5.6.4 Using Scientific Notation

Using Scientific Notation

Working with very large or very small numbers can be awkward. Since our number system is base ten, we can use powers of ten to rewrite very large or very small numbers to make them easier to work with. Consider the numbers 4,000 and 0.004.

Using place value, we can rewrite the numbers 4,000 and 0.004. We know that 4,000 means $4\times 1,000$ and 0.004 means $4\times \frac{1}{1000}$ .

If we write the 1,000 as a power of ten in exponential form, we can rewrite these numbers in this way:

4,000

$4\times 1,000$

$4\times {10}^3$

0.004

$4\times \frac{1}{1000}$

$4\times \frac{1}{{10}^3}$

$4\times {10}^{-3}$

When a number is written as a product of two numbers, where the first factor is a number greater than or equal to 1 but less than 10, and the second factor is a power of 10 written in exponential form, it is said to be in scientific notation.

A number is expressed in scientific notation when it is in this form :

$a\times {10}^n$, where $1\le │a│<10$ and $n$ is an integer.

It is customary in scientific notation to use × as the multiplication sign, even though we avoid using this sign elsewhere in algebra.

To convert a decimal to scientific notation:

Step 1. Move the decimal point so that the first factor is greater than or equal to 1 but less than 10.

Step 2. Count the number of decimal places, $n$, that the decimal point was moved.

Step 3. Write the number as a product with a power of 10. If the original number is

• greater than 1, the power of 10 will be ${10}^n$ .
• between 0 and 1, the power of 10 will be ${10}^{-n}$ .

Step 4. Check.

Example

Write in scientific notation:

1. 37,000

   Solution

2. 0.0052

   Solution

To convert scientific notation to decimal form:

Step 1. Determine the exponent, $n$, on the factor of 10.

Step 2. Move the decimal $n$ places, adding zeros if needed.

• If the exponent is positive, move the decimal point $n$ places to the right.
• If the exponent is negative, move the decimal point $n$ places to the left.

Step 3. Check.

Example

Write in decimal form:

1. $6.2\times {10}^3$

   Solution

2. $-8.9\times {10}^{-2}$

   Solution