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 notation 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 number called the coefficient,  multiplied by 10 raised to a certain power, called the exponent.. 

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

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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 non-significant trailing zeros. This  coefficient  is followed by a multiplication sign and then by 10 raised to the power necessary to reproduce the original number.  For a number with a large absolute value, the exponent 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 coefficient is between 1 and 10:

image

Example 4

Express each number in scientific notation.

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

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 for a number with a large absolute value, reverse the process, moving the decimal point to the right. Place zeros at 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

Skill-Building Exercise

Express each number in standard notation.

  1. 6.98 × 108

  2. 1.005 × 102

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

image

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

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:

image

In scientific notation, numbers with an absolute value greater than one have a positive power, while numbers with an absolute value between zero and one have a negative power.  The coefficient is positive for positive numbers, negative for negative numbers.

Example 6

Express each number in scientific notation.

  1. 0.000006567
  2. −0.0004004
  3. 0.000000000000123

 

Solution

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, moving the decimal point to the left the number of times indicated by the exponent and inserting zeros as necessary.  It is customary to put a zero to the left of the decimal point as well.

Example 7

Express each number in standard notation.

  1. 6.22 × 10−2
  2. 9.9 × 10−9

Solution

Skill-Building Exercise

Express each number in standard notation.

  1. 9.98 × 10−5

  2. 5.109 × 10−8

Many calculators can express numbers in scientific notation and toggle between scientific notation and floating point (regular) notation. The methods for dealing with 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

 

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