Scientific Notation

Scientific Notation
Often used to express
very large or
very small numbers.
Also used to maintain
correct number of
significant figures.
Form:
(# from 1 to 9.999) x 10exponent
800 = 8 x 10 x 10
= 8 x 102
2531 = 2.531 x 10 x 10 x 10
= 2.531 x 103
0.0014 = 1.4 / 10 / 10 / 10
= 1.4 x 10-3
Change to standard form.
000000187000000
.
.
1.87 x 10–5 = 0.0000187
3.7 x 108 = 370,000,000
7.88 x 101 = 78.8
2.164 x 10–2 = 0.02164
Change to scientific notation.
12,340 =
0.369 =
0.008 =
1,000,000,000 =
1.234 x 104
3.69 x 10–1
8 x 10–3
1 x 109
Using the Exponent Key
on a Calculator
EE
EXP
EE or EXP means “times 10 to the…”
How
How to
to type
type out
out 6.02
6.02 xx10
102323::
6
0
.
2
EE
2
3
Don’t do it like this…
6
WRONG!
0
.
yx
2
2
3
…or like this…
6
.
0
WRONG!
2
x
1
…or like this:
6
.
0
0
EE
2
3
TOO MUCH WORK.
2
x
1
0
yx
2
3
Also, know when to hit your (–) sign…
…before the number,
…after the number,
…or either one.
1.2 x 105
Example:

2.8 x 1013
Type this calculation in like this:
1
.
2
EE
5
2
.
8
EE
1

3
=
Calculator gives… 4.2857143 –09
or… 4.2857143 E–09
This is NOT written… 4.3–9
But instead is written… 4.3 x 10–9
or
4.3 E –9
7.5 x 10-6  - 8.7 x 10-4 = -6.525 x 10-9
report -6.5 x 10-9 (2 sig. figs.)
4.35 x 106  1.23 x 10-3
= 5.3505 x 103 or 5350.5
report 5.35 x 103 (3 sig. figs.)
5.76 x 10-16  9.86 x 10-4
= 5.84178499 x 10-13
report 5.84 x 10-13 (3 sig. figs.)
8.8 x 10  3.3 x 10
11
11
= 2.904 x 1023
report 2.9 x 1023 (2 sig. figs.)
6.022 x 1023  - 5.1 x 10-8 = -3.07122 x 1016
report -3.1 x 1016 (2 sig. figs.)
Scientific Notation
• Scientific Notation
• Converting Numbers to Scientific Notation
• How to Use a Scientific Calculator
Scientific Notation
We often use very small and very large numbers in chemistry.
Scientific notation is a method to express these numbers in a
manageable fashion.
Thus 0.000 000 1 cm can be written 1 x 10-7 cm.
Lets see why…
Scientific notation expresses a number as the product of two
factors, the first falling between 1 and 10 and
the second being a power of 10.
Method to express really big or small numbers.
Format is
Mantissa
Decimal part of
original number
6.02 x
Base Power
x
Decimal
you moved
23
10
We just move the decimal point around.
602000000000000000000000
Scientific Notation
Numbers are written in the form M x 10n, where the factor
M is a number greater than or equal to 1 but less than 10 and
n is a whole number.
5000 = 5 x 103 or
5 x (10 x 10 x 10)
5 x 1000
5000
5
Numbers > one have a positive exponent.
Numbers < one have a negative exponent.
xEE
10n
3
Converting Numbers to
Scientific Notation
0.00002205
1
2
3
4
2.205 x
-5
10
5
In scientific notation, a number is separated into two parts.
The first part is a number between 1 and 10.
The second part is a power of ten.
How to Use a Scientific Calculator
Divide: (5.44 x 107) .. (8.1 x 104)
07
04
671.604938
5.44
8.100
54400000.
How to enter this on a calculator:
5.44
EE
7
..
8.1
EE
4
ENTER
8.1
EXP
4
=
OR
5.44
EXP
7
..
671.6049383
rounded to 6.7 x 102
Davis, Metcalfe, Williams, Castka, Modern Chemistry, 1999, page 52
Rule for Multiplication
Calculating with Numbers Written in Scientific Notation
When multiplying numbers in scientific notation, multiply the
first factors and add the exponents.
Sample Problem: Multiply 3.2 x 10-7 by 2.1 x 105
(3.2) x (2.1) = 6.72
6.72 x 10-2
(-7) + (5) = -2 or 10-2
Exercise: Multiply 14.6 x 107 by 1.5 x 104
2.19 x 1012
Rule for Division
Calculating with Numbers Written in Scientific Notation
When dividing numbers in scientific notation, divide the first
factor in the numerator by the first factor in the denominator.
Then subtract the exponent in the denominator from the
exponent in the numerator.
Sample Problem: Divide 6.4 x 106 by 1.7 x 102
.
(6.4) . (1.7) = 3.76
3.76 x 104
(6) - (2) = 4 or 104
Exercise: Divide 2.4 x 10-7 by 3.1 x 1014
7.74 x 10-22
Rule for Addition and Subtraction
Calculating with Numbers Written in Scientific Notation
In order to add or subtract numbers written in scientific
notation, you must express them with the same power of 10.
Sample Problem: Add 5.8 x 103 and 2.16 x 104
27.4 x 103
2.74 x 104
Exercise: Add 8.32 x 10-7 and 1.2 x 10-5
1.28 x 10-5
(5.8 x 103) + (21.6 x 103) =
Using Scientific Notation
for Expressing the Correct Number of Significant Figures
Measurement
Number of significant
figures it contains
25 g
2
0.030 kg
2
1.240560 x 106 mg
7
6 x 104 sec
1
246.31 g
5
20.06 cm
4
1.050 m
4
Measurement
Number of significant
figures it contains
0.12 kg
2
1240560. cm
7
6000000 kg
1
6.00 x 106 kg
3
409 cm
3
29.200 cm
5
0.02500 g
4