1.4 Expressing Numbers: Scientific Notation

Learning Objective

  1. Express a large number or a small number in scientific notation.

The instructions for making a pot of coffee specified 3 scoops (rather than 12,000 grounds) because any measurement is expressed more efficiently with units that are appropriate in size. In science, however, we often must deal with quantities that are extremely small or incredibly large. For example, you may have 5,000,000,000,000 red blood cells in a liter of blood, and the diameter of an iron atom is 0.000000014 inches. Numbers with many zeros can be cumbersome to work with, so scientists use scientific notation.

Scientific notationA system for expressing very large or very small numbers in a compact manner. is a system for expressing very large or very small numbers in a compact manner. It uses the idea that such numbers can be rewritten as a simple number multiplied by 10 raised to a certain exponent, or powerThe exponent in a number expressed in scientific notation..

Let us look first at large numbers. Suppose a spacecraft is 1,500,000 miles from Mars. The number 1,500,000 can be thought of as follows:

That is, 1,500,000 is the same as 1.5 times 1 million, and 1 million is 10 × 10 × 10 × 10 × 10 × 10, or 106 (which is read as “ten to the sixth power”). Therefore, 1,500,000 can be rewritten as 1.5 times 106, or 1.5 × 106. The distance of the spacecraft from Mars can therefore be expressed as 1.5 × 106 miles.

100 = 1

101 = 10

102 = 100

103 = 1,000

104 = 10,000

and so forth

The convention for expressing numbers in scientific notation is to write a single nonzero first digit, a decimal point, and the rest of the digits, excluding any trailing zeros. This figure is followed by a multiplication sign and then by 10 raised to the power necessary to reproduce the original number. For example, although 1,500,000 can also be written as 15. × 105 (which would be 15. × 100,000), the convention is to have only one digit before the decimal point. How do we know to what power 10 is raised? The power is the number of places you have to move the decimal point to the left to make it follow the first digit, so that the number being multiplied is between 1 and 10:

Example 4

Express each number in scientific notation.

  1. 67,000,000,000
  2. 1,689
  3. 12.6

Solution

  1. Moving the decimal point 10 places to the left gives 6.7 × 1010.
  2. The decimal point is assumed to be at the end of the number, so moving it three places to the left gives 1.689 × 103.
  3. In this case, we need to move the decimal point only one place to the left, which yields 1.26 × 101.

Skill-Building Exercise

    Express each number in scientific notation.

  1. 1,492

  2. 102,000,000

  3. 101,325

To change scientific notation to standard notation, we reverse the process, moving the decimal point to the right. Add zeros to the end of the number being converted, if necessary, to produce a number of the proper magnitude.

Example 5

Express each number in standard, or nonscientific, notation.

  1. 5.27 × 104
  2. 1.0008 × 106

Solution

  1. Rather than moving the decimal to the left, we move it four places to the right and add zeros to give 52,700.
  2. Moving the decimal six places to the right gives 1,000,800.

Skill-Building Exercise

    Express each number in standard, or nonscientific, notation.

  1. 6.98 × 108

  2. 1.005 × 102

We can also use scientific notation to express numbers whose magnitudes are less than 1. For example, the number 0.006 can be expressed as follows:

10−1 = 1/10

10−2 = 1/100

10−3 = 1/1,000

10−4 = 1/10,000

10−5 = 1/100,000

and so forth

We use a negative number as the power to indicate the number of places we have to move the decimal point to the right to follow the first nonzero digit. This is illustrated as follows:

In scientific notation, numbers with a magnitude greater than one have a positive power, while numbers with a magnitude less than one have a negative power.

Example 6

Express each number in scientific notation.

  1. 0.000006567
  2. −0.0004004
  3. 0.000000000000123

Solution

  1. Move the decimal point six places to the right to get 6.567 × 10−6.
  2. Move the decimal point four places to the right to get −4.004 × 10−4. The negative sign on the number itself does not affect how we apply the rules of scientific notation.
  3. Move the decimal point 13 places to the right to get 1.23 × 10−13.

Skill-Building Exercise

    Express each number in scientific notation.

  1. 0.000355

  2. 0.314159

As with numbers with positive powers of 10, when changing from scientific notation to standard notation, we reverse the process.

Example 7

Express each number in standard notation.

  1. 6.22 × 10−2
  2. 9.9 × 10−9

Solution

  1. 0.0622
  2. 0.0000000099

Skill-Building Exercise

    Express each number in standard notation.

  1. 9.98 × 10−5

  2. 5.109 × 10−8

Although calculators can show 8 to 10 digits in their display windows, that is not always enough when working with very large or very small numbers. For this reason, many calculators are designed to handle scientific notation. The method for entering scientific notation differs for each calculator model, so take the time to learn how to do it properly on your calculator, asking your instructor for assistance if necessary. If you do not learn to enter scientific notation into your calculator properly, you will not get the correct final answer when performing a calculation.

Concept Review Exercises

  1. Why it is easier to use scientific notation to express very large or very small numbers?

  2. What is the relationship between how many places a decimal point moves and the power of 10 used in changing a conventional number into scientific notation?

Answers

  1. Scientific notation is more convenient than listing a large number of zeros.

  2. The number of places the decimal point moves equals the power of 10—positive if the decimal point moves to the left and negative if the decimal point moves to the right.

Key Takeaway

  • Large or small numbers are expressed in scientific notation, which use powers of 10.

Exercises

  1. Why is scientific notation useful in expressing numbers?

  2. What is the relationship between the power and the number of places a decimal point is moved when going from standard to scientific notation?

  3. Express each number in scientific notation.

    1. 0.00064
    2. 5,230,000
    3. −56,200
    4. 0.000000000220
    5. 1.0

  4. Express each number in scientific notation.

    1. 678
    2. −1,061
    3. 0.000560
    4. 0.0000003003
    5. 100,000,000

  5. Express each number in standard form.

    1. 6.72 × 104
    2. 2.088 × 10−4
    3. −3 × 106
    4. 9.98 × 10−7

  6. Express each number in standard form.

    1. 9.05 × 105
    2. 1.0 × 10−3
    3. 6.022 × 1023
    4. 8.834 × 10−12

  7. Complete the following table:

    Incorrect Scientific Notation Correct Scientific Notation
    54.7 × 104
    0.0066 × 103
    3,078 × 100

  8. Complete the following table:

    Incorrect Scientific Notation Correct Scientific Notation
    234.0 × 101
    36 × 10−4
    0.993 × 105

Answers

  1. Scientific notation is more convenient than listing a large number of zeros.

    1. 6.4 × 10−4
    2. 5.23 × 106
    3. −5.62 × 104
    4. 2.20 × 10−10
    5. 1.0 × 100

    1. 67,200
    2. 0.0002088
    3. −3,000,000
    4. 0.000000998
  7. Incorrect Scientific Notation Correct Scientific Notation
    54.7 × 104 5.47 × 105
    0.0066 × 103 6.6 × 100
    3,078 × 100 3.078 × 103