= Truncation toward zero
{title2=$\operatorname{trunc}(x)=\begin{cases}\lfloor x\rfloor&x\geq0\\\lceil x\rceil&x<0\end{cases}$}
Truncation toward zero removes the fractional magnitude of a <real number>, giving the <integer> obtained by applying the <floor function> to its <absolute value> and restoring the original sign; a magnitude below one gives zero. More precisely,
$$
\operatorname{trunc}(x)=\begin{cases}\lfloor x\rfloor&x\geq0,\\\lceil x\rceil&x<0.\end{cases}
$$
The result is zero for $-1<x<1$. For example, $\operatorname{trunc}(2.7)=2$ and $\operatorname{trunc}(-2.7)=-2$, whereas $\lfloor-2.7\rfloor=-3$. Python's `math.trunc` uses this towards-zero convention, so replacing it by `math.floor` changes calculations with negative noninteger inputs. https://docs.python.org/3.14/library/math.html#math.trunc[Python 3.14 math.trunc documentation: floor on positive inputs, ceiling on negative inputs]{ref}.
\Image[/truncation-toward-zero-and-signed-errors.png]
{title=<Truncation toward zero> is symmetric in sign, while the <fractional part> defined by the <floor function> is always nonnegative}
{description=Left: exact integer endpoint values distinguish the two sign-dependent truncation rules; a whole neighborhood of zero maps to zero. Right: $x-\operatorname{trunc}(x)$ takes the sign of a noninteger input, while $x-\lfloor x\rfloor$ lies in $[0,1)$ on both sides of zero. Open endpoints show excluded limiting values.}
For every <real number> $x$,
$$
|x-\operatorname{trunc}(x)|<1,\qquad \operatorname{trunc}(-x)=-\operatorname{trunc}(x).
$$
The discarded difference $x-\operatorname{trunc}(x)$ has the sign of $x$ when $x$ is noninteger. In contrast, the <fractional part> $x-\lfloor x\rfloor$ is always in $[0,1)$. At $x=-2.7$, these two differences are $-0.7$ and $0.3$. Thus specifying a signed fractional remainder requires specifying which integer-valued operation is used.
Unlike the <floor function> and <ceiling function>, <truncation toward zero> need not commute with adding an <integer>. At $x=-0.7$ and $j=1$, $\operatorname{trunc}(x+j)=\operatorname{trunc}(0.3)=0$, whereas $\operatorname{trunc}(x)+j=1$. This matters if negative coordinates are assigned to equal-width bins by converting a quotient to an <integer>: $x=-0.7$ belongs to the bin $[-1,0)$ under the <floor function>, but truncation gives zero. The appropriate operation follows from the declared bin <intervals>.
<Truncation toward zero> is a <nondecreasing function> with a <discontinuity of a function> at each nonzero <integer>. It has no <discontinuity of a function> at zero, because it is identically zero on $(-1,1)$. Its agreement with the <floor function> on the positive side and the <ceiling function> on the negative side explains the different closed endpoints in the diagram.
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