Square Root

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Square Root () (%: in J) is a monadic scalar function that returns the square root of its argument, that is, a number whose square is the argument. As there are two such numbers for any non-zero argument (negating one possible result gives another), the result returned is taken to have either a positive real part, or, if the real part is zero, a non-negative imaginary part; in particular, if the result is real then it will be zero or greater. Square Root may also be defined in terms of the Power function, by raising the argument to the power one-half. It is considered a case of the dyadic function Root (which uses the same glyph), with a default left argument of 2. Square Root is defined in J, NARS2000, dzaima/APL, and Extended Dyalog APL.


The radical symbol for the square root has been used in mathematics for centuries, and was suggested early in APL's development by Eugene McDonnell, who stated that "when I designed the APL printer trains for the IBM 1403 printer, I included the radical symbol √ on the train, in the expectation that one day this function would be added to APL."[1] It was featured in J following McDonnell's design but written with the ASCII bigraph %:, and was introduced with the radical symbol by NARS2000. More recently, it was added first to Extended Dyalog APL and then to dzaima/APL in 2018.

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APL built-ins [edit]
Primitives (Timeline) Functions
Monadic ConjugateNegateSignumReciprocalMagnitudeExponentialNatural LogarithmFloorCeilingFactorialNotPi TimesRollTypeImaginarySquare Root
Dyadic AddSubtractTimesDivideResiduePowerLogarithmMinimumMaximumBinomialComparison functionsBoolean functions (And, Or, Nand, Nor) ∙ GCDLCMCircularComplexRoot
Structural ShapeReshapeTallyDepthRavelEnlistTableCatenateReverseRotateTransposeRazeMixSplitEncloseNestCut (K)PairLinkPartitioned EnclosePartition
Selection FirstPickTakeDropUniqueIdentityStopSelectReplicateExpandSet functions (IntersectionUnionWithout) ∙ Bracket indexingIndexCartesian ProductSort
Selector Index generatorGradeIndex OfInterval IndexIndicesDealPrefix and suffix vectors
Computational MatchNot MatchMembershipFindNub SieveEncodeDecodeMatrix InverseMatrix DivideFormatExecuteMaterialiseRange
Operators Monadic EachCommuteConstantReplicateExpandReduceWindowed ReduceScanOuter ProductKeyI-BeamSpawnFunction axis
Dyadic BindCompositions (Compose, Reverse Compose, Beside, Withe, Atop, Over) ∙ Inner ProductDeterminantPowerAtUnderRankDepthVariantStencilCutDirect definition (operator)
Quad names Index originComparison toleranceMigration levelAtomic vector