## 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 \(4000\) and \(0.004.\) We know that \(4000\) means \(4\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}1000\) and \(0.004\) means \(4\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\frac{1}{1000}.\) If we write the \(1000\) as a power of ten in exponential form, we can rewrite these numbers in this way:

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.*

### Definition: Scientific Notation

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

where \(a\ge 1\) and \(a<10\) and \(n\) is an integer.

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

Scientific notation is a useful way of writing very large or very small numbers. It is used often in the sciences to make calculations easier.

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, \(4,\) by itself.

- The power of \(10\) is positive when the number is larger than \(1\text{:}\phantom{\rule{0.2em}{0ex}}4000=4\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{3}.\)
- The power of \(10\) is negative when the number is between \(0\) and \(1\text{:}\phantom{\rule{0.2em}{0ex}}0.004=4\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-3}.\)

## Example

Write \(37,000\) in scientific notation.

### Solution

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. | 3.70000 4 places |

Step 3: Write the number as a product with a power of 10. | \(3.7×{10}^{4}\) |

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 \({10}^{\mathrm{-n}}\)
| |

Step 4: Check. | |

\({10}^{4}\) is 10,000 and 10,000 times 3.7 will be 37,000. | |

\(37,000=3.7×{10}^{4}\) |

### How to 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 \({10}^{-n}.\)

- If the original number is:
- Check.

## Example

Write in scientific notation: \(0.0052.\)

### Solution

0.0052 | |

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

Count the number of decimal places the point was moved. | 3 places |

Write as a product with a power of 10. | \(5.2×{10}^{-3}\) |

Check your answer: \(\begin{array}{c}\hfill 5.2\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-3}\hfill \\ \hfill 5.2\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\frac{1}{{10}^{3}}\phantom{\rule{0.35em}{0ex}}\hfill \\ \\ \\ \hfill \phantom{\rule{0.15em}{0ex}}5.2\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\frac{1}{1000}\hfill \\ \hfill \phantom{\rule{0.25em}{0ex}}5.2\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}0.001\hfill \\ \hfill 0.0052\hfill \end{array}\) | |

\(0.0052=5.2\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-3}\) |

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