# Subtract

This page describes the dyadic arithmetic function. For negation of a single argument, see Negate. For subtraction of sets, see Without.
 `-`

Subtract (`-`), Minus, Subtraction, or Difference is a dyadic scalar function which gives the arithmetic difference of its arguments. Subtract shares the glyph <syntaxhighlight lang=apl inline>-</source> with the arithmetic function Negate, and its result is the left argument plus the negation of the right.

## Examples

<syntaxhighlight lang=apl>

```     ¯2 9 5 - 3 ¯4 6
```

¯5 13 ¯1

```     × 3 - 2 3 4.5  ⍝ Sign of difference
```

1 0 ¯1 </source> The second example computes a three-way comparison of each pair of arguments, with a result of <syntaxhighlight lang=apl inline>1</source> to indicate the left argument was greater, <syntaxhighlight lang=apl inline>0</source> to indicate the arguments are intolerantly equal, and <syntaxhighlight lang=apl inline>¯1</source> to indicate the right argument was greater.

## Properties

Subtraction is anti-commutative: swapping the arguments negates the result, or using Commute, <syntaxhighlight lang=apl inline>-⍨</source> $\Leftrightarrow$ <syntaxhighlight lang=apl inline>--</source>.

### Reduction and scan

Reduction with Subtract gives an alternating sum of the argument array, that is, elements are alternately added and subtracted to the result. The first element, third element, and so on are added to the final result while the second, fourth, and so on are subtracted. <syntaxhighlight lang=apl>

```     6 - 1 - 2
```

7

```     -/ 6 1 2
```

7 </source> In the absense of rounding error we have <syntaxhighlight lang=apl inline>-/v</source> $\Leftrightarrow$ <syntaxhighlight lang=apl inline>+/v×(⍴v)⍴1 ¯1</source> for a vector <syntaxhighlight lang=apl inline>v</source>.

An interesting property of the alternating difference is that it can be used as a divisibility test for division by 11, a counterpart to the better-known test for divisibility by 9. A number is divisible by 11 if and only if the sum of its digits is divisible by 11, so repeatedly taking the alternating sum of the digits determines divisibility by 11. <syntaxhighlight lang=apl>

```     11 | 946 943
```

0 8

```     -⌿ 10 (⊥⍣¯1) 946 943
```

11 8

```     (-⌿ 10 ⊥⍣¯1 ⊢)⍣≡ 946 943
```

0 2 </source>

Scan with subtraction produces a prefix of an alternating series. This property is one of the reasons why scan was designed to reduce prefixes rather than suffixes of the argument array. As an example, we can see that an alternating series using the powers of two begins to converge to a third: <syntaxhighlight lang=apl>

```     -\ ÷2*⍳6
```

0.5 0.25 0.375 0.3125 0.34375 0.328125 </source>