Round

Round[x]

gives the integer closest to x.

Round[x,a]

rounds to the nearest multiple of a.

Details

Mathematical function, suitable for both symbolic and numerical manipulation.
Round rounds numbers of the form x .5 toward the nearest even integer.
Round[x] returns an integer when x is any numeric quantity, whether or not it is an explicit number.
Round[x] applies separately to real and imaginary parts of complex numbers.
If a is not a real number, Round[x,a] is given by the formula Round[x,a]a Round[x/a]. »
For exact numeric quantities, Round internally uses numerical approximations to establish its result. This process can be affected by the setting of the global variable $MaxExtraPrecision.
Round automatically threads over lists.

Examples

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Basic Examples (3)

Round to the nearest integer:

Round to the nearest multiple of 10:

Plot the function over a subset of the reals:

Scope (25)

Numerical Evaluation (7)

Evaluate numerically:

Value at two consecutive half-integers:

Complex number inputs:

Single-argument Round always returns an exact result:

The two-argument form tracks the precision of the second argument:

Evaluate efficiently at high precision:

Round threads elementwise over lists:

Round can deal with real‐valued intervals:

Specific Values (6)

Values of Round at fixed points:

Value at 0:

Value at Infinity:

Evaluate symbolically:

Manipulate Round symbolically:

Find a value of x for which Round[x,2]=2:

Visualization (4)

Plot the Round function:

Visualize the two-argument form:

Plot Round in three dimensions:

Visualize Round in the complex plane:

Function Properties (4)

Round is defined for all real and complex inputs:

Round can produce infinitely large and small results:

Round is an odd function in its first argument:

Round is an even function in its second argument:

Differentiation and Integration (4)

First derivative with respect to x:

First derivative with respect to a:

Evaluate an integral:

Series expansion:

Applications (1)

Compute Fibonacci numbers:

Properties & Relations (6)

Negative numbers also round to the nearest integer:

Round[x,a] gives the multiple of a nearest to x:

In general, it can be expressed in terms of the one-argument form as follows:

Verify the formula:

Round[x,-a] is equal to Round[x,a]:

At midpoints, Round rounds toward even integers:

This is also true of the two-argument form, where it rounds toward even multiples:

Possible Issues (1)

Round does not automatically resolve the value:

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Round

Details

Examples

Basic Examples (3)