Section 3.1: Scientific Notation
This notation is particularly helpful for very extreme numbers (large or
small).
Recall:
102 =
103 =
104 =
Notice the pattern.
LARGE NUMBERS: Too many zeros are hard to keep track of and compare!
5,000,000 is the same as 5 x 1,000,000 = 5x106
(standard)
(scientific)
Scientific Notation consists of a number between 1 and 10 that is
multiplied by a power of 10.
EXAMPLE: State whether the following are in scientific notation:
a) 3 x 104
b) .05 x 106
c) 2.48 x 411
d) 1.357 x 109
e) 15 x 107
f) 5.28 x 210
Converting Scientific Notation to Standard Notation:
2.56 x 106 = 2.56 x 1,000,000
(Since our number system is base 10, each place value represents another power of ten. 106 requires that we make the number "bigger" (more extreme) by moving the decimal place 6 positions to the right.)
Practice: a) complete the above example
b) write in standard notation: 1.3567 x 102
Converting Standard Notation to Scientific Notation:
Example:
Given 748,000 where would we need to place the decimal for
scientific notation?
Now, what power of 10 do we need to multiply this by in order to get
our number back the way it was? We need to make our number bigger
by moving the decimal to the right 5 places so we choose an exponent
of positive 5.
Result:
Example:
a) Write in scientific notation: 40,050
b) Write in scientific notation: 82,360,000
Scientific Notation and SMALL NUMBERS
Numbers that look like 0.0000000000000000267 are also difficult to keep track of. This is a small number, but how does it compare to other very small numbers?
Scientific notation can help. Where would we need to place the decimal? How far to the left will we need to move it to put it back the way it was?
In this case, to indicate that the decimal needs to be moved to the left, a negative power of 10 is used. (for this number, the exponent needs to be ­17
Result:
EXAMPLES:
a) Write in standard notation: 3.09 x 10-4
b) Write in standard notation: 8.27 x 10-9
c) Write in scientific notation: 0.000000076
d) Write in scientific notation: 0.0624
In Scientific Notation:
• a positive exponent means the number is greater than 1.
• a negative exponent means the number is less than 1.
Now, let's revisit exponents and their notation:
103 105 =
Or we could have gotten this result by:
This leads us to the Product Rule of Exponents: am an = am+n
In other words: to multiply two powers with the same base,
• keep the base
• add the exponents
EXAMPLES: multiply and leave the answer in exponential form.
a) 104 103
b) m17 m9
c) 10­3 10­5
d) x3 x8 x4
Let's use our product rule for exponents to adjust numbers that are
not quite in scientific notation:
a) write in scientific notation: 11.567 x 1024
b) write in scientific notation: 16 x 105
Now we are ready to multiply numbers that are in scientific notation:
EXAMPLES: Multiply:
a) (5.8 x 102)(1.6 x 1010)
b) (1.23 x 10-5)(4.56 x 10-3)
c) (1.14 x 1010)(3.4 x 10-13)
Work with a partner to complete the Your Turn 6 Problem on page 208:
Another Example: It will vary from person to person, but on average a human body sheds 0.0033 pounds of skin cells each day. In the United States the average human life span is about 28,600 days. Use scientific notation and multiplication to determine the average number of pounds of skin cells shed by a U.S. citizen in their life time.