Balanced ternary

Balanced ternary is a type of numbering system with a base of 3.

The most common numbering system in use today is decimal. Decimal has a base of ten, so it has 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. Ternary has a base of three, so it has only 3 digits: 0, 1 and 2. Balanced ternary also has a base of three, but it uses the digits -1, 0 and 1.

Like decimal, balanced ternary uses a place value system. Each place value is equal to the previous place value multiplied by the base. In decimal, we have a one's place, a ten's place, a hundred's place, a thousand's place, and so on. In balanced ternary, we have a one's place, a three's place, a nine's place, a twenty-seven's place, and so on.

Balanced ternary allows you to write numbers less than zero without needing a dedicated minus sign; if the digit with the biggest place-value is negative, the entire number is negative. Another benefit is that when it comes to computers, there are much fewer rounding errors.

Comparing systems

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Most balanced ternary systems use the letter T or the symbol ! for the -1 digit. This article will use the letter T.

Base10 Ternary Balanced
Ternary
B. ternary
expanded
1 1 1 1x1
2 2 1t 1x3 + -1x1
3 10 10 1x3 + 0x1
4 11 11 1x3 + 1x1
5 12 1tt 1x9 + -1x3 + -1x1
6 20 1t0 1x9 + -1x3 + 0x1
7 21 1t1 1x9 + -1x3 + 1x1
8 22 10t 1x9 + 0x3 + -1x1
9 100 100 1x9 + 0x3 + 0x1
10 101 101 1x9 + 0x3 + 1x1
Base10 Ternary Balanced
Ternary
B. ternary
expanded
-1 -1 t -1x1
-2 -2 t1 -1x3 + 1x1
-3 -10 t0 -1x3 + 0x1
-4 -11 tt -1x3 + -1x1
-5 -12 t11 -1x9 + 1x3 + 1x1
-6 -20 t10 -1x9 + 1x3 + 0x1
-9 -21 t1t -1x9 + 1x3 + -1x1
-8 -22 t01 -1x9 + 0x3 + 1x1
-9 -100 t00 -1x9 + 0x3 + 0x1
-10 -101 t0t =1x9 + 0x3 + -1x1