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![[parent]](http://images.planetmath.org:8080/images/uparrow.png)
mantissa function
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(Definition)
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If we subtract from a real number  the greatest integer not exceeding  , we obtain a number  between 0 and 1, which can equal 0 if  is an integer. In other words,
where
 is the floor of  . Such a number  is called the mantissa of  . So we have for example
 ,
 ,
 ,
 ,
i.e. these numbers 2.7, 1.7, 0.7,  ,  at mutual distances an integer have the same mantissa (0.7). This is apparently always true -- thus the mantissa function
is periodic: its least period is 1.
The mantissa is identic with the mantissa used in the Briggsian logarithm calculations.
When  increases from an integer  towards the next integer  , its mantissa
 increases with the same speed from 0 tending to 1, but at  it falls back to 0.
Being a periodic function, the Fourier expansion of the function is easy to form:
This is valid for
 , since the series gives in the jump discontinuity points the arithmetic means (
 ) of left and right limits.
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"mantissa function" is owned by pahio.
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See Also: floor
| Also defines: |
mantissa, mantissa of real number |
This object's parent.
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Cross-references: arithmetic means, points, jump discontinuity, series, function, periodic function, Briggsian logarithm, distances, floor, number, integer, real number
There are 4 references to this entry.
This is version 4 of mantissa function, born on 2008-11-17, modified 2008-11-17.
Object id is 11261, canonical name is MantissaFunction.
Accessed 232 times total.
Classification:
| AMS MSC: | 26A09 (Real functions :: Functions of one variable :: Elementary functions) | | | 11-00 (Number theory :: General reference works ) |
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Pending Errata and Addenda
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![\begin{pspicture}(-5.5,-2.5)(5.5,3.5) \psaxes[Dx=1,Dy=1]{->}(0,0)(-4.5,-1.9)(4.5... ...1) \rput(2.5,2.5){$\mbox{Graph\; } y = x\!-\!\lfloor{x}\rfloor$} \end{pspicture}](http://images.planetmath.org:8080/cache/objects/11261/l2h/img23.png "\begin{pspicture}(-5.5,-2.5)(5.5,3.5) \psaxes[Dx=1,Dy=1]{->}(0,0)(-4.5,-1.9)(4.5... ...1) \rput(2.5,2.5){$\mbox{Graph\; } y = x\!-\!\lfloor{x}\rfloor$} \end{pspicture}")
