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Normalized 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.[1] Thus, a real number, when written out in normalized scientific notation, is as follows:

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

where n is an integer, &#123;\textstyle d_&#123;0&#125;,d_&#123;1&#125;,d_&#123;2&#125;,d_&#123;3&#125;,\ldots ,&#125; are the digits of the number in base 10, and &#123;\displaystyle d_&#123;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

&#123;\displaystyle 9.18082\times 10^&#123;2&#125;,&#125;

while the number −0.00574012 in normalized form is

&#123;\displaystyle -5.74012\times 10^&#123;-3&#125;.&#125;

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

&#123;\displaystyle \pm d_&#123;0&#125;.d_&#123;1&#125;d_&#123;2&#125;d_&#123;3&#125;\dots \times b^&#123;n&#125;,&#125;

where again &#123;\textstyle d_&#123;0&#125;\neq 0,&#125; and the digits, &#123;\textstyle d_&#123;0&#125;,d_&#123;1&#125;,d_&#123;2&#125;,d_&#123;3&#125;,\ldots ,&#125; are integers between &#123;\displaystyle 0&#125; and &#123;\displaystyle b-1&#125;.

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

  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.
Categories Computer arithmetic

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