# Balanced ternary

Balanced ternary is a type of numbering system. It is based on base 3.

The most common numbering system in use today is base 10, or decimal. With decimal, there are ten digits. these are 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. In ternary only 0, 1 and 2 are used. Balanced ternary is similar, but instead of using 0, 1 and 2, the numbers -1, 0 and 1 are used. The letter T is used to mean -1.

As with all numbering systems, each place is equal to the number Multiplied by the system's base raised to that positions location. For example with the base 10 number 1234, the value is 1x 103 (1000) + 2 x 102 (100) + 3 x 101 (10) + 4 x 100.(1) In base 3, these same values would be 27, 9, 3 and 1.

The benefits of using balanced ternary is that it is possible to write down numbers less than zero without the need to say whether a number is positive or negative. Another benefit is that when it comes to computers, there are much fewer rounding errors.

## Comparing systems

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
a
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