Normalized number: Difference between revisions
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In [[applied mathematics]], a number is '''normalized''' when it is written in [[scientific notation]] with one nonzero decimal digit before the decimal point.<ref>{{citation|title=A Student's Guide to the Mathematics of Astronomy|first1=Daniel|last1=Fleisch|first2=Julia|last2=Kregenow|publisher=Cambridge University Press|year=2013|isbn=9781107292550|page=35|url=https://books.google.com/books?id=oZFfAAAAQBAJ&pg=PT35}}.</ref> Thus, a [[real number]] when written out in normalized scientific notation is as follows: |
In [[applied mathematics]], a number is '''normalized''' when it is written in [[scientific notation]] with one nonzero decimal digit before the decimal point.<ref>{{citation|title=A Student's Guide to the Mathematics of Astronomy|first1=Daniel|last1=Fleisch|first2=Julia|last2=Kregenow|publisher=Cambridge University Press|year=2013|isbn=9781107292550|page=35|url=https://books.google.com/books?id=oZFfAAAAQBAJ&pg=PT35}}.</ref> Thus, a [[real number]] when written out in normalized scientific notation is as follows: |
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:<math>\pm d_0.d_1d_2d_3\dots\times 10^n</math> |
:<math>\pm d_0.d_1d_2d_3\dots\times 10^n</math> |
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where ''n'' is an [[integer]], <math>d_0,</math> <math>d_1,</math> <math>d_2</math>, <math>d_3</math>... are the [[Numerical digit|digits]] of the number in base 10, and <math>d_0</math> is not zero. That is, its leading digit (i.e. leftmost) is not zero and is followed by the decimal point. This is the form of [[scientific notation]]. An alternative style is to have the first non-zero digit ''after'' the decimal point. |
where ''n'' is an [[integer]], <math>d_0,</math> <math>d_1,</math> <math>d_2</math>, <math>d_3</math>... are the [[Numerical digit|digits]] of the number in base 10, and <math>d_0</math> is not zero. That is, its leading digit (i.e. leftmost) is not zero and is followed by the decimal point. This is the ''standard form'' of [[scientific notation]]. An alternative style is to have the first non-zero digit ''after'' the decimal point. |
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==Examples== |
==Examples== |
Revision as of 08:22, 6 December 2015
In applied mathematics, a number is normalized when it is written in scientific notation with one nonzero decimal digit before the decimal point.[1] Thus, a real number when written out in normalized scientific notation is as follows:
where n is an integer, , ... are the digits of the number in base 10, and is not zero. That is, its leading digit (i.e. leftmost) is not zero and is followed by the decimal point. This is the standard form of scientific notation. An alternative style is to have the first non-zero digit after the decimal point.
Examples
As examples, the number in normalized form is
- ,
while the number −0.00574012 in normalized form is
Clearly, any non-zero real number can be normalized.
Other bases
The same definition holds if the number is represented in another radix (that is, base of enumeration), rather than base 10. In base b a normalized number will have the form
where again and the "digits" , ... are integers between and .
In many computer systems, floating point numbers are represented internally using this normalized form for their binary representations; for details, see Normal number (computing) Converting a number to base two and normalizing it are the first steps in storing a real number as a binary floating-point number in a computer, though bases of eight and sixteen are also used. Although the point is described as "floating", for a normalised floating point number its position is fixed, the movement being reflected in the different values of the power.
References
- ^ Fleisch, Daniel; Kregenow, Julia (2013), A Student's Guide to the Mathematics of Astronomy, Cambridge University Press, p. 35, ISBN 9781107292550.