What is bijective base-6?

Bijective base-6 is a non-standard positional numeral system using base 6 and the six non-zero digits 1, 2, 3, 4, 5, and 6. Unlike standard senary (base-6) which uses digits 0 through 5, bijective base-6 has no zero digit, meaning every positive integer has a single unique representation without leading-zero ambiguity.

How do you count from 1 to 10 in bijective base-6?

Counting begins 1, 2, 3, 4, 5, 6. The next number (decimal 7) rolls over to 11 (1×6¹ + 1×6⁰ = 7). Decimal 8 is 12, decimal 9 is 13, and decimal 10 is 14.

Why is there no zero in bijective base-6?

In bijective numeration with base k, the digits are 1 through k instead of 0 through k-1. Eliminating zero creates an exact 1-to-1 bijection between positive integers and finite strings of digits. This is identical to how spreadsheet column titles work (A–Z, AA–AZ, etc.).

How do you convert decimal to bijective base-6?

To convert a positive decimal integer n to bijective base-6: repeatedly divide (n - 1) by 6. The remainder plus 1 produces the digit (from 1 to 6) from right to left, and the integer quotient becomes the new n until n reaches 0.

How does bijective base-6 addition work?

Addition is performed column by column from right to left. When the sum of column digits plus carry exceeds 6, the resulting digit is ((sum - 1) % 6) + 1, and the carry to the next column is floor((sum - 1) / 6).

Bijective Base-6 • Digits {1, 2, 3, 4, 5, 6} • No Zero

Bijective Base-6 Calculator & Explorer

An interactive educational tool to calculate and explore the bijective base-6 number system, which has no zero.

Live Conversion Explorer

Enter a decimal number to see its real-time representation across different systems.

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Bijective Math Calculator

Enter two bijective base-6 numbers to see the results for all operations.

Number 1 (Base-6)

Number 2 (Base-6)

Why Learn This System?

For an engineer, understanding alternative numeral systems like bijective base-6 offers practical advantages and deepens foundational knowledge.

What You Gain (Pros)

What You Miss (Cons)

System Comparison: Standard Base-6 vs. Bijective Base-6

Decimal	Standard Senary (Digits 0-5)	Bijective Base-6 (Digits 1-6)	Key Structural Difference
`1`	`1`	1	Identical representation
`5`	`5`	5	Identical representation
`6`	`10` (rolls over early)	6 (single digit!)	In bijective, 6 is a legitimate single digit
`7`	`11` (1×6 + 1)	11 (1×6 + 1)	First 2-digit bijective number
`12`	`20`	16	No zeros permitted in bijective: 1×6 + 6 = 12
`36`	`100`	56	5×6 + 6 = 36 (compact without leading zeros)
`42`	`110`	66	6×6 + 6 = 42 (all maximum digits before rollover)
`43`	`111`	111	First 3-digit bijective number (1×36 + 1×6 + 1)

How to Perform Addition

Addition is performed column by column, from right to left. The key difference from decimal is how you handle carrying over when a sum exceeds 6.

How to Perform Subtraction

Subtraction also works column by column. Borrowing is where it gets interesting: when you borrow from a column, you are borrowing a full 6, not 10.

How to Perform Multiplication

Multiplication follows the same long-multiplication algorithm as decimal, but it uses the bijective base-6 multiplication table. Each partial product is calculated and then summed using bijective addition rules.

Addition Table (1 to 24)

Multiplication Table (1 to 24)

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