Balanced ternary

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

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

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


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