3.3. Scientific Notation http://www.ck12.org
An electron’s mass is about 0.000 000 000 000 000 000 000 000 000 000 910 938 22 kg.
Write this number in scientific notation.
Solution: 9 .109 3822· 10 −^31
Example B
The Earth’s circumference is approximately 40,000,000 meters. What is the radius of the earth in scientific notation?
Solution: The relationship between circumference and radius isC= 2 πr.
4. 0 · 107 = 2 πr
r=^42 .π^0 · 107 ≈ 0. 6366 ...· 107 = 6. 366 · 106Note that the number of significant digits required depends on the context.
Example C
Simplify the following expression.
x= ( 4. 56 · 107 )·( 2. 89 · 108 )÷( 7. 15 · 10 −^15 )+ 216
Solution:x= 4. 56 · 2. 89 · 7. 15 · 100 + 216 = 310. 22556
Concept Problem Revisited
In order to represent an extremely large or small number you should count the number of moves necessary for the
decimal point to be directly after the first non-zero digit. This count will be the order of magnitude and will be used
as the exponent of 10 as a means of representing how large or small the number is.
Vocabulary
Order of magnitudeis formally the exponent in scientific notation. Informally it refers to size. Two objects or
numbers are of the same order of magnitude are relatively similar sizes. A marble and a planet are not of the same
order of magnitude, but Earth and Venus are.
Scientific notationis a means of representing a number as a product of a number between 1 and 10 and a power of
10.
Guided Practice
- Order the following numbers from least to greatest.
- 411 · 10 −^37. 837 · 10 −^49. 999 · 103 9. 5983 · 10 −^78. 0984 · 103
- Compute the following number and use scientific notation.
2 , 000 , 0003 · 3 , 0004 - Simplify the following expression.
( 4. 713 · 107 )+( 8. 985 · 105 )−( 4. 987 · 102 )·( 7. 3 · 10 −^6 )÷( 6. 74 · 10 −^9 )
Answers: - First consider the order of magnitude of each number. Small numbers have negative exponents. If two numbers
have the same order of magnitude, then compare the actual digits. - 5983 · 10 −^7 < 7. 837 · 10 −^4 < 5. 411 · 10 −^3 < 8. 0984 · 103 < 9. 999 · 103