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Normalized number

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Source updated February 19, 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. 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.

01Examples

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.

02Other 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

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

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.

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Sources and credits

This article is adapted from the Wikipedia article “Normalized number”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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Normalized number

01Examples

02Other bases

Sources and credits

Related topics

Significand

Normal number (computing)