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}
where 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 in 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}
References
- 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
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. 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.)
12/6/2018: Removed broken links, updated license, implemented new markup. (
McAdams, David E.)
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.)