BEAF Array Calculator
Enter a linear array using comma-separated positive integers. You can use square brackets, curly braces, or just the numbers.
Understanding Bowers Array Notation
Bowers' Exploding Array Function, commonly abbreviated as BEAF, is a large-number notation associated with mathematician and googologist Jonathan Bowers. It extends familiar arithmetic operations into increasingly powerful array-based constructions.
An array is written as a sequence of entries. In the linear notation used by this calculator, the entries are separated by commas. The number of entries and their values determine which operation or expansion rule applies.
Supported array forms
| Array | Interpretation in this calculator | Support |
|---|---|---|
[a] |
The value is a. | Numerical |
[a,b] |
ab | Numerical, bounded |
[a,b,c] |
a ↑c b, using Knuth up-arrows | Numerical, bounded |
[a,b,c,d] |
Recursive four-entry array expansion | Symbolic first step |
Three-entry arrays and up-arrows
For three-entry arrays, this calculator uses the convention [a,b,c] = a ↑c b. The third entry specifies the number of Knuth up-arrows.
[3,2,1]represents 3 ↑ 2, or 3 squared.[3,2,2]represents 3 ↑↑ 2, which equals 27.[3,3,2]represents 3 ↑↑ 3, which equals 7,625,597,484.[2,3,3]represents 2 ↑↑↑ 3.
This is the up-arrow convention for the three-entry subset described here; it should not be confused with every historical version of Bowers' array notation.
Four-entry recursive arrays
Four-entry arrays introduce recursive expansion. In the supported convention, two forms of the expansion are:
[a,b,c,d] → [a,[a,b−1,c,d],c−1,d]
[a,b,1,d] → [a,a,[a,b−1,1,d],d−1]
These rules create nested arrays. A nested expression can become much more complicated than an ordinary exponent or a fixed number of up-arrows. This calculator displays the first symbolic expansion rather than assigning an unsupported numerical value.
Example BEAF Expressions
Try these examples in the calculator to explore the supported subset.
| Input | Meaning | Expected output |
|---|---|---|
5 |
Single-entry array | 5 |
2,5 |
2 to the power of 5 | 32 |
3,2,2 |
3 with two up-arrows and height 2 | 27 |
3,3,2 |
3 tetrated to height 3 | 7,625,597,484 |
3,3,1,2 |
Four-entry recursive array | Symbolic expansion |
Frequently Asked Questions
What is the Bowers Exploding Array Function?
BEAF is a system for representing extremely large numbers using structured arrays and recursive rules. It extends beyond ordinary arithmetic, exponentiation, and fixed levels of hyperoperations.
What does BEAF stand for?
BEAF stands for Bowers Exploding Array Function. It is named after Jonathan Bowers, who developed the notation.
Can this calculator evaluate every BEAF expression?
No. It evaluates selected short linear arrays and shows a symbolic expansion for four-entry arrays. The complete BEAF framework includes more advanced structures that are not implemented here.
Why are some results displayed symbolically?
Recursive array expressions can grow beyond the practical limits of ordinary computer arithmetic. Showing the expansion symbolically is more accurate than presenting an incomplete or misleading numerical result.
Is BEAF larger than Graham's number?
BEAF can express numbers that vastly exceed Graham's number, depending on the array and the chosen conventions. A particular array must be specified before making a meaningful comparison.
What is the difference between BEAF and Knuth up-arrow notation?
Knuth up-arrow notation extends exponentiation by adding repeated levels of operations. BEAF uses array structures and recursive expansion rules, allowing constructions beyond a fixed number of up-arrows.
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