BASE 2BINARY
A switch.
Think off or on. A bit has two possible values. Groups of bits encode the data and instructions a computer uses.
01
Two choices per position
THE CASE FOR THREE
What if computing
started with 3?
Today’s computers build information from two choices: 0 and 1. Build with three—−1, 0, +1—and each digit can carry more information.
Start with a familiar byte. See what changes with trits.
Then discover why three is a mathematical sweet spot.
Start with a bit A 7-minute visual guide
RETHINK THE BUILDING BLOCK
3STATES. MORE POSSIBILITY.
−1NEGATIVE
0ZERO
+1POSITIVE
ONE TRITTHREE POSSIBLE VALUES
THE IDEA
The story in one line: binary builds with two-state digits; balanced ternary builds with three-state digits. Richer digits can mean fewer of them—and a different starting point for hardware.
01 / HOW COMPUTERS SPEAK
Everything starts
with a bit.
Numbers, words and photos become patterns inside a computer. So do the instructions that tell it what to do. In a traditional binary computer, those patterns are built from bits: digits that are either 0 or 1.
Hardware stores those patterns using two distinguishable states. The processor reads an instruction, performs the operation it specifies, then repeats. Logic circuits combine bits to add numbers, compare values and carry out that work. “Base 2” means two choices per digit. Balanced ternary changes that building block to three choices.
BASE 3BALANCED TERNARY
A switch with a middle.
Negative, zero or positive. A trit has three possible values. Each position offers one more choice for encoding information.
−10+1
Three choices per position
The signs are the digit values. The base counts the choices: two per bit, three per trit.
FROM ONE DIGIT TO A WHOLE GROUP
Eight bits. Five trits.
Almost the same room.
Computers commonly group eight bits into a byte. Why does that give 256 patterns? Each position has two choices, so every extra position doubles the total.
The small raised number counts the positions. 2⁸ means “multiply eight 2s together.” With three choices per position, five trits give 3⁵: multiply five 3s together.
8 BITS · 1 BYTEBINARY
10100101
Eight positions. Two choices in each.
28=256
“Two to the eighth power”
2 × 2 × 2 × 2 × 2 × 2 × 2 × 2
possible patterns
5 TRITSBALANCED TERNARY
−10+1−1+1
Five positions. Three choices in each.
35=243
“Three to the fifth power”
3 × 3 × 3 × 3 × 3
possible patterns
Same job. Three fewer positions.
Need unique codes for 243 items? Binary needs 8 bits—seven bits only give 128 patterns. Balanced ternary needs 5 trits. Each trit does more of the work.
Close does not mean identical: 243 is 13 patterns short of 256. To preserve every possible byte code in one fixed-length ternary group, you need six trits and an explicit encoding. In Trinary.ai, those six trits make one tryte, with 729 patterns.
NOW FLIP THE COMPARISON
Same positions.
More possibilities.
We just used fewer trits for a similar number of codes. Now give both code locks the same number of positions. Watch how the extra choice in every trit compounds.
1 position8 positions
Binary2⁶
64possible codes
Ternary3⁶
729possible codes
11.39× as many codes, with the same number of digit positions.
The idea scales: unique codes for a million items need 20 bits or 13 trits. The gain is in the number of digit positions needed to carry the information.
WHY THIS MATTERS INSIDE A COMPUTER
Fewer digits to store.
Fewer symbols to move.
Computing means storing information, moving it between memory and the processor, and doing work on it. A machine built around three-state digits can represent the same information with fewer digits.
That creates an opportunity to use fewer storage locations and move fewer symbols. It can help reduce hardware area or energy if the third state is cheap and reliable enough. Changing the way a number is written on a binary computer does not create those hardware gains; the machine itself has to be designed for three states.
02 / THE SWEET SPOT
Why stop at three?
Ask e.
So why add just one more state? Why not ten or a hundred? More choices shrink the digit count, but those choices also have to be built and kept distinguishable. The next question is how much complexity each digit is worth.
2.71828…
EULER’S NUMBER · PRONOUNCED “OY-LER”
Imagine each position on a code lock costs one token for every value it can display. A two-state position costs two tokens. A three-state one costs three.
In this simple model, the most economical base is e ≈ 2.718. You can’t give a digit 2.718 distinct states. Of the whole-number choices, three gives the best result.
THE SAME 12-TOKEN BUDGETILLUSTRATIVE MODEL
Binary
2 · 2 · 2 · 2 · 2 · 2
64 codesTernary
3 · 3 · 3 · 3
81 codesSix digits × 2 tokens, or four digits × 3 tokens. Same budget. More combinations with three.
Three is close to the mathematical sweet spot. That’s the intuition behind the case for ternary.
THE COST OF REPRESENTING INFORMATIONLOWER IS BETTER ↓
0.46%
above the ideal. Three is the most economical whole-number base in this model.
Curious about the math?
To cover N possibilities, base b needs about ln(N) / ln(b) digits. If each digit costs b tokens, total cost is b × ln(N) / ln(b). For the same N, compare b / ln(b). Its minimum is at b = e. The graph shows this cost divided by e; it ignores whole-digit rounding and hardware details.
THE ASSUMPTION MATTERS
This is a model of representation cost. It assumes digit cost grows directly with the number of states. It is not a measurement of chip area, energy or speed. Real devices determine whether the mathematical advantage survives.
03 / NEGATIVE. ZERO. POSITIVE.
Three brings
its own balance.
Fewer digits are one advantage. The choice of −1, 0, +1 brings another: every digit can subtract, do nothing, or add. Here’s how those choices build actual numbers.
Binary place values double: 1, 2, 4, 8… Balanced ternary place values triple: 1, 3, 9… Multiply each digit by its place value, then add the results.
It represents positive and negative numbers with the same set of digits, without a separate minus-sign digit. Binary computers handle signed numbers too, commonly using two’s complement.
In Trinary.ai’s native convention, six balanced trits make one tryte: 729 patterns, representing −364 through +364.
MAKE A NUMBERBALANCED TERNARY
Use − and + to explore from −13 to +13
NINESTHREESONES
+10−1
9 + 0 − 1 = 8
A negative digit subtracts its place value.
Here, eight is simply nine minus one.
04 / MATH MEETS MATTER
So why did
binary win?
We’ve seen the case for three: richer digits, fewer positions and naturally signed numbers. Binary became dominant because two states are straightforward to build reliably—and the industry got extraordinarily good at it.
Two states are easier
to tell apart.
Circuits use ranges of voltages, rather than perfect numbers. Keeping “low” and “high” far apart leaves room for electrical noise. Fit a third level into the same voltage range and the gaps shrink.
Simple switches became
an extraordinary industry.
Two-state logic fits switching circuits well. Decades of investment in manufacturing, tools and software made binary systems exceptionally capable and economical.
A third state has
to earn its place.
Three reliable states can require extra circuitry or tighter control. If their physical cost outweighs the information gained, the theoretical advantage disappears.
THE REAL HARDWARE QUESTIONHow much more can
How much more can
a three-state digit cost?
A trit can carry about 1.585 bits’ worth of information, when its states are equally likely. In a simple storage-only comparison, it wins on area if it costs less than 1.585 times the area of a bit.
Same area · 1×Twice the area · 2×
18.0% lessstorage area for the same information
Illustrative large-storage model. Excludes rounding, error protection, wiring and read/write circuits; says nothing about energy or speed.
A DIFFERENT STARTING POINT
Binary built the present.
Three deserves a future.
Start with a bit and every position gives two choices. Move to a balanced trit and every position gives three. The result is more information per digit, fewer digits for the same job, and a natural way to work with positive and negative numbers.
Euler’s number helps explain why three is a compelling balance between digit count and digit complexity. Trinary.ai is exploring native ternary computing across memory and compute to turn that mathematical opportunity into a practical engineering advantage.
Explore Trinary.aiThe evidence behind the explanation Sources & assumptions
Bits, bytes and programs
Binary machines encode instructions and data as bit patterns. Eight bits form a byte with 256 patterns. Five balanced trits have 243 patterns, so they cannot preserve all byte codes. Six trits have 729 patterns. The eight-versus-five comparison is exact for giving 243 items unique codes.
MIT OpenCourseWare, Designing an Instruction Set ↗NIST, Byte ↗Information per digit
A digit with b equally likely states carries log₂(b) bits of information. For three states, that’s approximately 1.585. Six positions have 2⁶ = 64 or 3⁶ = 729 patterns. A million possibilities need ceil(log₂ 1,000,000) = 20 bits or ceil(log₃ 1,000,000) = 13 trits.
Shannon, A Mathematical Theory of Communication (1948) ↗Euler’s number and the cost model
The continuous minimum of b / ln(b) is e. Among integer bases b ≥ 2, base 3 is the minimum. Compared with binary, ternary uses about 5.4% fewer idealized cost units in this model. This is separate from the 36.9% reduction in digit count for large representations.
UC San Diego, CSE 20, lecture 1, pp. 18–20 ↗Physical signals and reliability
Discrete symbols are implemented with ranges of physical signals. Noise margins and circuit costs matter. None of the examples on this page establishes measured performance for a Trinary.ai device or a complete native software stack.
MIT OpenCourseWare, The Digital Abstraction ↗