Normalized number: Difference between revisions

Content deleted Content added

Inline

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:

\pm d_{0}.d_{1}d_{2}d_{3}\dots \times 10^{n}

where n is an integer, ${\textstyle d_{0},d_{1},d_{2},d_{3},\ldots ,}$ are the digits of the number in base 10, and $d_{0}$ 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

9.18082\times 10^{2},

while the number −0.00574012 in normalized form is

-5.74012\times 10^{-3}.

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

\pm d_{0}.d_{1}d_{2}d_{3}\dots \times b^{n},

where again ${\textstyle d_{0}\neq 0,}$ and the digits, ${\textstyle d_{0},d_{1},d_{2},d_{3},\ldots ,}$ are integers between $0$ and $b-1$ .

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.

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.

[1] 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.

[1]

@@ Line 1: / Line 1: @@
+{{Confused|Normal number}}
-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 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:
-:<math>\pm d_0.d_1d_2d_3\dots\times 10^n</math>
+:<math>\pm d_0 . d_1 d_2 d_3 \dots \times 10^n</math>
-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 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.
 ==Examples==
-As examples, the number <math>x=918.082</math> in normalized form is
+As examples, the number 918.082 in normalized form is
-:<math>9.18082\times10^2</math>,
+:<math>9.18082 \times 10^2,</math>
-while the number &minus;0.00574012 in normalized form is
+while the number {{val|-0.00574012}} in normalized form is
-:<math>-5.74012\times 10^{-3}.</math>
+:<math>-5.74012 \times 10^{-3}.</math>
 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
-:<math>\pm d_0.d_1d_2d_3\dots\times b^n,</math>
+:<math>\pm d_0 . d_1 d_2 d_3 \dots \times b^n,</math>
-where again <math>d_0\not=0,</math> and the "digits" <math>d_0,</math> <math>d_1,</math> <math>d_2</math>, <math>d_3</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>.
+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.
+==See also==
-In many computer systems, [[floating point]] numbers are represented internally using this normalized form for their [[binary representation]]s; for details, see [[Normal number (computing)]]
+*[[Significand]]
-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.
+*[[Normal number (computing)]]
 ==References==

Latest revision as of 09:16, 19 February 2024

Examples[edit]

Other bases[edit]

See also[edit]

References[edit]