Download Scientific Notation

• --
•
•
Scientific Notation
Chemists use scientific notation to write very small and very large -numbers.
Scientific notation allows a very large or very small number to be written as
a number between 1 and 10 multiplied by a power of 10. By expressing
numbers in this way, scientific notation makes calculating easier.
The most common number that is raised to an exponent is 10. For any
positive whole number n, 10n is 1 followed by n zeros; for example, 102 = 100;
103 = 1000; and 106 = 1,000,000. The exponent tells you how many zeros
come after the 1. Any number raised to the zeroth power is 1, so 10° = 1.
When 10 is raised to a negative power, the exponent tells you how many
places after the decimal point to place the 1; for example, 10-1 = 0.1; 10-3 =
0.001; and 10-5 = 0.00001.
Expressing Numbers in Scientific Notation
To express a number such as 137,000,000 in scientific notation, first write
down the significant digits, with a decimal point after the first digit: 1 .37.
Then count how many places you moved the decimal point. To get 1.37 from
137,000,000, the unwritten decimal point was moved 8 places to the left.
Thus, you need to multiply 1.37 by 108: 137,000,000 = 1.37 xlO 8 .
Similarly, to express a small number such as 0.00290 in scientific notation,
move the decimal point 3 places to the right and multiply by 10 -3. The result is
2.90 x 10-3.
,
Multiply and Divide in Scientific Notation
To multiply two numbers written in scientific notation, first multiply the
factors that are not powers of 10. Then multiply the powers of 10 by adding
their exponents. If the answer is not in scientific notation, rewrite it in
scientific notation. Remember to round your answer to the correct number of
significant figures.
Example 1
Multiply 3.7 x 102 by 5.1 x 103
Solution
First, multiply 3.7 by 5.1 to get 18.87. Then multiply 102 by 103 to get 10s.
Then round 18.87 x 105 to two significant figures and rewrite the answer in
scientific notation. This gives 1.9 x 106.
To divide numbers written in scientific notation, first divide the factors that
are not powers of 10. Then divide the powers of 10 by subtracting
exponents. If the answer is not in scientific notation, rewrite it in scientific
notation.
Example 2
Divide 2.3 x 10-3 by 4.6 x 10-6
Solution
Divide 2.3 by 4.6 and 10-3 by 10-6. This gives 0.50 x 10-3-(-6) = 0.50 x 103. In
scientific notation, the answer is 5.0 x 102.
Add and Subtract in Scientific Notation
To add or subtract numbers written in scientific notation, first change the
numbers so that they are both expressed to the same power of 10. Then add
or subtract the factors. The exponent does not change for an addition or
subtraction. If your answer is not in scientific notation, rewrite it in scientific
notation.
Example 3
Add 9.67 x 102 and 8.5 x 103.
Solution
First, convert the numbers so that the powers of 10 are the same. For
example, change 8.5 x 103 to 85 x 102. Then add. This gives 94.67 x 102.
This is not in scientific notation; rewrite it as 9.467 x 103. Rounding to the
tenths place (because 8.5 x 103 is the least exact number in the calculation)
gives 9.5 x 103.
Example 4
Subtract 8.08 x 10-6 from 2.07 x 10-5.
Solution
First, convert the numbers so that the powers of 10 are the same. For
example, change 2.07 x 10-5 to 20.7 x 10-6. Then subtract. This gives 12.62 x
10-6. In scientific notation, this is 1.262 x 10-5. Rounding to the hundredths
place (because 2.07 x 10-5 is the least exact number in the calculation) gives
1.26 x 10-5.