IntegerPart

IntegerPart[x]

gives the integer part of x.

Details

Mathematical function, suitable for both symbolic and numerical manipulation.
IntegerPart[x] in effect takes all digits to the left of the decimal point and drops the others.
IntegerPart[x]+FractionalPart[x] is always exactly x.
IntegerPart[x] returns an integer when x is any numeric quantity, whether or not it is an explicit number.
For exact numeric quantities, IntegerPart internally uses numerical approximations to establish its result. This process can be affected by the setting of the global variable $MaxExtraPrecision.
IntegerPart applies separately to real and imaginary parts of complex numbers.
IntegerPart automatically threads over lists.

Examples

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Basic Examples (3)

Find the integer part of a real number:

Find the integer part of a negative real number:

Plot over a subset of the reals:

Scope (21)

Numerical Evaluation (5)

Evaluate numerically:

Complex number inputs:

Evaluate efficiently at high precision:

IntegerPart threads elementwise over lists:

IntegerPart can deal with real‐valued intervals:

Specific Values (6)

Values of IntegerPart at fixed points:

Value at zero:

Value at Infinity:

Evaluate symbolically:

Manipulate IntegerPart symbolically:

Find a value of x for which the IntegerPart[x]=1:

Visualization (4)

Plot the IntegerPart function:

Plot scaled IntegerPart functions:

Plot IntegerPart in three dimensions:

Visualize IntegerPart in the complex plane:

Function Properties (3)

IntegerPart is defined for all real and complex inputs:

IntegerPart can produce infinitely large and small results:

IntegerPart is an odd function:

Differentiation and Integration (3)

First derivative with respect to x:

Definite integrals of IntegerPart:

Series expansion:

Applications (8)

Iso-curves become full‐dimensional regions for piecewise constant functions:

Fibonacci numbers:

Implement a divide-and-conquer‐type recursion relation:

Find the 1000000 digit of the fraction 1/99^2 in base 10:

Compare with RealDigits functionality:

Find the day of the week in the Gregorian calendar:

Birthday of Leonard Euler:

Compare with DateString:

Implement the Frisch continuous-but-nowhere-differentiable function:

Properties & Relations (5)

Simplify expressions containing IntegerPart:

Symbolically expand for complex arguments:

IntegerPart is idempotent:

Use PiecewiseExpand to canonicalize:

Reduce equations containing IntegerPart:

Possible Issues (3)

Numerical decision procedures with default settings cannot simplify this expression:

Use Simplify to resolve:

Machine‐precision numericalization of IntegerPart can give wrong results:

Use arbitrary-precision evaluation instead:

Because the answer is exact, raising the internal precision does not remove the message:

Symbolic preprocessing of functions containing IntegerPart can be time consuming:

As a discontinuous function, IntegerPart can cause numerical algorithms to converge slowly:

Neat Examples (1)

Build a nondecreasing sequence of integers where each number occurs times [more info]:

Generate the sequence up to 5:

Group the same numbers:

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IntegerPart

Details

Examples

Basic Examples (3)