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The fractional part of a real number may be defined in two ways, which differ only for negative numbers.
Definition 1:
The fractional part of a real number is most commonly defined as
![{\displaystyle \{x\}:={\rm {frac}}(x):=x-\lfloor x\rfloor ,\quad x\in \mathbb {R} ,\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/e48632fd4567d097fdf07459cfa0fe2d4de63dab)
where
is the floor function (i.e. the integer nearest to
).
With this definition, we have
![{\displaystyle x=\lfloor x\rfloor +\{x\}={\rm {floor}}(x)+{\rm {frac}}(x),\quad x\in \mathbb {R} .\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/ba0bbacd6a437c08704f614bd0df5182efd219d4)
Definition 2:
The fractional part of a real number is sometimes defined as
![{\displaystyle \{x\}:={\rm {frac}}(x):=\operatorname {sgn}(x)(|x|-\lfloor |x|\rfloor )=x-[x],\quad x\in \mathbb {R} ,\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/c8e82b76a53db4dd8617fcb8a009868fc28e2764)
where
-
is the sign function,
-
is the absolute value,
-
is the floor function (i.e. the integer nearest to
),
-
is the ceiling function (i.e. the integer nearest to
) and
-
is the integer part (i.e. the integer nearest to 0).
With this definition, we have
![{\displaystyle x=[x]+\{x\}={\rm {int}}(x)+{\rm {frac}}(x).\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/febf0944c840363ee6a87c04d3db6d8761db1d75)
Fractional part of a complex number
The fractional part of a complex number
is defined as
![{\displaystyle {\rm {frac}}(z)={\rm {frac}}(x+iy):={\rm {frac}}(x)+i\,{\rm {frac}}(y),\quad z\in \mathbb {C} ,\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/08cab577965fef0e863169c7bd05d79b1b9373b4)
where the fractional part definition may be one of the above two for real numbers.
Definition 1:
With this definition, we have
![{\displaystyle z=\lfloor z\rfloor +\{z\}={\rm {floor}}(z)+{\rm {frac}}(z),\quad z\in \mathbb {C} ,\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/03f2e0a61f06b8ad8d3afb9429264090a7d48328)
with
Definition 2:
With this definition, we have
![{\displaystyle z=[z]+\{z\}={\rm {int}}(z)+{\rm {frac}}(z),\quad z\in \mathbb {C} ,\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/08ce90c68a7561c1e66541eda20d14cc294d821c)
with
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