Download Scientific Notation and Significant Figures

Answer in complete sentences.
1. What is the volume in this
graduated cylinder (in mL)?
2. What is the uncertainty?
3. What could you do to find a more
accurate value for the volume?
4. Do you think this graduated
cylinder is accurate? Why or why
not?
5. If I read volumes of 8.11, 8.12,
and 8.10 mL from this picture,
would my answer be accurate,
precise, both, or neither? Explain
your answer.
• An engineer was responsible for
calculating amount of water that
overflowed from a dam. He measured
all of the water runoff going into the
reservoir (1.2 million cubic feet per
year), the rainfall (860 cubic feet per
year), and the capacity of the reservoir
(3.8 million cubic feet). He did some
fancy calculations. He reported to his
boss that the overflow from the dam
would be 350,246.2544330 cubic feet
per year.
• What’s wrong here?
Significant
Figures and
Scientific
Notation
I. Significant Figures
aka: Significant Digits
• A. Nonzero integers count as significant
figures
– Ex. Any number that is NOT zero (1, 2, 3, 4, 5,
6, 7, 8, 9)
– 345
– 597.2
– 145.456
• Zeros
– B. Leading zeros that come before all the
nonzero digits do NOT count as significant
figures
– Ex: 0.0025 has two sig. fig. The zeros are
“leading” and do not count.
– 0.23
– 0.0004
– 0.03564
– C. Captive zeros are between nonzero digits
and DO count as sig. fig.
– Ex: 1.008 has four sig. fig. The zeros are
captive and DO count.
– 10,004
– 1.000006
– 1,000,000,000,000,567
– D. Trailing zeros are to the right end of the
number and DO count as sig. fig. if the
number contains a decimal point.
– Ex.: 100 has only one sig. fig. because the
trailing zeros DO NOT have a decimal point.
– Example: 1.00 has three sig. fig. because
the trailing zeros DO have a decimal point.
– 1.000000
– 3,000,000
– 3.00000
– 30.00
– 300
– 300.
• E. Exact numbers
– Any number found by counting has an
infinite number of significant figures.
– Ex: I have 3 apples. The 3 has an infinite
number of significant figures.
– 50 people
– 100 baseballs
Which are exact numbers?
1. The elevation of Breckenridge,
Colorado is 9600 feet.
2. There are 12 eggs in a dozen.
3. One yard is equal to 0.9144 meters.
4. The attendance at a football game was
52,806 people.
5. The budget deficit of the US
government in 1990 was $269 billion.
6. The beaker held 25.6 mL of water.
Practice – Copy the number and identify the
number of significant figures.
1.
2.
3.
4.
5.
6.
7.
8.
256
647.9
647.0
321.00
4005
0.45
0.00045
nine
9. 200.
10. 200.0
11. 0.009009
12. -500
13. 100.007
14. -500.0
15. -500.
16. 1.30x1032
How many significant figures?
1. A student’s extraction procedure yields
0.0105 g of caffeine.
2. A chemist records a mass of 0.050080 g
in an analysis.
3. In an experiment, a span of time is
determined to be 8.050 x 10-3 s.
4. Rewrite 8.050 x 10-3 so it has three
significant figures.
• The sample of gold contained
1,200,000,000,000,000,000,000,000,000
atoms.
• How do we keep track of ALL those
zeros?
• In chemistry, some numbers are HUGE!
II. Scientific Notation
(aka: Exponential Notation)
• 8,000,000 =
• 0.00012 =
• Integer must be 1≤x<10
• Positive exponent: number > 1
• Negative exponent: number < 1 (but > 0!)
•
•
•
•
•
•
•
4,500,000
3,950,000,000
230
230.
0.00000045
-0.002
0.00781
Copy the number and rewrite
in scientific notation
a.
b.
c.
d.
e.
f.
g.
h.
100,000
-5,000,000
450,000,000,000
1,300
0.01
0.00 005
-0.0 045
0.00 000 000 000 000 023
Remember…
• A negative exponent is a tiny number but
is bigger than 0 (NOT a negative
number!)
• A big exponent is a HUGE number.
• A negative number can have either a
positive exponent or a negative
exponent.
Round to three sig. fig. and express in
exponential notation.
1.
2.
3.
4.
745,000
0.00054000
540,321,324
0.143589
Homework
• Handout: Significant Figures and
Scientific Notation
III. Rules for Sig. Fig. in
Mathematical Operations
• A. Multiplication and Division
– The number of sig. fig. in the results should
be the same as the number of sig. fig. in the
least precise measurement used in the
calculation.
– Example: 4.56 x 1.4 = 6.38  6.4
• B. Addition and Subtraction
– The result should have the same number of
decimal places as the least precise
measurement used in the calculation.
– Example: 12.11 + 18.0 + 1.013 = 31.123 
31.1 (one decimal place)
• 13 x 1.000 = 13.000 =
• 23.45 x 400 = 9380 =
• 5000 / 3.12 = 1602.56410256…
• 14 + 3.567 = 17.567
• 56.2 + 23.988 = 80.188
• 100 – 1.9995 = 98.0005
IV. Rounding
• Calculate first, then round
• Example: round 4.348 to two sig. fig.
– 4.3
– Never round until your final answer!
Assignment
• P. 32 #8