Scientific Notation

SCIENTIFIC NOTATION

Definition of Scientific Notation

Scientific Notation is defined as the most convenient way of writing very small and very large numbers using the powers of ten. This allows for easier reading, writing, and calculation of such numbers.

Rules for Proper Scientific Notation

To properly express a number in scientific notation, the number must adhere to the following rule:

  1. It should be represented as a number between 1 and 10.

  2. This number must be multiplied by a power of ten.

Example of Improper Scientific Notation

An example of improper scientific notation would be:
23 imes 10^{5}
Why is this improper? Because 23 is not between 1 and 10.

Conversion Example

As a demonstration, consider the number 137,000,000:

  • This can be rewritten in scientific notation as:
    1.37 imes 10^{8}

Practice Problems

Your Turn

Rewrite the following numbers into scientific notation:

  1. 347,000

  2. 902,000,000

  3. 61,400

  4. 5,903,574,000

  5. 365

Example of Scientific Notation in a Real-World Context

In the United States, there are approximately 15,000,000 households that utilize private wells for their water supply. This number can be expressed in scientific notation as:
1.5 imes 10^{7}

Understanding Negative Exponents

Explanation of Negative Exponents

A negative exponent in scientific notation results in a smaller number. Here’s how this is demonstrated:

  • 10^{4} = 10 imes 10 imes 10 imes 10

  • 10^{3} = 10 imes 10 imes 10

  • 10^{2} = 10 imes 10

  • 10^{1} = 10

  • 10^{0} = 1

  • Following this pattern, it can be seen that:

    • 10^{-1} = rac{1}{10} = 0.1

    • 10^{-2} = rac{1}{10^2} = 0.01

Observed Pattern

From this, a pattern emerges that shows how negative exponents decrease the magnitude of the number.

Additional Practice Problems

Your Turn

Rewrite the following decimals into scientific notation:

  1. 0.000882

  2. 0.00000059

  3. 0.00004

  4. 0.15

  5. 0.06027

Example of Scientific Notation in Biological Measurements

The nucleus of a human cell has a diameter of approximately:
7 imes 10^{-6} meters.

Conversion to Standard Notation

To convert this scientific notation to standard notation, this would be written out as:

  • 0.000007