Binary vs Ternary Computing

Overview

Binary is not the theoretically optimal choice for information representation—base-3 (ternary) is more efficient mathematically. Yet binary dominates all modern computing. This apparent paradox reveals a fascinating tension between mathematical optimality and physical practicality.

Source

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Key Concept

Mathematical optimality favors ternary; physical reality favors binary.

The optimal base for information representation is Euler’s number $e\approx 2.718$. For integer bases, base-3 achieves the minimum value of the radix economy function $\frac{b}{\ln (b)}$, making it theoretically most efficient by 5-6% over binary.

Yet binary dominates because physical switching devices naturally have two stable states, and reliably distinguishing two states in noisy environments is far easier than distinguishing three.

Details

Radix Economy

When measuring efficiency as the product of base $b$ times the number of digits needed:

Base	Efficiency	Notes
$e$	0.368	Theoretical maximum (non-integer)
3	0.366	Most efficient integer base
2	0.347	Ties with base-4

Two ternary “trits” can represent 9 different values; two binary bits only 4. A number requiring 42 bits needs only 27 trits.

Why Physics Favors Binary

Transistor Physics: Transistors naturally operate as switches with two stable states—“on” (conducting) and “off” (non-conducting). Creating three-state devices requires distinguishing three voltage levels.

Noise Margins: Binary systems define:

• Logical “0” as 0-20% of supply voltage
• Logical “1” as 80-100% of supply voltage
• 60% noise margin between states

Ternary must divide voltage into thirds:

• Binary inverters: 389.5mV noise margins
• Ternary inverters: 150mV noise margins

Error Correction: Checking two states is straightforward; distinguishing three states reliably under thermal noise, voltage fluctuations, and electromagnetic interference is exponentially harder.

Historical Precedent

The Soviet SETUN computer (1958) successfully implemented ternary logic, proving feasibility. However, the practical advantages of binary’s noise immunity, combined with path dependency and network effects, cemented binary dominance.

Implications

1. Engineering trumps mathematics when physical constraints dominate
2. Noise immunity is often more valuable than theoretical efficiency
3. Binary’s naturalness may reflect deeper physics (quantum measurement)
4. Quantum computing might eventually transcend this tradeoff

The Deeper Question

Binary’s dominance may reflect quantum mechanics’ fundamental yes/no, measured/unmeasured structure. Quantum measurements project superposition states onto eigenstates, yielding binary outcomes. Perhaps binary encoding works well because it reflects the actual quantum substrate of reality.

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Appendix

Created: 2024-12-31 | Modified: 2024-12-31

See Also

• Definition - Information Theory
• Definition - Margolus-Levitin Theorem
• Definition - It From Bit

Related MOCs

• MOC - Computer Science - Theoretical computation foundations
• MOC - Information Theory - Information theory and physics
• MOC - Physics - Physical constraints on computing

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