# 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(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(8)

Iso-curves become fulldimensional regions for piecewise constant functions:

Fibonacci numbers:

Implement a divide-and-conquertype 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:

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: