# 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 × 10b where a is the mantissa or significand and b is the exponent. Example: 1.7 × 103 = 1700.

In many computer languages numbers are written in E notation. E notation is in the form ab where aE+b = a × 10b 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 × 103 = 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 × 105 1.575 E+05 157.5 × 103 4 5
0.000000474 4.74 × 10-7 4.74 E -7 474 × 10-9 3 -7
-4726.4 -4.7264 × 103 -4.7264 E 3 -4.7264 × 103 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 in scientific notation can be added, subtracted, multiplied and divided.

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 × 105 + 3.42 × 104
= 2.75 × 105 + 0.342 × 105.
With the exponents being identical, the significands can now be added.
2.75 × 105 + 0.342 × 105
= 3.092 × 105
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 × 105

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 × 105 · 3.42 × 104
= 2.75 · 3.42 × 105 + 4
= 9.405 × 109
≈ 9.41 × 109

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 × 105 ÷ 3.42 × 104
= 2.75 ÷ 3.42 × 105 - 4
≈ 0.80409356725146198830409356725146 × 101
≈ 0.804 × 101
≈ 8.04 × 100
1. McAdams, David E.. All Math Words Dictionary, scientific notation. 2nd Classroom edition 20150108-4799968. pg 161. Life is a Story Problem LLC. January 8, 2015. Buy the book

• 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/.

McAdams, David E. Scientific Notation. 5/3/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/s/scientificnotation.html.

### Revision History

5/3/2019: Changed equations and expressions to new format. (McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)