## Converting From Decimal Notation to Scientific Notation

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Remember working with place value for whole numbers and decimals? Our number system is based on powers of 10. We use tens, hundreds, thousands, and so on. Our decimal numbers are also based on powers of tens—tenths, hundredths, thousandths, and so on. Consider the numbers 4,000 and \(0.004\). We know that 4,000 means \(4\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}1,000\) and 0.004 means \(4\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\frac{1}{1,000}\).

If we write the 1000 as a power of ten in exponential form, we can rewrite these numbers in this way:

\(\begin{array}{cccc}4,000\hfill & & & \phantom{\rule{4em}{0ex}}0.004\hfill \\ 4\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}1,000\hfill & & & \phantom{\rule{4em}{0ex}}4\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\frac{1}{1,000}\hfill \\ 4\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{3}\hfill & & & \phantom{\rule{4em}{0ex}}4\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\frac{1}{{10}^{3}}\hfill \\ & & & \phantom{\rule{4em}{0ex}}4\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-3}\hfill \end{array}\)

When a number is written as a product of two numbers, where the first factor is a number greater than or equal to one 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.*

### Scientific Notation

A number is expressed in **scientific notation** when it is of the form

\(\begin{array}{cccc}& & & a\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{n}\phantom{\rule{0.2em}{0ex}}\text{where}\phantom{\rule{0.2em}{0ex}}1\le a<10\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}n\phantom{\rule{0.2em}{0ex}}\text{is an integer}\hfill \end{array}\)

It is customary in scientific notation to use as the \(\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\) multiplication sign, even though we avoid using this sign elsewhere in algebra.

If we look at what happened to the decimal point, we can see a method to easily convert from decimal notation to **scientific notation**.

In both cases, the decimal was moved 3 places to get the first factor between 1 and 10.

\(\begin{array}{cccc}\text{The power of 10 is positive when the number is larger than 1:}\hfill & & & 4,000=4\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{3}\hfill \\ \text{The power of 10 is negative when the number is between 0 and 1:}\hfill & & & 0.004=4\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-3}\hfill \end{array}\)

### Example: How to Convert from Decimal Notation to Scientific Notation

Write in scientific notation: 37,000.

### Solution

### Convert from decimal notation to scientific notation

- Move the decimal point so that the first factor is greater than or equal to 1 but less than 10.
- Count the number of decimal places,
*n*, that the decimal point was moved. - 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}.

- greater than 1, the power of 10 will be 10
- Check.

## Example

Write in scientific notation: \(0.0052.\)

### Solution

The original number, \(0.0052\), is between 0 and 1 so we will have a negative power of 10.

Move the decimal point to get 5.2, a number between 1 and 10. | |

Count the number of decimal places the point was moved. | |

Write as a product with a power of 10. | |

Check. | |

\(\begin{array}{}\phantom{\rule{3em}{0ex}}5.2\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-3}\hfill \\ \phantom{\rule{3em}{0ex}}5.2\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\frac{1}{{10}^{3}}\hfill \\ \phantom{\rule{3em}{0ex}}5.2\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\frac{1}{1000}\hfill \\ \phantom{\rule{3em}{0ex}}5.2\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}0.001\hfill \end{array}\) | |

\(\phantom{\rule{2em}{0ex}}0.0052\) |