Normalized number: Difference between revisions
→Other bases: binary representation: the article is not about the representation of binary FP numbers, which is misleading; thus just link on "binary" instead of "binary representation". |
Undid revision 1208850579 by 86.27.248.11 (talk) First, not all rational numbers can be written with a finite number of digits in some basis. Moreover, including an infinite number of digits for normalized numbers may be useful from a theoretical point of view (e.g. for math proofs or specifications). |
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{{Confused|Normal number}} |
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In [[applied mathematics]], a number is '''normalized''' when it is written in [[scientific notation]] with one non-zero 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 non-zero 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|bibcode=2013sgma.book.....F |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_1 d_2 d_3 \dots \times 10^n</math> |
:<math>\pm d_0 . d_1 d_2 d_3 \dots \times 10^n</math> |
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where ''n'' is an [[integer]], <math display="inline">d_0, d_1, d_2, d_3, \ldots,</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. |
where ''n'' is an [[integer]], <math display="inline">d_0, d_1, d_2, d_3, \ldots,</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. Simply speaking, a number is ''normalized'' when it is written in the form of ''a'' × 10<sup>''n''</sup> where 1 ≤ |''a''| < 10 without leading zeros in ''a''. 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== |
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:<math>9.18082 \times 10^2,</math> |
:<math>9.18082 \times 10^2,</math> |
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while the number |
while the number {{val|-0.00574012}} in normalized form is |
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:<math>-5.74012 \times 10^{-3}.</math> |
:<math>-5.74012 \times 10^{-3}.</math> |
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where again <math display="inline">d_0 \neq 0,</math> and the digits, <math display="inline">d_0, d_1, d_2, d_3, \ldots,</math> are integers between <math>0</math> and <math>b - 1</math>. |
where again <math display="inline">d_0 \neq 0,</math> and the digits, <math display="inline">d_0, d_1, d_2, d_3, \ldots,</math> are integers between <math>0</math> and <math>b - 1</math>. |
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In many computer systems, [[Floating-point arithmetic|floating-point]] numbers are represented internally using this normalized form for their |
In many computer systems, [[Binary number|binary]] [[Floating-point arithmetic|floating-point]] numbers are represented internally using this normalized form for their representations; for details, see [[normal number (computing)]]. Although the point is described as ''floating'', for a normalized floating-point number, its position is fixed, the movement being reflected in the different values of the power. |
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==See also== |
==See also== |
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*[[Significand]] |
*[[Significand]] |
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*[[Normal number (computing)]] |
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==References== |
==References== |
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Latest revision as of 09:16, 19 February 2024
In applied mathematics, a number is normalized when it is written in scientific notation with one non-zero 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. Simply speaking, a number is normalized when it is written in the form of a × 10n where 1 ≤ |a| < 10 without leading zeros in a. This is the standard form of scientific notation. An alternative style is to have the first non-zero digit after the decimal point.
Examples[edit]
As examples, the number 918.082 in normalized form is
while the number −0.00574012 in normalized form is
Clearly, any non-zero real number can be normalized.
Other bases[edit]
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, binary floating-point numbers are represented internally using this normalized form for their representations; for details, see normal number (computing). Although the point is described as floating, for a normalized floating-point number, its position is fixed, the movement being reflected in the different values of the power.
See also[edit]
References[edit]
- ^ Fleisch, Daniel; Kregenow, Julia (2013), A Student's Guide to the Mathematics of Astronomy, Cambridge University Press, p. 35, Bibcode:2013sgma.book.....F, ISBN 9781107292550.