Normalized number

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:

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

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

As examples, the number 918.082 in normalized form is

${\displaystyle 9.18082\times 10^{2},}${\displaystyle 9.18082\times 10^{2},}

while the number −0.00574012 in normalized form is

${\displaystyle -5.74012\times 10^{-3}.}${\displaystyle -5.74012\times 10^{-3}.}

Clearly, any non-zero real number can be normalized.

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

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

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

• Significand
• Normal number (computing)

• Computer arithmetic