Magnitude

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Magnitude (|), or Absolute Value, is a monadic scalar function which gives the absolute value of a real or complex number. Magnitude shares the glyph | with the dyadic arithmetic function Residue.

Examples

      |0 1 2 ¯1 ¯2
0 1 2 1 2
 
      |0J2 ¯3J¯4
2 5

Properties

The magnitude of any number is a non-negative real number.

For real numbers, the magnitude equals the original number times (or divided by, for non-zero numbers) its sign. 

      v←0 1E¯100 20 1E300 ¯1E¯100 ¯20 ¯1E300
      (|v)≡v××v
1
      (|v)=v÷×v
0 1 1 1 1 1 1

For complex numbers, the magnitude is defined as the Euclidean distance from the number 0 on the complex plane. 

      Dist←{0.5*⍨+.×⍨9 11○⍵} ⍝ Square root of square sum of real and imaginary parts
      Dist¨ 0 1J2 ¯3J4
0 2.236067977 5
      |0 1J2 ¯3J4
0 2.236067977 5
Works in: Dyalog APL

Any real or complex number is equal to the product of its signum and magnitude. 

      (⊢ ≡ ××|) 0 1 1E¯300 ¯2.5 0J3.5 ¯3J¯4
1
Works in: Dyalog APL

External links

Documentation

APL built-ins [edit]
Primitives (Timeline) Functions
Scalar
Monadic Conjugate ∙ Negate ∙ Signum ∙ Reciprocal ∙ Magnitude ∙ Exponential ∙ Natural Logarithm ∙ Floor ∙ Ceiling ∙ Factorial ∙ Not ∙ Pi Times ∙ Roll ∙ Type ∙ Imaginary ∙ Square Root ∙ Round
Dyadic Add ∙ Subtract ∙ Times ∙ Divide ∙ Residue ∙ Power ∙ Logarithm ∙ Minimum ∙ Maximum ∙ Binomial ∙ Comparison functions ∙ Boolean functions (And, Or, Nand, Nor) ∙ GCD ∙ LCM ∙ Circular ∙ Complex ∙ Root
Non-Scalar
Structural Shape ∙ Reshape ∙ Tally ∙ Depth ∙ Ravel ∙ Enlist ∙ Table ∙ Catenate ∙ Reverse ∙ Rotate ∙ Transpose ∙ Raze ∙ Mix ∙ Split ∙ Enclose ∙ Nest ∙ Cut (K) ∙ Pair ∙ Link ∙ Partitioned Enclose ∙ Partition
Selection First ∙ Pick ∙ Take ∙ Drop ∙ Unique ∙ Identity ∙ Stop ∙ Select ∙ Replicate ∙ Expand ∙ Set functions (Intersection ∙ Union ∙ Without) ∙ Bracket indexing ∙ Index ∙ Cartesian Product ∙ Sort
Selector Index generator ∙ Grade ∙ Index Of ∙ Interval Index ∙ Indices ∙ Deal ∙ Prefix and suffix vectors
Computational Match ∙ Not Match ∙ Membership ∙ Find ∙ Nub Sieve ∙ Encode ∙ Decode ∙ Matrix Inverse ∙ Matrix Divide ∙ Format ∙ Execute ∙ Materialise ∙ Range
Operators Monadic Each ∙ Commute ∙ Constant ∙ Replicate ∙ Expand ∙ Reduce ∙ Windowed Reduce ∙ Scan ∙ Outer Product ∙ Key ∙ I-Beam ∙ Spawn ∙ Function axis ∙ Identity (Null, Ident)
Dyadic Bind ∙ Compositions (Compose, Reverse Compose, Beside, Withe, Atop, Over) ∙ Inner Product ∙ Determinant ∙ Power ∙ At ∙ Under ∙ Rank ∙ Depth ∙ Variant ∙ Stencil ∙ Cut ∙ Direct definition (operator) ∙ Identity (Lev, Dex)
Quad names Index origin ∙ Comparison tolerance ∙ Migration level ∙ Atomic vector
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