Base 10 to base 6 converter
We probably count in base 10 because we have ten fingers, but there is nothing special about that number. Type a number in either box and the converter translates it into the other base.
Key formula
d₃d₂d₁d₀ (base 6) = d₃·6³ + d₂·6² + d₁·6 + d₀
Try it
How to use the simulator
In base 10 each position is worth ten times the one to its right (ones, tens, hundreds). In base 6 each position is worth six times as much: ones, sixes, 36s, 216s... and only the digits 0 to 5 are used. The converter works both ways, so you can type in either box.
Controls
- Base 10 box: type a decimal number to see it in base 6.
- Base 6 box: type a number using only the digits 0 to 5.
- Example buttons: hand-picked numbers with interesting results.
Things to try
- Type 6, 36 and 216: you get 10, 100 and 1000, the powers of six.
- Try 346, Grace's pendulum count: in base 6 it's 1334.
- Try typing a 6 or a 7 in the base 6 box. They don't exist in that system, just as base 10 has no single digit for ten.
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How a base works
In a positional system, each digit is worth more depending on where it sits. In base 10, the number 346 means 3 hundreds, 4 tens and 6 ones: 3·10² + 4·10 + 6. In base 6 only the digits 0 to 5 are used, and each position is worth six times the one before: ones, sixes, thirty-sixes, two hundred sixteens…
To go from base 10 to base 6, divide the number by 6 over and over and write down the remainders; read from bottom to top, they form the base-6 number. To go back, multiply each digit by its power of 6 and add them up.
Other bases in history
The Babylonians used base 60, which is where the 60 minutes in an hour and the 360 degrees in a circle come from. The Maya counted in base 20. Computers work in base 2, and programmers often use base 16 to write those numbers compactly.
Base 6 has real advantages: since 6 is divisible by both 2 and 3, fractions like 1/3 have an exact form (0.2 in base 6), which isn't true in base 10. Linguists have documented languages in New Guinea that count in base 6.
Worked example
Example: the year 2026 in base 6
- 2026 ÷ 6 = 337, remainder 4.
- 337 ÷ 6 = 56, remainder 1. 56 ÷ 6 = 9, remainder 2. 9 ÷ 6 = 1, remainder 3. 1 ÷ 6 = 0, remainder 1.
- Reading the remainders bottom to top: 2026 = 13214 in base 6.
- The other way: 1334 in base 6 = 1·216 + 3·36 + 3·6 + 4 = 346. Check it in the converter.
In Project Hail Mary
The novel features an alien species that counts in base 6, and understanding its number system is one of the first steps toward communicating with it.
Frequently asked questions
Why do we count in base 10?
The most widely accepted explanation is that we count on our fingers. Many number words in different languages come from words for hand or finger.
What is the best base?
It depends on the use. Base 2 is simplest for circuits; base 12 or base 6 make dividing by 2, 3 and 4 easy; base 10 has the advantage that everyone knows it.
How do you write fractions in base 6?
Just as in base 10, but each place to the right of the point is worth 1/6, 1/36, 1/216… For example, 0.3 in base 6 is 3/6 = 0.5 in base 10.