Transcript Scientific Notation

Significant Figures
Rules for Determining the Number of Significant
Figures
1. All non-zero digits are significant.
2. Zeros located between 2 non-zero digits are
significant.
3. Leading zeros (those at the start of a number)
are never significant.
4. Trailing zeros (those at the end of a number)
are never significant unless they are preceded
by a decimal point somewhere in the number.
Practice Problems
• How many significant figures are present in each
of the following measurements?
1.
2.
3.
4.
5.
6.
7.
8.
5.13
100.01
0.0401
0.0050
220,000
1.90 x 103
153.000
1.0050
1. 5.13 contains 3 significant figures. All non-zero digits are significant.
2. 100.01 contains 5 significant figures. All non-zero digits are significant. All
zeros located between two non-zero digits are significant.
3. 0.0401 contains 3 significant figures. Only the '401' digits are significant.
Leading zeros (those at the start of a number) are not significant.
4. 0.0050 contains 2 significant figures. Only the '50' digits are significant.
Leading zeros are never significant. Trailing zeros (those at the end of a
number) are significant when they are found to the right of the decimal point.
5. 220,000 contains 2 significant figures. Only the "22" digits are significant.
Since there is no decimal point in the number, the trailing zeros are not
significant.
6. 1.90 x 103 contains 3 significant figures. When a number is written in
scientific notation, ignore the "x 10power" and look only at the first number.
In this case 1.90 contains 3 significant figures since trailing zeros to the right
of the decimal point are significant.
7. 153.000 contains 6 significant figures. Trailing zeros that are to the right of a
decimal point are significant.
8. 1.0050 contains 5 significant figures. Zeros found between non-zero digits are
always significant. Trailing zeros that are to the right of a decimal point are
significant, too.
Arithmetic and Significant Figures
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Rules for Multiplication and Division
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In calculations involving measurements, only the
measurements are considered when determining
the correct number of significant figures for the
answer. Ignore exact values such as conversion
factors.
The answer must contain the same number of
significant figures as the measurement with the
fewest significant figures.
Example:
• 11.510 g/7.85 mL = 1.46624204 g/mL = 1.47
g/mL
• The answer is rounded to three significant
figures because 7.85 contains 3 significant
figures while 11.510 contains five significant
figures.
• Rules for Addition and Subtraction
– In calculations involving measurements, only the
measurements are considered when determining the
correct number of significant figures for the answer.
– The answer must contain the same number of
decimal places as there are in the measurement with
the fewest decimal places.
Example:
• 125.1 g + 1.300 g + 0.27 g = 126.670 g = 126.7
g
• The answer is rounded to 1 decimal place
because the three masses that are being
added have 1, 3, and 2 decimal places,
respectively. Report the answer to the fewest
number of decimal places.
Practice Problems
• Perform the following calculations using
the rules that apply to calculations
involving measurements (i.e. apply the
rules for significant figures.
1. 2.501 + 12.40 - 3.996 =
2. 25.3 x 1.0 x 2.75 =
3. (2.503 - 2.303)/2.303 =
4. 15.00/1.50 =
1.
2.
3.
4.
2.501 + 12.40 - 3.996 = 10.905 = 10.91 (Report your
answer to 2 decimal places because 12.40 has the
fewest (2) decimal places)
25.3 x 1.0 x 2.75 = 69.575 = 7.0 x 101 (Report to 2
significant figures because 1.0 has the fewest (2)
significant figures)
(2.503 - 2.303)/2.303 = 0.200/2.303 = 0.086843 =
0.0868 (Determine the number of decimal places to
use in the numerator using the rules for addition and
subtraction. Then count the number of significant
figures in the numerator and denominator and use the
fewest to report your answer.)
15.00/1.50 = 10.0 (Report your answer to 3
significant figures because 1.50 has the fewest (3)
significant figures.)
Scientific Notation
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Science deals with both very large and very small numbers. For example:
Earth’s diameter is about 13,000,000 meters.
The radius of a hydrogen atom is 0.00000000012 meters.
Consequently, scientists and engineers use the so called scientific notation or exponential
notation ("shorthand" way) to write very large or very small numbers involving powers
of ten. Thus
– 1 = 100
10 = 101
100 = 102
1000 = 103
10,000 = 104
100,000 = 105
1,000,000 = 106
– 0.1 = 1/10 = 10-1
0.01 = 1/100 = 10-2
0.001 = 1/1000 = 10-3
0.0001 = 1/10,000 = 10-4
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In general, any number X can be written as the
product of another number N and a power of
ten.
It's important to remember that 1 < N <10. In
other words, N MUST be at least 1 but less than
10.
The general format for a number written in
scientific notation will be:
N x 10power
Examples:
20 = 2 x 10 = 2 x 101
3500 = 3.5 x 1000 = 3.5 x 103
0.0055 = 5.5 x 0.001 = 5.5 x 10-3
Converting a Number into Proper Scientific Notation
• Re-write those digits as a number with 1 digit in
front of the decimal point and the rest of the digits
after the decimal point (i.e. as a number greater
than or equal to 1 but less than 10)
• Look at the new number you have written. Count
the number of places you must move the decimal
point in order to get back to where the decimal
point was originally located.
• If you have to move the decimal point to the right
to get the original number, then write the exponent
as a positive number.
• If you have to move the decimal point to the left to
get the original number, then write the exponent as
a negative number.
Examples:
• Write 22 650 000 in proper scientific notation:
– Write all significant figures as a number > 1 but <10:
22,650,000 = 2.265 x 10?
– To get back to the original number the decimal place
must be moved 7 places to the right so the exponent will
be positive 7.
22,650,000 = 2.265 x 107
• Write 0.0004050 in proper scientific notation:
– Write all significant figures as a number > 1 but < 10:
0.0004050 = 4.050 x 10?
– To get back to the original number, the decimal place
must be moved 4 places to the left so the exponent will
be negative 4.
0.0004050 = 4.050 x 10-4
Practice Problems
• Express the following numbers using
proper scientific notation.
1. 13,000,000
2. 7500.3
3. 209,000
4. 0.00970
5. 0.0000605
6. 0.00300
1.3 x 107
7.5003 x 103
2.09 x 105
9.70 x 10-3 Notice that the trailing zero is kept
because it is a significant figure.
5. 6.05 x 10-5
6. 3.00 x 10-3 Notice that both trailing zeros are
kept because they are significant figures.
1.
2.
3.
4.
Converting from Scientific Notation to Decimal Notation
In order to convert a number written in scientific notation to
one written in standard or decimal notation, follow these
steps.
• Write the number down without the "x 10power" part.
• Use the sign and numerical value of the exponent (power)
to determine the direction and number of places to move
the decimal place.
• Move the decimal point to the right if the exponent is
positive.
• Move the decimal point to the left if the exponent is
negative.
• Remember, numbers with an exponent of 0 are between 1
and 10. Numbers with a positive exponent are greater than
or equal to 10 while those with a negative exponent are
between zero and 1.
Examples:
• Convert 6.53 x 104 into decimal (standard) notation.
• Write the number without the "x 104" and add
some extra zeros after in order to move the decimal
point.
6.53 x 104 becomes 6.5300
• Since the exponent is positive, move the decimal 4
places to the right.
6.53 x 104 becomes 65300
Notice that the decimal place doesn't actually
appear in this case; it is understood to be at the
end of the number. In science, however, placing a
decimal point after the last zero in a number
greater than or equal to 10 indicates that the zeros
are significant figures.
• Convert 2.50 x 10-3 into decimal (standard)
notation.
• Write the number without the "x 10-3" part and put
some extra zeros in front of the number.
2.50 x 10-3 becomes 0002.50
• Since the exponent is negative, move the decimal 3
places to the left.
2.50 x 10-3 becomes 0.00250
Remember that the number should have the same
number of significant figures as the original
number.
Practice Problems
• Convert the following numbers from
scientific notation to standard (decimal)
notation.
1. 3.0900 x 103
2. 6.55 x 10-5
3. 2.455 x 102
4. 1.9 x 10-4
5. 8.008 x 102
6. 2.05 x 10-3
1.
2.
3.
4.
5.
6.
3090.0 Notice that the number written in scientific
notation included a positive exponent. Therefore, the
decimal was moved to the right and a number greater
than 10 was obtained. Also notice that the original
number had 5 significant figures so my answer must
also have 5 significant figures.
0.0000655 Notice that the number written in
scientific notation included a negative exponent.
Therefore, the decimal was moved to the left and a
number between 0 and 1 was obtained.
245.5
0.00019
800.8
0.00205
Use three significant figures
for your answer
Use two significant figures
for your answer