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

[math]\displaystyle{ \pm d_0 . d_1 d_2 d_3 \dots \times 10^n }[/math]

where n is an integer, [math]\displaystyle{ d_0, d_1, d_2, d_3, \ldots, }[/math] are the digits of the number in base 10, and [math]\displaystyle{ 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 × 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

As examples, the number 918.082 in normalized form is

[math]\displaystyle{ 9.18082 \times 10^2, }[/math]

while the number −0.00574012 in normalized form is

[math]\displaystyle{ -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]\displaystyle{ \pm d_0 . d_1 d_2 d_3 \dots \times b^n, }[/math]

where again [math]\displaystyle{ d_0 \neq 0, }[/math] and the digits, [math]\displaystyle{ d_0, d_1, d_2, d_3, \ldots, }[/math] are integers between [math]\displaystyle{ 0 }[/math] and [math]\displaystyle{ b - 1 }[/math].

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

• Significand
• Normal number (computing)