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f_{}(operatorname{T}) = -1, f_{}(0) = 0 quad and f_{}(1) = 1,The symbols 0 and 1 appear twice in the equalities f_{}(0) = 0 and f_{}(1) = 1, but these instances do not represent the same thing. The right hand side 0 and 1 mean the integers inZ, but the instances inside f's parentheses (which belong to mathcal{D}_{3}) should be thought of as being nothing more than symbols.

vleft( d_n ldots d_0 right) ~=~ sum_{i=0}^{n} f_{} left( d_{i} right) 3^{i}.

vleft( d_n d_{n-1} ldots d_0 right) ~=~ f_{} left( d_{n} right) 3^{n} + vleft( d_{n-1} ldots d_0 right)

vleft( 0 d_n ldots d_0 right) = vleft( d_n ldots d_0 right)

begin{alignat}{10}

vleft( 1 0 1 operatorname{T} right) = f_{}left( 1 right) 3^{3} + vleft( 0 1 operatorname{T} right) = (1) 27 + vleft( 1 operatorname{T} right) = 27 + 2 = 29.

10bal3 = 1 × 31 + 0 × 30 = 3dec 10𝖳bal3 = 1 × 32 + 0 × 31 + (−1) × 30 = 8dec −9dec = −1 × 32 + 0 × 31 + 0 × 30 = 𝖳00bal3 8dec = 1 × 32 + 0 × 31 + (−1) × 30 = 10𝖳bal3

−{{sfrac|2|3}}dec = −1 + {{sfrac|1|3}} = −1 × 30 + 1 × 3−1 = 𝖳.1bal3.

{| class="wikitable"! Decimal! style="text-align: right" | −0.9! style="text-align: right" | −0.8! style="text-align: right" | −0.7! style="text-align: right" | −0.6! style="text-align: right" | −0.5! style="text-align: right" | −0.4! style="text-align: right" | −0.3! style="text-align: right" | −0.2! style="text-align: right" | −0.1! style="text-align: right" | 0! Balanced Ternary

010𝖳}}

𝖳.{{overline

10𝖳0}}

𝖳.{{overline

𝖳}} or 𝖳.{{overline

𝖳𝖳11}}

0.{{overline

𝖳11𝖳}}

0.{{overline

| 0

! Decimal! style="text-align: right" | 0.9! style="text-align: right" | 0.8! style="text-align: right" | 0.7! style="text-align: right" | 0.6! style="text-align: right" | 0.5! style="text-align: right" | 0.4! style="text-align: right" | 0.3! style="text-align: right" | 0.2! style="text-align: right" | 0.1! style="text-align: right" | 0! Balanced Ternary

0𝖳01}}

1.{{overline

𝖳010}}

1.{{overline

1}} or 1.{{overline

11𝖳𝖳}}

0.{{overline

1𝖳𝖳1}}

0.{{overline

| 0

In decimal or binary, integer values and terminating fractions have multiple representations. For example, {{sfrac|1|10}} = 0.1 = 0.1{{overline|0}} = 0.0{{overline|9}}. And, {{sfrac|1|2}} = 0.12 = 0.1{{overline|0}}2 = 0.0{{overline|1}}2. Some balanced ternary fractions have multiple representations too. For example, {{sfrac|1|6}} = 0.1{{overline|𝖳}}bal3 = 0.0{{overline|1}}bal3. Certainly, in the decimal and binary, we may omit the rightmost trailing infinite 0s after the radix point and gain a representations of integer or terminating fraction. But, in balanced ternary, we can't omit the rightmost trailing infinite −1s after the radix point in order to gain a representations of integer or terminating fraction.Donald KnuthBOOK, Knuth, Donald, Donald Knuth, The art of Computer Programming, 2, 195–213, Addison-Wesley, 199711}} (round to 0, and truncate to 0) and 1.{{overlineradix, Rounding#Double rounding>double rounding is also equivalent to directly rounding to the final precision, unlike with an even radix.The basic operations—addition, subtraction, multiplication, and division—are done as in regular ternary. Multiplication by two can be done by adding a number to itself, or subtracting itself after a-trit-left-shifting.An arithmetic shift left of a balanced ternary number is the equivalent of multiplication by a (positive, integral) power of 3; and an arithmetic shift right of a balanced ternary number is the equivalent of division by a (positive, integral) power of 3.

Conversion to and from a fraction

{| class="wikitable" style="text-align: center;"!Fraction!!colspan="2"|Balanced ternary

	1

1		1}}	1.{{overline|𝖳}}

1		0.1

1		0.{{overline|1𝖳}}

1		0.{{overline|1𝖳𝖳1}}

1		1}}	0.1{{overline|𝖳}}

1		0.{{overline|0110𝖳𝖳}}

1		0.{{overline|01}}

1		0.01

1		0.{{overline|010𝖳}}

{| class="wikitable" style="text-align: center;"!Fraction!!colspan="2"|Balanced ternary

1		0.{{overline|01𝖳11}}

1		0.0{{overline|1𝖳}}

1		0.{{overline|01𝖳}}

1		0.{{overline|01𝖳0𝖳1}}

1		0.0{{overline|1𝖳𝖳1}}

1		0.{{overline|01𝖳𝖳}}

1		0.{{overline|01𝖳𝖳𝖳10𝖳0𝖳111𝖳01}}

1		1}}	0.01{{overline|𝖳}}

1		0.{{overline|00111𝖳10100𝖳𝖳𝖳1𝖳0𝖳}}

1		0.{{overline|0011}}

The conversion of a repeating balanced ternary number to a fraction is analogous to converting a repeating decimal. For example (because of 111111bal3 = ({{sfrac|36 − 1|3 − 1}})dec):

Irrational numbers

As in any other integer base, algebraic irrationals and transcendental numbers do not terminate or repeat. For example: !Decimal !! Balanced ternary

| sqrt{mathrm{1T}}=mathrm{1.11T1TT00T00T01T0T00T00T01TTldots}

| sqrt{mathrm{10}}=mathrm{1T.T1TT10T0000TT1100T0TTT011T0ldots}

| sqrt{mathrm{1TT}}=mathrm{1T.1T0101010TTT1TT11010TTT01T1ldots}

| varphi=frac{1 + sqrt{mathrm{1TT}}}{mathrm{1T}}=mathrm{1T.T0TT01TT0T10TT11T0011T10011ldots}

| tau=mathrm{1T0.10TT0T1100T110TT0T1TT000001}ldots

| pi=mathrm{10.011T111T000T011T1101T111111}ldots

| e=mathrm{10.T0111TT0T0T111T0111T000T11T}ldots

The balanced ternary expansions of pi is given in OEIS as {{OEIS link|A331313}}, that of e in {{OEIS link|A331990}}.

Conversion from ternary

Unbalanced ternary can be converted to balanced ternary notation in two ways:

• Add 1 trit-by-trit from the first non-zero trit with carry, and then subtract 1 trit-by-trit from the same trit without borrow. For example,
• : 0213 + 113 = 1023, 1023 − 113 = 1T1bal3 = 7dec.
• If a 2 is present in ternary, turn it into 1T. For example,
• : 02123 = 0010bal3 + 1T00bal3 + 001Tbal3 = 10TTbal3 = 23dec

{| class="wikitable floatright" style=" text-align: center"! Balanced || Logic || Unsigned

	| 2

	| 1

	| 0

If the three values of ternary logic are false, unknown and true, and these are mapped to balanced ternary as T, 0 and 1 and to conventional unsigned ternary values as 0, 1 and 2, then balanced ternary can be viewed as a biased number system analogous to the offset binary system.If the ternary number has n trits, then the bias b is which is represented as all ones in either conventional or biased form.Douglas W. Jones, Ternary Number Systems, October 15, 2013.As a result, if these two representations are used for balanced and unsigned ternary numbers, an unsigned n-trit positive ternary value can be converted to balanced form by adding the bias b and a positive balanced number can be converted to unsigned form by subtracting the bias b. Furthermore, if x and y are balanced numbers, their balanced sum is {{nowrap|x + y − b}} when computed using conventional unsigned ternary arithmetic. Similarly, if x and y are conventional unsigned ternary numbers, their sum is {{nowrap|x + y + b}} when computed using balanced ternary arithmetic.

Conversion to balanced ternary from any integer base

We may convert to balanced ternary with the following formula: where, For instance,

Addition, subtraction and multiplication and division

The single-trit addition, subtraction, multiplication and division tables are shown below. For subtraction and division, which are not commutative, the first operand is given to the left of the table, while the second is given at the top. For instance, the answer to 1 âˆ’ T = 1T is found in the bottom left corner of the subtraction table.

Multi-trit addition and subtraction

Multi-trit addition and subtraction is analogous to that of binary and decimal. Add and subtract trit by trit, and add the carry appropriately.For example:

Multi-trit multiplication

Multi-trit multiplication is analogous to that of binary and decimal.

Multi-trit division

Balanced ternary division is analogous to that of binary and decimal.However, 0.5dec = 0.1111...bal3 or 1.TTTT...bal3. If the dividend over the plus or minus half divisor, the trit of the quotient must be 1 or T. If the dividend is between the plus and minus of half the divisor, the trit of the quotient is 0. The magnitude of the dividend must be compared with that of half the divisor before setting the quotient trit. For example, Another example, Another example,

Square roots and cube roots

The process of extracting the square root in balanced ternary is analogous to that in decimal or binary. begin{cases}mathrm{T10}cdot x+1, & y=mathrm{T} mathrm{1T0}cdot x+1, & y=1end{cases}As in division, we should check the value of half the divisor first. For example, In the early days of computing, a few experimental Soviet computers were built with balanced ternary instead of binary, the most famous being the Setun, built by Nikolay Brusentsov and Sergei Sobolev. The notation has a number of computational advantages over traditional binary and ternary. Particularly, the plus–minus consistency cuts down the carry rate in multi-digit multiplication, and the rounding–truncation equivalence cuts down the carry rate in rounding on fractions. In balanced ternary, the one-digit multiplication table remains one-digit and has no carry and the addition table has only two carries out of nine entries, compared to unbalanced ternary with one and three respectively. Knuth wrote that "Perhaps the symmetric properties and simple arithmetic of this number system will prove to be quite important some day,"