Transcript Scientific Notation

Scientific Notation
Scientific Notation
In the sciences, one frequently needs to use numbers that have a lot of digits. For
example:
Distance to the Sun: 150000000000 km
Speed of Light: 300000000 m/s
Mass of a Proton: 0.00000000000000000000000000167 kg
Seconds in One Year: 31536000 s
It would be very cumbersome if every time a very large or a very small number was
expressed, all of its digits had to be written out.
Scientific notation is one method used to compactly express numbers that have
numerous digits. Scientific notation allows the previous examples to be expressed as:
Distance to the Sun: 1.5 × 1011 km
Speed of Light: 3 × 108 m/s
Mass of a Proton: 1.67 × 10-27 kg
Seconds in One Year: 3.1536 × 107 s
The key to understanding scientific notation is to realize that a
number expressed in scientific notation is always in the form
a × 10n
where 1 ≤ |a|< 10
and n is an integer
Standard Notation
Scientific Notation
123
1.23 × 102
20693
2.0693 × 104
0.08273
8.273 × 10-2
5.6
5.6 × 100
-0.792
-7.92 × 10-1
971.56
9.7156 × 102
a × 10n
To convert numbers from standard notation to scientific
notation:
1) Determine the appropriate value of a.
2) Determine the value of n.
Example: Convert 382 to scientific notation:
1) What are the possible values of a?
Possible values: 382., 38.2, 3.82, 0.382, etc
The only potential value of a that fits the criterion of 1 ≤ |a|< 10
is 3.82.
a = 3.82
2) What is the value of n?
n is the exponent on the base 10. Its value affects the location of the
decimal.
To determine n, ask yourself, “How many times does a have to be
multiplied or divided by 10 to reconstruct the original number
expressed in standard from?”.
Whatever that number is, it is the absolute value of n.
In our example, 3.82 needs to be multiplied by 10 two times to get
back to 382.
n=2
Finally, if you had to multiply, the value of n will remain positive; if you
had to divide, the value of n must be made negative.
In our example we had to multiply so n = 2 remains n = 2 and the final
answer is:
382 = 3.82 × 102
Examples
Standard
Form
a
tentative
n
Multiply or
Divide?
n
Scientific
Notation
8700
8.7
3
Multiply
3
8.7 × 103
0.56
5.6
1
Divide
-1
5.6 × 10-1
-0.00492
-4.92
3
Divide
-3
-4.92 × 10-3
579000
5.79
5
Multiply
5
5.79 × 105
105.9
1.059
2
Multiply
2
1.059 × 102