IntegerPart

The integer part of x: x truncated toward 0.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageIntegerPart!↗

IntegerPart(x)x truncated toward 0.

Typed signatures

Type	Package	Applies
(number) -> number	analytic	always

Domain: Elementary functions

Details

• Truncates toward 0, unlike Floor (which rounds toward $-\infty$): $\mathrm{IntegerPart}(-2.4) = -2$, where $\mathrm{Floor}(-2.4) = -3$.
• Exact at an exact argument -- a rational reduces to an exact integer, and so does a symbolic constant like Pi ($\mathrm{IntegerPart}(\pi) = 3$, not a decimal).
• At a concretely complex argument, truncates the real and imaginary parts separately.
• $x = \mathrm{IntegerPart}(x) + \mathrm{FractionalPart}(x)$ always -- see FractionalPart.

Examples

Basic #

live ↗

In$\mathrm{IntegerPart}(2.4)$Out$2$

Truncates toward 0, unlike Floor

live ↗

In$\mathrm{IntegerPart}(-2.4)$Out$-2$

live ↗

In$\mathrm{IntegerPart}(\frac{7}{2})$Out$3$

live ↗

In$\mathrm{IntegerPart}(-\frac{7}{2})$Out$-3$

Scope #

Exact numeric arguments stay exact

live ↗

In$\mathrm{IntegerPart}(π)$Out$3$

Threads element-wise over a list

live ↗

In$\mathrm{IntegerPart}(\bigl\lbrack2.5, -2.5\bigr\rbrack)$Out$\bigl\lbrack2, -2\bigr\rbrack$

Complex numbers: real and imaginary parts truncated separately

live ↗

In$\mathrm{IntegerPart}(2.5+3.7i)$Out$2+3i$

The integer part truncates toward zero: $-\frac54\to-1$

live ↗

In$\mathrm{IntegerPart}(-\frac{5}{4})$Out$-1$

Zero is its own integer part

live ↗

In$\mathrm{IntegerPart}(0)$Out$0$

Truncation on a bounded domain: a branch per integer

live ↗

In$\mathrm{PiecewiseExpand}(\mathrm{IntegerPart}(x), 0\lt x\lt4)$Out$\mathrm{Piecewise}(\bigl\lbrack\bigl\lbrack1, 1≤x∧x\lt2\bigr\rbrack, \bigl\lbrack2, 2≤x∧x\lt3\bigr\rbrack, \bigl\lbrack3, 3≤x\bigr\rbrack\bigr\rbrack, 0)$

live ↗

In$\mathrm{IntegerPart}(e+π)$Out$5$

Properties #

$x = \mathrm{IntegerPart}(x) + \mathrm{FractionalPart}(x)$

live ↗

In$\mathrm{IntegerPart}(-\frac{7}{2})+\mathrm{FractionalPart}(-\frac{7}{2})=-\frac{7}{2}$Out$\mathrm{True}$

The same with a mixed expression: no piecewise rewriting

live ↗

In$\mathrm{PiecewiseExpand}(\mathrm{IntegerPart}(x^2+\sqrt{x}), 0\lt x\lt2)$Out$\mathrm{IntegerPart}(x^2+\sqrt{x})$

An integer is its own integer part

live ↗

In$\mathrm{IntegerPart}(1)$Out$1$

Known value · Wolfram MathematicalFunctionData, IntegerPart

$-1$ is unchanged

live ↗

In$\mathrm{IntegerPart}(-1)$Out$-1$

Known value · Wolfram MathematicalFunctionData, IntegerPart

Truncates $2.3$ to 2

live ↗

In$\mathrm{IntegerPart}(\frac{23}{10})$Out$2$

Known value · Wolfram MathematicalFunctionData, IntegerPart

A negative integer is unchanged

live ↗

In$\mathrm{IntegerPart}(-3)$Out$-3$

Known value · Wolfram MathematicalFunctionData, IntegerPart

Truncation toward zero gives $-3$ for $-\pi$

live ↗

In$\mathrm{IntegerPart}(-π)$Out$-3$

Known value · Wolfram MathematicalFunctionData, IntegerPart

Negative values round up toward zero: $-2.7\to-2$

live ↗

In$\mathrm{IntegerPart}(-\frac{27}{10})$Out$-2$

Known value · Wolfram MathematicalFunctionData, IntegerPart

Simplify cannot reduce $\lfloor x\rfloor$ difference even with $x\ge0$

live ↗

In$\mathrm{Simplify}(\mathrm{IntegerPart}(x+1)-\mathrm{IntegerPart}(x), x≥0)$Out$\mathrm{IntegerPart}(x+1)-\mathrm{IntegerPart}(x)$

Nested integer parts: the inner one is expanded first, then the outer one over its pieces

live ↗

In$\mathrm{PiecewiseExpand}(\mathrm{IntegerPart}(x+\frac{1}{3}+\mathrm{IntegerPart}(1-x/2)^3), x∈ℝ∧-2\lt x\lt3)$Out$\mathrm{Piecewise}(\bigl\lbrack\bigl\lbrack1, 2/3≤x∧x\lt5/3∨-1/3≤x≤0\bigr\rbrack, \bigl\lbrack2, 5/3≤x∧x\lt8/3\bigr\rbrack, \bigl\lbrack3, 8/3≤x\bigr\rbrack\bigr\rbrack, 0)$

live ↗

In$\mathrm{IntegerPart}(i)$Out$i$

Known value · Wolfram MathematicalFunctionData, IntegerPart

live ↗

In$\mathrm{IntegerPart}(-i)$Out$-i$

Known value · Wolfram MathematicalFunctionData, IntegerPart

live ↗

In$\mathrm{IntegerPart}(\frac{23}{10}-ie)$Out$2-2i$

Known value · Wolfram MathematicalFunctionData, IntegerPart

Possible issues #

An expression that is exactly 0 gives 0

live ↗

In$\mathrm{IntegerPart}(1+2π-(1+π)^2+π^2)$Out$0$

About

Examples	25
Checked against	Wolfram Language 23 agree2 differ
Crosswalk	Wolfram Language