gives the integer part of x.


  • 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.


open allclose all

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  (27)

Numerical Evaluation  (5)

Evaluate numerically:

Complex number inputs:

Evaluate efficiently at high precision:

IntegerPart threads elementwise over lists:

IntegerPart can deal with realvalued 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  (9)

IntegerPart is defined for all real and complex inputs:

IntegerPart can produce infinitely large and small results:

IntegerPart is an odd function:

IntegerPart is not an analytic function:

It has both singularities and discontinuities:

IntegerPart is nondecreasing:

IntegerPart is not injective:

IntegerPart is not surjective:

IntegerPart is neither non-negative nor non-positive:

IntegerPart is neither convex nor concave:

Differentiation and Integration  (3)

First derivative with respect to x:

Definite integrals of IntegerPart:

Series expansion:

Applications  (9)

Iso-curves become fulldimensional regions for piecewise constant functions:

Fibonacci numbers:

Implement a divide-and-conquertype 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:

Consider the IntegerPart of the earthquake magnitudes recorded in the US from 1935 to 1989:

The integer parts of the magnitudes recorded on a Richter scale can be modeled with a ParetoDistribution:

Compare the histogram of the magnitudes with the fitted distribution:

Find the probability of an earthquake with magnitude at least 6 on the Richter scale:

Find the average magnitude:

Simulate the next 30 earthquakes:

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:

Machineprecision 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:

Wolfram Research (1996), IntegerPart, Wolfram Language function,


Wolfram Research (1996), IntegerPart, Wolfram Language function,


Wolfram Language. 1996. "IntegerPart." Wolfram Language & System Documentation Center. Wolfram Research.


Wolfram Language. (1996). IntegerPart. Wolfram Language & System Documentation Center. Retrieved from


@misc{reference.wolfram_2024_integerpart, author="Wolfram Research", title="{IntegerPart}", year="1996", howpublished="\url{}", note=[Accessed: 22-May-2024 ]}


@online{reference.wolfram_2024_integerpart, organization={Wolfram Research}, title={IntegerPart}, year={1996}, url={}, note=[Accessed: 22-May-2024 ]}