Normalized number

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A real number when written out in the normalized form is as follows:

\pm d_0.d_1d_2d_3\dots\times 10^n

where n is an integer, d_0, d_1, d_2, d_3... 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. This is the form of scientific notation. An alternative style is to have the first non-zero digit after the decimal point.

As examples, the number x=918.082 in normalized form is

9.18082\times10^2,

while the number −0.00574012 in normalized form is

-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

\pm d_0.d_1d_2d_3\dots\times b^n,

where again d_0\not=0, and the "digits" d_0, d_1, d_2, d_3... are integers between 0 and b-1.

Converting a number to base two and normalizing it are the first steps in storing a real number as a binary floating-point number in a computer, though bases of eight and sixteen are also used. Although the point is described as "floating", for a normalised floating point number its position is fixed, the movement being reflected in the different values of the power.

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