Round

Rounds x to the nearest integer, or to the nearest multiple of a.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaRounding↗MathWorldNearestIntegerFunction↗Rounding↗WikidataQ663208↗Wolfram LanguageRound✓↗

Round(x)rounds x to the nearest integer, a tie to the even one.

SageMathround(x)SymPyround(x)mpmathnint(x)Juliaenumeratio_round(x)Rustround(x)

Round(x, a)rounds to the nearest multiple of the step a.

Round(x, a?)Rounds x to the nearest integer, or to the nearest multiple of a.

Typed signatures

Type	Package	Applies
(x: real | signed_infinity | ~oo, step: real?) -> real | signed_infinity | ~oo	analyticreplaces compute-engine	always
(x: real | signed_infinity | ~oo, step: real?) -> real | signed_infinity | ~oo	compute-engine	always
(any, any?) -> any	structuresreplaces analytic	always

Domain: Arithmetic

Details

• Rounds to the nearest integer, with ties (an exact .5) going to the even one, as in Wolfram, Sage, SymPy, mpmath and Python: $\mathrm{Round}(2.5) = 2$, $\mathrm{Round}(3.5) = 4$ and $\mathrm{Round}(-2.5) = -2$.
• A second argument a rounds to the nearest multiple of a, as in Wolfram: $\mathrm{Round}(226, 10) = 230$ and $\mathrm{Round}(3.14159, 1/100) = 157/50$. A step of 1 agrees with the 1-argument form.
• Threads element-wise over a list.
• Floor and Ceil take a step the same way.
• Differs from Rust's round, and from compute-engine's own default, in the tie rule: both send a tie away from zero, so $\mathrm{Round}(5/2) = 3$ there where it is $2$ here. Only exact ties differ.
• Python's round(x, n) counts digits: its round(x, 2) is $\mathrm{Round}(x, 1/100)$ here.
• Ties follow the engine's roundingTies setting, which every engine we ship sets to to-even; compute-engine's default is away from zero.

Examples

3 more kept as data, checked against the oracles

Basic #

live ↗

In$\mathrm{round}(\frac{7}{2})$Out$4$

live ↗

In$\mathrm{round}(3.5)$Out$4$

Integers pass through unchanged

live ↗

In$\mathrm{round}(5)$Out$5$

A second argument rounds to the nearest multiple of it: a step of 1/100

live ↗

In$\mathrm{round}(3.141\,59, \frac{1}{100})$Out$\frac{157}{50}$

Threads element-wise over a list

live ↗

In$\mathrm{round}(\bigl\lbrack1.5, -1.5, 2.4\bigr\rbrack)$Out$\bigl\lbrack2, -2, 2\bigr\rbrack$

live ↗

In$\mathrm{round}(2.4)$Out$2$

live ↗

In$\mathrm{round}(2.6)$Out$3$

live ↗

In$\mathrm{round}(5.37)$Out$5$

live ↗

In$\mathrm{round}(-3.7)$Out$-4$

Scope #

Stays symbolic

live ↗

In$\mathrm{round}(x)$Out$\mathrm{round}(x)$

To the nearest multiple of ten

live ↗

In$\mathrm{round}(226, 10)$Out$230$

An exact argument rounds to an exact rational

live ↗

In$\mathrm{round}(\sqrt{2}, \frac{1}{10\,000})$Out$\frac{7\,071}{5\,000}$

Evaluates at an exact constant: $\mathrm{Round}(\pi) = 3$

live ↗

In$\mathrm{round}(π)$Out$3$

$100e \approx 271.83$ rounds to 272

live ↗

In$\mathrm{round}(100e)$Out$272$

Rounds the real and imaginary parts separately

live ↗

In$\mathrm{round}(5.37-1.3i)$Out$5-i$

The step can be any size: 226 is nearest to 200 among the multiples of 100

live ↗

In$\mathrm{round}(226, 100)$Out$200$

Applications #

Rounding a price to the nearest cent: 19.995 rounds up to 20

live ↗

In$\mathrm{round}(19.995, \frac{1}{100})$Out$20$

Properties #

1234 is nearest to 1200 among the multiples of 100

live ↗

In$\mathrm{round}(1\,234, 100)=1\,200$Out$\mathrm{True}$

A step of 1 matches the 1-argument form

live ↗

In$\mathrm{round}(3.141\,59, 1)=\mathrm{round}(3.141\,59)$Out$\mathrm{True}$

Possible issues #

Rounds a tie to the even integer, as banker's rounding does: 2.5 rounds to 2, not 3

live ↗

In$\mathrm{round}(2.5)$Out$2$

...and by the same rule, -2.5 rounds to -2, not -3

live ↗

In$\mathrm{round}(-2.5)$Out$-2$

The same tie-breaking rule on an exact rational, not just a float: 5/2 rounds to 2

live ↗

In$\mathrm{round}(\frac{5}{2})$Out$2$

An exact half rounds to the even integer as well: 1/2 rounds to 0

live ↗

In$\mathrm{round}(\frac{1}{2})$Out$0$

Round(ComplexInfinity) is ComplexInfinity: rounding an infinite magnitude with no direction is still that magnitude, the same convention as Ceil and Floor

live ↗

In$\mathrm{round}(\tilde∞)$Out$\tilde∞$

Known value · Wolfram Round[ComplexInfinity]

About

Examples	27
Checked against	Wolfram Language 27 agreeSageMath 16 agree4 errorSymPy 25 agree2 errormpmath 17 agreeJulia 17 agreeRust 11 agree5 differ
Crosswalk	WikipediaMathWorld ×2WikidataWolfram LanguageSageMathSymPympmathJuliaRust

Implementation

mappedsympypackages/oracle/src/mappings.ts

SymPy's round, like Wolfram's, sends a tie to the even integer and rounds a complex number part by part.

mappedsympypackages/oracle/src/mappings.ts

mappedmpmathpackages/oracle/src/mappings.ts

mpmath rounds half to even.

mappedsagepackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts

mappedrustpackages/oracle/src/mappings.ts

Ties go away from zero, as num-rational's round and std's do.

See also: Floor, Ceil, Clamp