Normalized number

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A real number is called normalized, if it is in the form:

\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.

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 2 and normalizing it are the first steps in storing a real number as a floating-point number in a computer.

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