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.

+Base10TernaryBalancedTernaryB. ternaryexpanded1111x1221t1x3 +  -1x1310101x3 + 0x1411111x3 + 1x15121tt1x9 + -1x3 + -1x16201t01x9 + -1x3 + 0x17211t11x9 + -1x3 + 1x182210t1x9 + 0x3 + -1x191001001x9 + 0x3 + 0x1101011011x9 + 0x3 + 1x1+Base10TernaryBalancedTernaryB. ternaryexpanded -1-1t-1x1 -2-2t1-1x3 +  1x1 -3-10t0-1x3 + 0x1 -4-11tt-1x3 + -1x1 -5-12t11-1x9 + 1x3 + 1x1 -6-20t10-1x9 + 1x3 + 0x1 -9-21t1t-1x9 + 1x3 + -1x1 -8-22t01-1x9 + 0x3 + 1x1 -9-100t00-1x9 + 0x3 + 0x1 -10-101t0t=1x9 + 0x3 + -1x1

Ternary, Balanced