Main events in the history of balanced ternary number system
The idea of number systems with either positive and negative digits had very venerable age. Some historians found its roots in ancient Indian Vedic mathematics. For example, vinculum was used to reduce single digits larger than 5. Because 19=20-1 and 87=100-13, we can wrote 21 instead 19, 113 instead 87 and so on. Vinculum representation is very useful in Vedic multiplication and division methods.
The best-known application of balanced ternary notation is in mathematical puzzles that have to do with weighing.
|
For example the task is to weigh load. It’s mass is 33 kg. Let’s represent decimal number 33 in balanced ternary notation: 3310 = 27 + 9 - 3 + 0 = 111103. It means that two weights (27 kg and 9 kg) should be placed on the one scale, and load with weight (3 kg) should be placed on the other. If it possible to use one more weight (1 kg), we can measure any load which mass is less that 40 kg. Practically, for task solving representation should be found in balanced ternary notation of integer notation, equal to load mass.
|
It is considered, that the famous French mathematician Claude Gaspar Bachet de Meziriac (1581-1638) was the first who proposed such a puzzle ñ1612. In fact Leonardo di Pisa Fibonacci (c1170-c1250) knew this puzzle few centuries earlier.
There are some evidences that the great astronomer and mathematician Johannes Kepler (1571-1630) also implicitly used a balanced ternary scheme for some purposes.
John Colson (1680-1760), who became the fifth Lucasian Professor of Mathematics at Cambridge University in 1739, published in 1726 brief essay on "negativo-affirmative arithmetick" [1], in which he at first investigated number systems with either negative and positive digits.
Sir John Leslie (1766-1832) in his Philosophy of Arithmetic [2] proposed some methods of calculating in any base with either signed or unsigned digits.
In 1840 the great French mathematician Augustin Louis Cauchy (1789-1857) discussed signed-digit numbers in various bases [3].
The same year his compatriot, well-known engineer, inventor and mathematician Leon Louis Chretien Lalanne (1811-1892) investigated the special properties of balanced ternary system [4].
Only 100 years later ternary system attracted the notice of early computer designers. In 1950, Claude E. Shannon (1916-2001) published an account of symmetrical signed-digit systems, including ternary and other bases [5].
The earliest published discussion of this idea appeared in the book High-speed Computing Devices, a survey of computer technologies compiled by the staff of Engineering Research Associates [6].
At about the same time Herbert R. J. Grosch (b. 1918) proposed a ternary architecture for the Whirlwind computer project at MIT. Unfortunately ternary arithmetic was not tested in Whirlwind computer, which lately became the core of control system for a military radar network. Forty years later Grosh wrote in his memoires:
|
A world industry was to grow up around core memory. Forrester assigned his patents to MIT, which got rich from licenses granted to a hundred companies, including IBM. And Jay himself did very well, and richly deserved it. At the time we talked at the JCC, he was considering going to three dimensions, relying on a touchier ratio: that two coincident pulses would cause no response, but three would flip the polarity. I reminded him that the real payoff was how many cores he could pack into a cubic foot, that a 3D structure would be much harder to wire either by hand or mechanically, and that marginal cores would be more frequent.
He already knew all that, of course, but it got us talking. I said what might be genuinely gainful would be to store a ternary digit in each core, and calculate in base-three rather than binary fashion. There were materials – some kinds of permalloy, as I remember – that had north, south and neutral stable magnetic states. I told him I had taught my Poughkeepsie evening classes at IBM about a special kind of base-three arithmetic I called "signed ternary", in which zero was in the middle of the number range. In this curious system there was no need for algebraic signs, no problem about the sign of zero, and you rounded perfectly by dropping digits.
Jay being a stiff type, I refrained from calling the ternary digits "tits", a name which had been the source of much boyish amusement in the Poughkeepsie classes.
Twenty years later I was utterly amazed to be asked to comment on a technical letter from two Argentines, who as graduate students at MIT had exhumed the report I had done for Forrester on base-three computing, and were proposing to try it with hardware. It would have been a great idea if something as tiny and as cheap as the core-memory rings could have been made with three stable states; the embodiment with two cores per tit which they proposed was not attractive. And somebody in Kiev had tried it, anyhow [7].
|
The last words of Herb Grosch are mistake – it was Nickolay P. Brusentsov from the Moscow State University who not only "had tried it, anyhow", but really build the first ternary computer Setun' in 1959.
In 1973 Gideon Frieder and his colleagues at the State University of New York described 3-base computer TERNAC, and created a software emulator of it. It seems that it was the last attempt to exploit the idea of ternary computing [8, 9].
"Perhaps the prettiest number system of all is the balanced ternary notation", writes famous scientist Donald E. Knuth in his The Art of Computer Programming [10].
We were inspired by ternary number system as well as Mr. Knuth, so we used our imagination. You can see the result of our creative search.
Literature
1. Colson, John. A short account of negativo-affirmative arithmetick // Philosophical Transactions of the Royal Society of London. 34. 1726. P. 161-173.
2. Leslie, John. The Philosophy of Arithmetic, Exhibiting a Progressive View of the Theory and Practice of Calculation, with Tables for the Multiplication of Numbers as Far as One Thousand. Edinburgh: William and Charles Tait, 1820.
3. Cauchy, Augustin. Sur les moyens d'eviter les erreurs dans les calculs numeriques // Comptes rendus hebdomadaires des seances de l'Academie des sciences. 11. 1840. P. 789-798.
4. Lalanne, Leon. Note sur quelques propositions d'arithmologie elementaire // Comptes rendus hebdomadaires des seances de l'Academie des sciences. 11. 1840. P. 903-905.
5. Shannon, Claude E. A symmetrical notation for numbers // The Americal mathematical monthly. Vol. 57. N 2. Feb. 1950. P. 90-93.
6. High Speed Computing Devices. Engineering Research Associates. N.-Y.: McGraw Hill, 1950. P. 287-289.
7. Grosch, Herbert R. J. Computer: Bit Slices From a Life. Novato, CA: Third Millennium Books, 1991. P. 151-152.
8. Knuth, Donald E. The Art of Computer Programming. Vol. 2: Seminumerical Algorithms. Second edition. Reading, Mass: Addison-Wesley, 1981. P. 190-193.
9. Frieder G., Fong A., Chow C. Y. A balancedternary computer // Conference Record of the 1973 International Symposium on Multiple-valued Logic, 1973. P. 68-88.
10. Epstein G., Frieder G., Rine D. Ñ. The development of multiple-valued logic as related to computer science // Computer. Vol. 7. N 9. September 1974. P. 20-32.
See also:
Gardner, Martin. The "tyranny of 10" overthrown with the ternary number system // Scientific American. 210(5). 1964. P. 118-124.
