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Scientific Notation
C3.1
Scientific Notation
A number is written in scientific notation if it is
written as the product of a coefficient n between
1 and 10 and an integer power of 10 (exponent).
A number written in scientific notation has the
form
n 10
r
where 1 < n < 10 and r is an integer.
Use of scientific notation
Express very small and very large numbers
2.0 x 108 instead of 200,000,000
3.5 x 10-7 instead of 0.00000035
Indicates the precision of a number
What is meant if two cities are said to be
separated by a distance of 3,000 miles?
What do we mean by 3,000 miles?






A distance between 2,999 and 3001 miles?
A distance between 2,990 and 3010 miles?
A distance between 2,900 and 3100 miles?
A distance between 2,000 and 4000 miles?
Scientific notation allows us to express all these
quantities with the precision or uncertainty being
explicit;
3.000 x 103
or 3.00 x 103
or 3.0 x 103
or 3 x 103
Sample Problems
Write the following in scientific notation
to 2 sig. figs.
230230
,000,000
,000
 2
.3? 10
,000
8
Write the following in scientific notation
to 4 sig. figs.
 ? ´10
230,230
000,,000
000,000
= 2.300
8
Sample Problems
Write the following in scientific notation.
5
0
.
00004583

?
0.00004583  4.583  10
Properties for Exponents
review
If a is a real number, and r and s are
integers, then
r s
aa  a a a?
rr
ss
Properties for Exponents
review
If a is any nonzero real number, and r is a
positive integer, then
1
a

?
a  r
a

r
r
Note:
Kg / L = Kg. L-1
Properties for Exponents
review
If a is any nonzero real number, and r and s
are any two integers, then
r r
a a r s

a

?
s s
aa
Multiplication

To multiply two numbers in scientific notation:
– Multiply the coefficients.
– Add the exponents.
– Examples:
»
»
4
4
3.0 x 10 x 2.0 x 10
8
= 6.0 x 10
3
5
4.0 x 10 x 3.0 x 10
2
3
= 12 x 10 = 1.2 x 10
Division

To divide two numbers in scientific notation:
– Divide the coefficients.
– Subtract the exponents.
– Examples:
9.0 x 105 =
3.0 x 103
3.0 x 102
-2-1
4.5 x 104 =
5.0
0.50x x10
10
9.0 x 105
Use a calculator to perform the indicated operation. Write your
result in correct scientific notation.
3
5
( 9.1x10 ) x ( 4.2x10 )
1) Enter 9.1 in your calculator.
2) Press the key marked EXP or EE on your calculator. If
this is written above another key, then you will have to
press SHIFT or 2nd before pressing the EXP or EE key.
3) Enter the value of the exponent.
4) Press the times key.
5) Enter 4.2
6) Repeat steps 2 and 3.
7) Press Enter or = shows 0.3822 = 3.8 x 10-1
Addition

To add two numbers in scientific notation:
(3.5 x 105) + (6.4 x 106)
-Make the exponents the same
-Then add the coefficients



(3.5 x 105) + (6.4 x 106) =
(0.35 x 106) + (6.4 x 106)=
6.75 x 106 =
6.8 x 106
Subtraction




To subtract two numbers in scientific notation:
(6.4 x 106) - (3.5 x 105)
-Make the exponents the same
-Then subtract the coefficients
(6.4 x 106) - (3.5 x 105) =
(64 x 105) - (3.5 x 105) =
6
5
6.1
x
10
60.5 x 10 =