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(32)

Numerical Evaluation(8)

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 can deal with realvalued intervals:

Compute the elementwise values of an array using automatic threading:

Or compute the matrix Round function using MatrixFunction:

Compute average-case statistical intervals using Around:

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:

Wolfram Research (1988), Round, Wolfram Language function, https://reference.wolfram.com/language/ref/Round.html (updated 2007).

Text

Wolfram Research (1988), Round, Wolfram Language function, https://reference.wolfram.com/language/ref/Round.html (updated 2007).

CMS

Wolfram Language. 1988. "Round." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2007. https://reference.wolfram.com/language/ref/Round.html.

APA

Wolfram Language. (1988). Round. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Round.html

BibTeX

@misc{reference.wolfram_2024_round, author="Wolfram Research", title="{Round}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/Round.html}", note=[Accessed: 16-September-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_round, organization={Wolfram Research}, title={Round}, year={2007}, url={https://reference.wolfram.com/language/ref/Round.html}, note=[Accessed: 16-September-2024 ]}