# Magnitude

 `|`

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