# Scientific Notation

Pronunciation: /ˌsaɪ ənˈtɪf ɪk noʊˈteɪ ʃən/ Explain
Scientific notation is a way to write
real numbers
that is very useful for very large and very small numbers. It is also useful in communicating the number of
significant digits
of a quantity and its order of magnitude. A number expressed in scientific
notation is in the form `a × 10`^{b}
were `a` is the mantissa or significand
and `b` is the exponent. *Example:* 1.7×10^{3} = 1700.

In many computer languages
numbers are written in E notation. E notation is in the form `aE±b`
where `aE+b = a × 10`^{b} and `aE-b = a × 10`^{-b}.
*Example:* 2.3E-02 = 2.3×10^{-2} = 0.023.

Another notation similar to scientific notation is engineering notation.
Engineering notation is the same as scientific notation except that the exponent is always a multiple of
three, and the mantissa is between -999.9 and
999.9. *Example:*
23.56×10^{3} = 23560.

More examples of numbers in these notations are:

Decimal format | Scientific notation | E notation | Engineering notation | Implied significant digits | Order of magnitude |

1575000000 | 1.575 × 10^{5} | 1.575 E+05 | 157.5 × 10^{3} | 4 | 5 |

0.000000474 | 4.74 × 10^{-7} | 4.74 E -7 | 474 × 10^{-9} | 3 | -7 |

-4726.4 | -4.7264 × 10^{3} | -4.7264 E 3 | -4.7264 × 10^{3} | 5 | 3 |

Table 1: Examples of scientific, E and engineering notation |

### Operations

Since a number expressed in scientific notation is a real number, all the properties and operations
on real numbers apply. Numbers is scientific notation can be added, subtracted, multiplied and divided.

#### Addition and subtraction

To add two numbers in scientific notation, one must first allow for differences in the order of magnitude.
this can be done by converting the numbers to the same exponent.

`2.75 × 10`^{5} + 3.42 × 10^{4}

`= 2.75 × 10`^{5} + 0.342 × 10^{5}.

With the exponents being identical, the significands can now be added.

`2.75 × 10`^{5} + 0.342 × 10^{5}

`= 3.092 × 10`^{5}

The result shows too many significant digits. There are only three significant digits including
the digit before the decimal point. Round the number to the nearest significant digit.

`≈ 3.09 × 10`^{5}

To subtract two numbers in scientific notation, add the first number (minuend) to the additive
inverse (negative) of the second number (subtrahend).

#### Multiplication and division

To multiply two real numbers in scientific notation, multiply the
significands and round the result to the number of significant digits, then add the exponents.
The final step is to normalize the number, if needed.

`2.75 × 10`^{5} · 3.42 × 10^{4}

`= 2.75 · 3.42 × 10`^{5+4}

`= 9.405 × 10`^{9}

`≈ 9.41 × 10`^{9}

To divide two real numbers in scientific notation, divide the
significands and round the result to the number of significant digits, then subtract the exponents.
The final step is to normalize the number, if needed.

`2.75 × 10`^{5} ÷ 3.42 × 10^{4}

`= 2.75 ÷ 3.42 × 10`^{5-4}

`≈ 0.80409356725146198830409356725146 × 10`^{1}

`≈ 0.804 × 10`^{1}

`≈ 8.04 × 10`^{0}

### More Information

- Hamilton, Dr. Douglas P.; Asbury, Mike; Muhlberger, Curran. Scientific Notation. janus.astro.umd.edu. Astronomy Workshop. 11/25/2009. http://janus.astro.umd.edu/astro/scinote/.

### Cite this article as:

McAdams, David E. Scientific Notation. 7/18/2018. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/s/scientificnotation.html.
### Revision History

7/18/2018: Changed title to common format. (

McAdams, David E.)
6/2/2011: Added examples, engineering notation. (

McAdams, David E.)
11/25/2009: Initial version. (

McAdams, David E.)