# 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(31)

### 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 realvalued 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(10)

Round[x] is defined for all real and complex inputs:

Round[x,a] is defined for a!=0:

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:

Round is not an analytic function:

It has both singularities and discontinuities:

Round is nondecreasing:

Round is not injective:

Round is not surjective:

Round is neither non-negative nor non-positive:

Round is neither convex nor concave:

### Differentiation and Integration(4)

First derivative with respect to x:

First derivative with respect to a:

Evaluate an integral:

Series expansion:

## Applications(2)

Compute Fibonacci numbers:

Click the bars to hear the name of the country and its rounded GDP per capita:

## 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: 