# FractionalPart

gives the fractional part of x.

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

# Examples

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## Basic Examples(3)

Find the fractional part of a real number:

Find the fractional part of a negative real number:

Plot over a subset of the reals:

## Scope(23)

### Numerical Evaluation(5)

Evaluate numerically:

Complex number inputs:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Evaluate efficiently at high precision:

### Specific Values(6)

Values of FractionalPart at fixed points:

Value at zero:

Value at Infinity:

Evaluate symbolically:

Manipulate FractionalPart symbolically:

Find a value of x for which the FractionalPart[x]=0.5:

### Visualization(4)

Plot the FractionalPart function:

Plot scaled FractionalPart functions:

Plot FractionalPart in three dimensions:

Visualize FractionalPart in the complex plane:

### Function Properties(4)

FractionalPart is defined for all real and complex inputs:

FractionalPart is an odd function:

FractionalPart can be made periodic on the reals by adding one to its value on the negative reals:

### Differentiation and Integration(4)

First derivative with respect to x:

Second derivative with respect to x:

Evaluate an integral:

Series expansion:

## Applications(6)

Plot fractional parts of powers:

Plot fractional parts of powers of a Pisot number:

Iterate the shift map with a rational initial condition and plot the result:

Irrational initial condition:

See the degradation in precision for approximate real numbers:

Make a Bernoulli polynomial periodic and plot it:

## Properties & Relations(3)

Convert FractionalPart to Piecewise:

Denest FractionalPart functions:

## Possible Issues(2)

Guard digits influence the result of FractionalPart:

Numerical decision procedures with default settings cannot simplify this expression: Using a larger setting for \$MaxExtraPrecision gives the expected result:

## Neat Examples(1)

Convergence of the Fourier series of FractionalPart: