Scientific Notation

Adapted from `Scientific Arithmetic' by Davison E. Soper at the University of Oregon.

We use scientific notation to help with the arithmetic of large and small numbers.

1,000,000 = 106
0.000001 = 1/1,000,000 = 10-6
1,230,000 = 1.23 × 106
0.00000123 = 1.23 × 10-6

The same number can take different forms:

1,230,000 = 1.23 × 106 = 12.3 × 105 = 0.123 × 107
(The form 1.23 × 106 is usually preferred, because the constant in front (1.23) is between 1 and 10.)

To multiply, you add the exponents:

(1.2 × 106) × (2 × 105) = (1.2 × 2) × 10(6+5) = 2.4 × 1011

To divide, you subtract the exponents:

(4.2 × 1012) ÷ (2 × 108) = (4.2 ÷ 2) × 10(12-8) = 2.1 × 104

To add or subtract numbers in scientific notation, you have to make the exponents the same first:

(1.2 × 106) + (2 × 105) = (1.2 × 106) + (0.2 × 106) = 1.4 × 106

Test yourself: Select the right answer to each question, and click on it to check.

  1. What is 456,000,000 in scientific notation?
    1. 4.56 × 103
    2. 4.56 × 104
    3. 4.56 × 106
    4. 4.56 × 108
    5. 4.56 × 109

  2. What is 8.88 × 103 in ordinary notation?
    1. 88.8
    2. 888
    3. 8,880
    4. 88,800
    5. 888,000

  3. What is (2 × 106) × (3 × 108)?
    1. 6 × 106
    2. 6 × 108
    3. 6 × 1010
    4. 6 × 1012
    5. 6 × 1014

  4. What is (3 × 106) × (3 × 10-4)?
    1. 9 × 10-2
    2. 9 × 100
    3. 9 × 102
    4. 9 × 104
    5. 9 × 106

  5. What is (6 × 107) ÷ (3 × 105)?
    1. 2 × 102
    2. 2 × 103
    3. 2 × 104
    4. 2 × 105
    5. 2 × 106

  6. What is (3 × 106) ÷ (2 × 10-2)?
    1. 6 × 108
    2. 1.5 × 108
    3. 6 × 104
    4. 1.5 × 104
    5. 5 × 108

  7. What is (2 × 106) + (3 × 104)?
    1. 5 × 106
    2. 2.03 × 1010
    3. 2.3 × 106
    4. 2.03 × 106
    5. 2.003 × 106
Joshua E. Barnes (barnes@ifa.hawaii.edu)

Last modified: August 15, 2001
http://www.ifa.hawaii.edu/~barnes/ast110/scinote.html