The ancient Roman computer
I was recently assembling a slide deck on assistive technology and the sort of gradient that exists between assistive devices and human enhancement. We already have the technology to build exoskeletons that can give their user increased strength, or allow them to work longer with less fatigue (albeit only in specific types of tasks). These are physical enhancements, the same way that e.g. reading glasses on a person with normal vision act as magnifiers, and give them better close-in vision. The existence of physical enhancements begs the question of whether we can also have cognitive enhancements.
Donald Norman describes the use of Arabic numerals as a “cognitive artifact”, a tool that we use because it makes mathematical operations easier. One alternative in Western civilization is the Roman numeral system, which uses letters to represent values. The Roman numerals are I (1), V (5), X(10), L(50), C (100), D (500), and M (1000). So far so good, but Roman numerals are not a place-based system. Instead, the value of a number is a summation based on rules:
- Read from left to right, summing up values.
- Iif the number you are looking at is bigger than the number to its right, add it to the total.
- If the number you are looking at is smaller than the number to its right, subtract the number you’re looking at from the one to its right and add that.
So VI is 6 (5 + 1), but IV is 4 (5 -1), and VC is 95. The current year is MMXVII (1000 + 1000 + 10 + 5 + 1 + 1), which is OK to decode, but 1998 was MCMXCVIII or 1000 + (1000 – 100) + (100 – 10) + 5 + 1 + 1 + 1. Looking at the Arabic expansion, the -100 and +100 cancel out, so it could be reduced to 1000 + 1000 – 10 + 5 + 1 + 1 + 1, so MXMVIII. There’s a trick to make addition of Roman numerals easier, which Norman describes, but doing multiplication is something of a nightmare.
I recently got to thinking about writing a Roman numeral library for a programming language, so that you could put this sort of thing in a program. This has no doubt been done already, so I’m not going to redo it, but my first thought was that I’d just write a translation from Roman to Arabic numerals (which pretty much all programming languages use already) and back, and then I’d do the math in Arabic numerals. That’s not a Roman numeral library, though. That’s an Arabic library with a Roman presentation layer.
Unfortunately, on most computer architectures, there’s no other way to do it. The binary ALU in processors is a place-value based system. It’s essentially Arabic numerals for people with one finger, where you carry to the next higher place as soon as you have more than one of something. An implementation of Roman numerals on pretty much any CPU will be turned into an Arabic-like representation before it gets operated on.
This got me thinking about what it would be like to attempt to build an ALU, or a simple handheld four-function calculator, that operated in Roman numerals natively. That is to say, the voltages in the various “bits” would represent I, V, X, L, C, D, and M, instead of a place-based binary representation. This immediately runs into a LOT of problems. For starters, the cool thing about binary is that you operate the transistors in switching mode, either on or off, rather than linear mode. Current either flows with minimal resistance, or doesn’t flow. In linear mode, to get multiple voltages, the transistors act as resistors, dissipating some of the power as heat. After that, you have to deal with the context-based values, rather than place-based values. Imagine that we have an adder, with two inputs, A and B, and we feed it A = I, B = V. Clearly, the output has to be, uh… well, it has to be 4, but there isn’t a numeral for that. There’s no single-digit Roman numeral that is 4. But lets say that we don’t care about not having a representation, and that this is just an analog computer. And so that our heads don’t explode trying to keep things straight, lets say that the voltage for ‘4’ is a little less than the voltage for V, but 4 times the voltage for I. This is nice because the voltage for IIII is the same as the voltage for IV. So far so good, but now we give our adder A = V, B = I, and it has to put out ‘6’, which we also don’t have a representation for. We’re clever, it’s analog, the voltage that represents ‘6’ is the voltage for ‘V’ plus the voltage for ‘I’. So if A > B, the output is A + B, if A < B, the output is A-B, and if A = B, the output is A + B (VV is 10, I suppose, but so is X). This is all doable, I’m sure, probably with op-amps, some of which would be acting as comparators, but it’s complex. More complexity is more transistors and components, and so more waste heat and chances to get the wiring wrong.
Annoyingly, that’s not the end of our problems. In the last couple of paragraphs, there are some ambiguous representations of numbers. VV is 10, but so is X. In Arabic numerals, 9 is 9, and 9 is the only digit that’s 9. Arabic also has the nice property that you can guess about how big the representation of the result is, based on the size of the operands. There’s no way to add a 4 digit number and a four digit number (assuming they’re both positive), and get a 2 digit number, or a 12 digit number. Adding II to MXMVIII gets you XX. Multiply two Arabic numbers, and the result will be about the length of the two numbers stuck end to end, +/-1. Division will be the difference in length of the numbers (I’m handwaving here and ignoring fractions). Multiply two Roman numerals, and your guess is as good as mine (As Norman points out, the written length of an Arabic numeral is also a rough gauge of it’s value. 1999 (length 4) is bigger than 322 (length 3), but MXMVIII (length 7) is less than MM (length 2)). There may be some regularities, but I’m not about to sit down and work them out. If anyone does, the bound on the Roman computer word size would be whatever number is longest and less than whatever size of MACHINE_MAX_INT you feel like implementing in the silicon space and power-hungry representation that the machine uses. As a result, a natively Roman calculator would potentially have very long “words” or “bytes”, much of which would go unused most of the time.
Roman numerals also didn’t have a standard representation for fractions, although they apparently sometimes used a duodecimal system for them (yes, two different number systems, one for integers and one for fractions). Roman numerals also had no formal representation at all for zero, although the Romans clearly understood the idea of zero (You have ten dinars, and you owe me ten dinars. I collect your debt, how many tunics can you buy now?). They used “nulla” or “nihil” sometimes, and represented it with an N, but they wouldn’t have written 500 as VNN. They also didn’t represent negative numbers at all, although there were probably cases where they understood a subtraction to result in something that wasn’t even zero (You have 10 dinars and owe me 15 dinars. I collect your debt. Clearly you don’t have -5 dinars, but I also clearly don’t consider myself fully repaid).
I’ve heard that the Roman conception of math and numbers was highly geometric, rather than arithmetical, so they did have things like the square root of 2, which is the length of the diagonal of a unit square (you know, a square with sides of length I). Pythagoras figured that one out, at roughly the same time people were using “nulla”. Instead of running on abstractions of the representations of numbers, a properly Native Roman Computer might have been a mechanical device, a calculator that operated by using finely crafted physical representations of circles and triangles to perform its operations. Interestingly, the closest thing we have to a calculator from that time period is exactly that.