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About

Bijective base-26 numeration maps every positive integer to a unique string over the alphabet {a…z} with no zero digit. This is the same system used by spreadsheet applications to label columns: 1 → a, 26 → z, 27 → aa, 702 → zz, 703 → aaa. Getting the mapping wrong in code generation, serialization, or data export pipelines produces silent off-by-one errors that corrupt downstream datasets. This tool computes both directions of the bijection and maintains a session history so you can audit conversions without re-entering values. It assumes input n ≥ 1; the mapping is undefined for zero and negative integers.

Formulas

The bijective base-26 system encodes a positive integer n as a string s = c_kc_k−1…c₁ where each character c_i ∈ {a, b, …, z} maps to values 1 - 26.

n = k∑i=1 c_i ⋅ 26^{i − 1}

The encoding algorithm iterates:

r = (n − 1) mod 26
c = r + 1 → character a…z
n = n − 126 (floor)

Repeat until n = 0, then reverse the collected characters.

Where n = input positive integer, r = remainder after bijective modulo, c = character value (1 - 26), k = resulting string length.

Reference Data

Number	Alpha String	Equivalent Expression
1	a	1
2	b	2
13	m	13
26	z	26
27	aa	26 + 1
28	ab	26 + 2
52	az	26 + 26
53	ba	2 ⋅ 26 + 1
256	iv	9 ⋅ 26 + 22
702	zz	26 ⋅ 26 + 26
703	aaa	1 ⋅ 26² + 1 ⋅ 26 + 1
1152	arh	1 ⋅ 676 + 18 ⋅ 26 + 8
16384	xfd	Excel max column
18278	zzz	26³ + 26² + 26
18279	aaaa	First 4-letter string
18432	aafx	1 ⋅ 26³ + 1 ⋅ 26² + 6 ⋅ 26 + 24
475254	zzzzz	Max 5-letter string
1000000	bdwgn	6-digit milestone

Frequently Asked Questions

Standard base-26 uses digits 0-25 and requires a zero placeholder. Bijective base-26 uses digits 1-26 (mapped to a - z) with no zero. This means there is no ambiguity between, say, "a" and "aa" - they represent distinct values (1 and 27). Every positive integer maps to exactly one string, and every non-empty lowercase string maps to exactly one positive integer.

Because the digit set is {1, 2, …, 26} rather than {0, 1, …, 25}. Subtracting 1 before taking the modulus shifts the range so that n = 26 correctly yields remainder 25 (mapping to z) instead of 0. Without this offset, multiples of 26 would produce an empty character.

The tool accepts integers up to 20 digits. JavaScript loses integer precision beyond 2⁵³ − 1 (9,007,199,254,740,991), which is a 16-digit number producing a 11-character alpha string. Inputs beyond this threshold may yield incorrect results due to floating-point truncation. For cryptographic or arbitrary-precision needs, use a BigInt-based implementation.

Yes. Microsoft Excel, Google Sheets, and LibreOffice Calc all label columns using bijective base-26: A=1, Z=26, AA=27, and so on. Excel's maximum column is XFD=16384. The only difference is case - spreadsheets use uppercase. This tool outputs lowercase to match the npm lower-alpha convention.

Yes. Enter multiple values separated by commas, spaces, or newlines. The tool will process each value independently and display all results in a table. This is useful for generating column labels for data export scripts or mapping index arrays to alphabetic identifiers.

Toggle the conversion direction using the swap button between the input fields. In alpha-to-number mode, enter a lowercase string (e.g., arh) and the tool returns 1152. The reverse formula is: accumulate result = result × 26 + (character value) for each character left to right.