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^(th) 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:

Introduced in 1996
 (3.0)