🎓 Chapter 3: Scalar Math & Relational Logic
In conventional languages, performing math on list items requires iterating with a for loop. In APL, all primitive functions automatically extend across arrays of any shape!
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1. Scalar Extension
When a binary operation combines a single scalar on one side with a vector or matrix on the other side, APL automatically applies the scalar operation to every single element of the array.
⍝ Multiply scalar 10 by vector 1..5
10 × ⍳ 5---
2. Relational Functions (< ≤ = ≥ > ≠)
APL relational functions evaluate element-wise and return a boolean array containing 1 for True and 0 for False.
| Function | Name | Example | Result |
| :--- | :--- | :--- | :--- |
| < | Less Than | 3 < 5 | 1 |
| ≤ | Less Than or Equal | 5 ≤ 5 | 1 |
| = | Equal To | 4 = 7 | 0 |
| ≥ | Greater Than or Equal | 9 ≥ 2 | 1 |
| > | Greater Than | 2 > 8 | 0 |
| ≠ | Not Equal To | 5 ≠ 5 | 0 |
Let's test creating a boolean mask on a vector:
⍝ Create a boolean mask of elements > 5
Numbers ← 3 8 2 9 5 7 1
Mask ← Numbers > 5
Mask---
3. Array Masking (Filtering without if statements!)
Because 1 represents True and 0 represents False, multiplying a numeric array by a boolean mask keeps values where the mask is 1 and zeroes out values where the mask is 0!
This is known as Branchless Array Filtering.
⍝ Zero out numbers that are not > 5
Numbers × (Numbers > 5)---
4. Logical Functions (∧ ∨ ~ ⍲ ⍱)
APL supports full boolean logic:
- And (
∧):1 ∧ 1 = 1,1 ∧ 0 = 0 - Or (
∨):1 ∨ 0 = 1,0 ∨ 0 = 0 - Not (
~):~ 1 = 0,~ 0 = 1
⍝ Find numbers that are BOTH > 3 AND < 8
Numbers × (Numbers > 3) ∧ (Numbers < 8)← Chapter 2: Vectors, Matrices & Reshape (⍴, ⍳) Chapter 4: Reductions (/) & Prefix Scans (\) →