Scientific Notation
Many of the numbers that are encountered in science are either very large or very
small. For example, the distance from the earth to the sun is 93,000,000 miles, and a
hydrogen atom weighs 0.0000000000000000000000017 gram. Scientific notation
provides a convenient way of writing very large and very small numbers. In scientific
notation the distance from the earth to the sun is 9.3
Ã— 10^{7} miles and a hydrogen atom
weighs 1.7 Ã— 10^{24} gram. In scientific notation the times symbol, , is used to indicate
multiplication. Converting a number from scientific notation to standard notation
is simply a matter of multiplication.
Example 1
Scientific notation to standard notation
Write each number using standard notation.
a) 7.62 Ã— 10^{5}
b) 6.35 Ã— 10^{4}
Solution
a) Multiplying a number by 10^{5} moves the decimal point five places to the right:
b) Multiplying a number by 10^{4} or 0.0001 moves the decimal point four places to
the left:
The procedure for converting a number from scientific notation to standard
notation is summarized as follows.
Strategy for Converting from Scientific Notation
to Standard Notation
1. Determine the number of places to move the decimal point by examining
the exponent on the 10.
2. Move to the right for a positive exponent and to the left for a negative
exponent.
A positive number in scientific notation is written as a product of a number between
1 and 10, and a power of 10. Numbers in scientific notation are written with only
one digit to the left of the decimal point. A number larger than 10 is written with a
positive power of 10, and a positive number smaller than 1 is written with a negative
power of 10. Numbers between 1 and 10 are usually not written in scientific
notation. To convert to scientific notation, we reverse the strategy for converting
from scientific notation.
Strategy for Converting from Standard Notation
to Scientific Notation
1. Count the number of places (n) that the decimal point must be moved so
that it will follow the first nonzero digit of the number.
2. If the original number was larger than 10, use 10^{n}.
3. If the original number was smaller than 1, use 10^{n}.
Example 2
Standard notation to scientific notation
Convert each number to scientific notation.
a) 934,000,000
b) 0.0000025
Solution
a) In 934,000,000 the decimal point must be moved eight places to the left to get it
to follow 9, the first nonzero digit.
934,000,000 = 9.34 Ã— 108 Use 8 because 934,000,000
> 10.
b) The decimal point in 0.0000025 must be moved six places to the right to get the
2 to the left of the decimal point.
0.0000025 = 2.5 Ã— 10^{6} Use 6 because 0.0000025
< 1.
We can perform computations with numbers in scientific notation by using the
rules of exponents on the powers of 10.
Example 3
Using scientific notation in computations
Evaluate by first converting each number to scientific notation.
Solution





Commutative and associative properties 

= 0.5 Ã— 10^{5} 


= 5 Ã— 10^{1} Ã— 10^{5} 
Write 0.5 in scientific notation. 

= 5 Ã—10^{4} 

