WOLFRAM

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)Summary of the most common use cases

Round to the nearest integer:

Out[1]=1
Out[2]=2

Round to the nearest multiple of 10:

Out[1]=1

Plot the function over a subset of the reals:

Out[1]=1

Scope  (32)Survey of the scope of standard use cases

Numerical Evaluation  (8)

Evaluate numerically:

Out[1]=1
Out[12]=12
Out[7]=7
Out[3]=3

Value at two consecutive half-integers:

Out[1]=1

Complex number inputs:

Out[1]=1

Single-argument Round always returns an exact result:

Out[1]=1

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

Out[2]=2

Evaluate efficiently at high precision:

Out[1]=1
Out[2]=2

Round can deal with realvalued intervals:

Out[1]=1

Compute the elementwise values of an array using automatic threading:

Out[1]=1

Or compute the matrix Round function using MatrixFunction:

Out[2]=2

Compute average-case statistical intervals using Around:

Out[1]=1

Specific Values  (6)

Values of Round at fixed points:

Out[1]=1

Value at 0:

Out[1]=1

Value at Infinity:

Out[1]=1

Evaluate symbolically:

Out[1]=1

Manipulate Round symbolically:

Out[1]=1

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

Out[1]=1
Out[2]=2

Visualization  (4)

Plot the Round function:

Out[1]=1

Visualize the two-argument form:

Out[1]=1

Plot Round in three dimensions:

Out[1]=1

Visualize Round in the complex plane:

Out[1]=1

Function Properties  (10)

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

Out[1]=1
Out[2]=2

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

Out[3]=3

Round can produce infinitely large and small results:

Out[1]=1

Round is an odd function in its first argument:

Out[1]=1

Round is an even function in its second argument:

Out[1]=1

Round is not an analytic function:

Out[1]=1

It has both singularities and discontinuities:

Out[2]=2
Out[3]=3

Round is nondecreasing:

Out[1]=1
Out[2]=2

Round is not injective:

Out[1]=1
Out[2]=2

Round is not surjective:

Out[1]=1
Out[2]=2

Round is neither non-negative nor non-positive:

Out[1]=1

Round is neither convex nor concave:

Out[1]=1

Differentiation and Integration  (4)

First derivative with respect to x:

Out[1]=1

First derivative with respect to a:

Out[1]=1

Evaluate an integral:

Out[1]=1

Series expansion:

Out[1]=1

Applications  (2)Sample problems that can be solved with this function

Compute Fibonacci numbers:

Out[1]=1
Out[2]=2

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

Out[2]=2

Properties & Relations  (6)Properties of the function, and connections to other functions

Negative numbers also round to the nearest integer:

Out[1]=1

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

Out[1]=1

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

Out[2]=2

Verify the formula:

Out[3]=3

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

Out[1]=1

At midpoints, Round rounds toward even integers:

Out[1]=1
Out[2]=2

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

Out[3]=3
Out[1]=1
Out[1]=1

Possible Issues  (1)Common pitfalls and unexpected behavior

Round does not automatically resolve the value:

Out[1]=1
Out[2]=2
Wolfram Research (1988), Round, Wolfram Language function, https://reference.wolfram.com/language/ref/Round.html (updated 2007).
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).

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.

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

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: 21-January-2025 ]}

@misc{reference.wolfram_2024_round, author="Wolfram Research", title="{Round}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/Round.html}", note=[Accessed: 21-January-2025 ]}

BibLaTeX

@online{reference.wolfram_2024_round, organization={Wolfram Research}, title={Round}, year={2007}, url={https://reference.wolfram.com/language/ref/Round.html}, note=[Accessed: 21-January-2025 ]}

@online{reference.wolfram_2024_round, organization={Wolfram Research}, title={Round}, year={2007}, url={https://reference.wolfram.com/language/ref/Round.html}, note=[Accessed: 21-January-2025 ]}