.487 = 22.5268
= 22.53
i)
Calculations & Significant Digits
In multiple step problems if addition or subtraction AND
multiplication or division is used the rules for rounding
are based off of multiplication and division (it “trumps”
the addition and subtraction rules).
There is no uncertainty in a conversion factor; therefore
they do not affect the degree of certainty of your answer.
The answer should have the same number of SF as the
initial value.
a. Convert 25 meters to millimeters.
?mm→ 25 m
1
X
1000 mm = 25 000 mm
1m
2SF
b. Convert 0.12L to mL.
?mL→ 0.12L
1
X
1000 m = 120 mL
1L
2SF
Unit conversions & Significant Digits
• Sometimes it is more convenient to express a value
in different units.
• When units change, basically the number of
significant digits does not.
E.g. 1.23 m = 123 cm = 1230 mm = 0.00123 km
• Notice that these all have 3 significant digits
• This should make sense mathematically since you are
multiplying or dividing by a term that has an infinite
number of significant digits.
conversion factors= infinite # of sig. digits
E.g. 123 cm x 10 mm / cm = 1230 mm
• Try question E on the handout
i)
E)
Answers
1.0 cm = 0.010 m
ii) 0.0390 kg = 39.0 g
iii) 1.7 m = 1700 mm or 1.7 x 103 mm
• A shocking number of patients die every year in United
States hospitals as the result of medication errors, and
many more are harmed. One widely cited estimate
(Institute of Medicine, 2000) places the toll at 44,000 to
98,000 deaths, making death by medication
"misadventure" greater than all highway accidents, breast
cancer, or AIDS. If this estimate is in the ballpark, then
nurses (and patients) beware: Medication errors are the
forth to sixth leading cause of death in America.
Real World Connections :
• Information from the website “Medication Math for
the Nursing Student” at
http://www.alysion.org/dimensional/analysis.htm#
problems