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:

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 ${\displaystyle d_{0}}$ 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 × 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

while the number −0.00574012 in normalized form is

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

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