The Babylonians used separate combinations of two symbols to represent every single number from 1 to 59. That sounds pretty confusing, doesn’t it? Our decimal system seems simple by comparison, with just 10 digits from 0 to 9 to express every number imaginable.
Computers need only two digits: 0 and 1. But that binary system is not the only option for these machines. In the past, experts developed calculating machines that worked with three digits, a ternary system, that they hoped would allow for more efficient information processing. Yet today ternary computers are just hobby projects. How did this come to be?
In principle, any number can be represented by any number system, whether the latter be base 10, base 60, base 3 or base 2. The math works perfectly in each case.
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In the usual decimal system, the number 17 (that is, a 1 followed by a 7) indicates that you have to calculate 10 + 7 × 1 (17 = 1 × 10 + 7 × 1).
If you want to express 17 in base 3, it looks like this: 1710 = 1 × 32 + 2 × 31 + 2 × 30 = 1223 . (The lower numbers symbolize the base that is used.) In binary notation, the number is 1710 = 1 × 24 + 0 × 23 + 0 × 22 + 0 × 21 + 1 × 20 = 10,0012.
Comparing these numbers, the familiar decimal notation is the most efficient by far: only two digits are needed to write down 17, which could be advantageous for a computer. On the other hand, the machine would have to work with a base of 10 different digits, which is problematic in practice.
In conventional computers, the 0’s and 1’s are represented by the state of a particular piece of electrical hardware, such as a capacitor or transistor. Many of these devices simply translate a state—such as turning the transistor on or off—into either a 0 or 1 in a way that does not support the gradations or degrees that one would require for base 10 computing. Imagine designing a switch that had 10 different positions between the on and off state—this approach would get complicated quickly.
For practical reasons, decimal notation can therefore be ruled out as a basis for a computer. There are too many digits. But is there a better number system for information processing than the binary system that is ubiquitous today?
The Perfect Basis for Computers
To answer that question, you have to find a compromise between the representation length l of a number n and the number b of digits used in a base. To do this, you can look at the product of the two quantities, b × l, and ask yourself for which base this is the smallest. The length of a number n represented by the base b corresponds approximately to the quotient log(n) / log(b). The question is therefore: For which base b is the product b × log(n) / log(b) smallest?
If you remember your school days, you may still know how to calculate the optimum value of b: You derive the named product in terms of b, set it equal to zero and then solve the equation for b. Alternatively, you can draw the equation—that is y = b × [log(n) / log(b)]—on a piece of paper and identify the point on the curve that is closest to the x-axis. Whichever method you choose, assuming you don’t make a mistake, the optimum value for b = e ≈ 2.718, Euler’s number.
An irrational result may seem confusing at first glance. How can you construct a number system for a base that is irrational? If you consider all numbers, including irrational values such as pi (π ), then a number system with base e may have advantages. But if you want to represent integers such as 1, 2 or 3 in this way, it quickly becomes complicated. That’s why it makes sense to round Euler’s number: so among the integers, 3 seems to be the optimal base to process information.
Ternary System: In Perfect Balance
A number system with three digits such as 0, 1 and 2 is known as a ternary system. This, however, can also be represented in what mathematicians consider a more balanced way using the three numbers –1, 0 and 1.
In this “balanced ternary” system, the number 1710 looks like this:
1710 = 1 × 33 + (–1) × 32 + 0 × 31 + (–1) × 30 = 1(–1)0(–1)
Experts found the balanced ternary system particularly attractive because of its symmetry. For example, in the second volume of his book series The Art of Computer Programming, computer scientist Donald E. Knuth described it as “the prettiest number system of all.”
The ternary system is not only inspiring in theory. In 1840 English inventor Thomas Fowler built a calculating machine that worked with the balanced ternary system, a mechanical computer that calculates with the numbers –1, 0 and 1. It used a different logic than today’s computers. If you combine two digits in the ternary system, you can not only get 0 or 1 (which are often equated with true or false in the binary system) but also a third result.
Some calculations can be shortened as a result: for example, if you want to compare two numbers with each other, you can use the ternary system to find out in just one step whether one number is smaller, larger or equal to the other. In the binary system, on the other hand, two calculation steps are required: first you check whether they are different and then which one is larger or smaller.
A Ternary Computer behind the Iron Curtain
Fowler’s mechanical device was not the only computer to count to three. At the beginning of the cold war, the Soviet Union tried to develop its first electronic computers. Because it was very difficult to get hold of transistors (the electronic components on which conventional computers are based), the Soviets looked for other options to achieve their goal.
In 1958 the first electronic ternary computer, called Setun, was built at Moscow State University. It used magnetic cores and diodes to process information in ternary digits, or “trits.” Over the years, around 50 Setun computers were produced.
But ternary computers did not catch on, in part because of the hardware and existing conventions. It is difficult to code electronic components with three different states. With Setun, the researchers needed to use two magnetic components per trit—but had they worked in binary, they could have encoded twice as many bits with these components.
All computers now work with transistors. These have two inputs through which either current can flow (encoding a 1) or not (a 0) and an output that either passes on current (a 1) or not (a 0). By cleverly connecting transistors together, logic gates can be built to perform all calculable operations.
Many hobbyists develop ternary computers—but these are entirely for fun. Because the ternary machines and binary computers process completely different digits and use different logic, the two cannot be connected. That is somehow a pity, I think, even if a ternary computer could do no more than conventional devices.
This article originally appeared in Spektrum der Wissenschaft and was reproduced with permission.