The widespread and persistent myth that it is easier to multiply and divide with Hindu-Arabic numerals than with Roman ones.

Last Sunday the eminent British historian of the twentieth century, Richard Evans, tweeted the following:

Let’s remember we use Arabic numerals – 1, 2, 3 etc. Try dividing MCMLXVI by XXXIX ­– Sir Richard Evans (@Richard Evans36)

There was no context to the tweet, a reply or whatever, so I can only assume that he was offering a defence of Islamic or Muslim culture against the widespread current attacks by drawing attention to the fact that we appropriated our number system along with much else from that culture. I would point out, as I have already done in my nineteenth-century style over long title, that one should call them Hindu-Arabic numerals, as although we appropriated them from the Islamic Empire, they in turn had appropriated them from the Indians, who created them.

As the title suggests, in his tweet Evans is actually guilty of perpetuating a widespread and very persistent myth concerning the comparative utility of the Hindu-Arabic number system and the Roman one when carrying out basic arithmetical calculations. Although I have taken Professor Evans’ tweet as incentive to write this post, I have thought about doing so on many occasions in the past when reading numerous similar comments. Before proving Professor Evans wrong I will make some general comments about the various types of number system that have been used historically.

Our Hindu-Arabic number system is a place-value decimal number system, which means that the numerals used take on different values depending on their position within a given number if I write the Number of the Beast, 666, the three sixes each represents a different value. The six on the far right stands for six times one, i.e. six, its immediate neighbour on the left stands for six time ten, i.e. sixty, and the six on the left stands for six times one hundred, i.e. six hundred, so our whole number is six hundred and sixty six. It is a decimal (i.e. ten) system going from right to left the first numeral is a multiple of 100 (for those who maths is a little rusty, anything to the power of zero is one), the second numeral is a multiple of 101, the third is a multiple of 102, the forth is a multiple of 103, the fifth is a multiple of 104, and so on and so fourth. If we have a decimal point the first numeral to the right of it is 10-1 (i.e. one tenth), the second 10-2 (i.e. one hundredth), the third 10-3 (i.e. one thousandth), and so on and so forth. This is a very powerful system of writing numbers because it comes out with just ten numerals, one to nine and zero making it very economical to write.

The Hindu-Arabic number system developed sometime in the early centuries CE and our first written account of it is from the Indian mathematician, Brahmagupta, in his Brāhmasphuṭasiddhānta (“Correctly Established Doctrine of Brahma“) written c. 628 CE. It came into Europe via Al-Khwārizmī’s treatise, On the Calculation with Hindu Numerals from 825 CE, which only survives in the 12th-century Latin translation, Algoritmi de numero Indorum. After its initial introduction into Europe in the high Middle Ages the Hindu-Arabic system was only really used on the universities to carry out computos, that is the calculation of the dates on which Easter falls. Various medieval scholars such as Robert Grosseteste John of Sacrobosco wrote elementary textbooks explaining the Hindu-Arabic system and how to use it. The system was reintroduced for trading purposes by Leonard of Pisa, who had learnt it trading with Arabs in Spain, in his book the Liber Abbaci in the thirteenth century but didn’t really take off until the introduction of double-entry bookkeeping in the fourteenth century.

The Hindu-Arabic system was not the earliest place-value number system. That honour goes to the Babylonians, who developed a place-value system about 1700 2100* BCE but was not a decimal system but a sexagesimal system, that is base sixty, so the first numeral is a multiple of 600, the second a multiple of 601, the third a multiple of 602, and so on and so fourth. Fractions work the same, sixtieths, three thousand six-hundredths (!), and so on and so fourth. Mathematically a base sixty system is in some senses superior to a base ten one. The Babylonian system suffered from the problem that it did not have distinct numerals but a stroke list system with two symbols, one for individual stroke and a second one for ten stokes:

Babylonian Numerals Source: Wikimedia Commons

Babylonian Numerals
Source: Wikimedia Commons

The Babylonian system also initially suffered from the fact that it possessed no zero. This meant that to take the simple case, apart from context there was no way of knowing if a single stroke stood for one, sixty, three thousand six hundred or whatever. The problem gets even more difficult for more complex numbers. Later the Babylonians developed a symbol for zero. However the Babylonian zero was just a placeholder and not a number as in the Hindu-Arabic system.

The Babylonian sexagesimal system is the reason why we have sixty minutes in an hour, sixty seconds in a minute, sixty minutes in a degree and so forth. It is not however, contrary to a widespread belief the reason for the three hundred and sixty degrees in a circle; this comes from the Egyptian solar years of twelve thirty day months projected on to the ecliptic, a division that the Babylonian then took over from the Egyptians.

The Greeks used letters for numbers. For this purpose the Greek alphabet was extended to twenty-seven letters. The first nine letters represented the numbers one to nine, the next nine the multiples of ten from ten to ninety and the last nine the hundreds from one hundred to nine hundred. For the thousands they started again with alpha, beta etc. but with a superscript subscript prime mark. So twice through the alphabet takes you to nine hundred thousand nine hundred and ninety-nine. If you need to go further you start at the beginning again with two subscript primes. Interestingly the Greek astronomers continued to use the Babylonian sexagesimal system, a tradition in the astronomy that continued in Europe down to the Renaissance.

We now turn to the Romans, who also have a simple stroke number system with a cancelled stroke forming an X as a bundle of ten strokes. The X halved horizontally through the middle gives a V for a bundle of five. As should be well known L stands for a bundle of fifty, C for a bundle of one hundred and M for a bundle of one thousand given us the well known Roman numerals. A lower symbol placed before a higher one reduces it by one, so LX is sixty but XL is forty. Of interest is the well-known IV instead of IIII for four was first introduced in the Middle Ages. The year of my birth 1951 becomes in Roman numerals MCMLI.

When compared with the Hindu-Arabic number system the Greek and Roman systems seem to be cumbersome and the implied sneer in Professor Evans’ tweet seems justified. However there are two important points that have to be taken into consideration before forming a judgement about the relative merits of the systems. Firstly up till the Early Modern period almost all arithmetic was carried out using a counting-board or abacus, which with its columns for the counters is basically a physical representation of a place value number system.

Rechentisch/Counting board (engraving probably from Strasbourg) Source: Wikimedia Commons

Rechentisch/Counting board (engraving probably from Strasbourg)
Source: Wikimedia Commons

The oldest surviving counting board dates back to about 300 BCE and they were still in use in the seventeenth century.

An early photograph of the Salamis Tablet, 1899. The original is marble and is held by the National Museum of Epigraphy, in Athens. Source: Wikimedia Commons

An early photograph of the Salamis Tablet, 1899. The original is marble and is held by the National Museum of Epigraphy, in Athens.
Source: Wikimedia Commons

A skilful counting-board operator can not only add and subtract but can also multiply and divide and even extract square roots using his board so he has no need to do written calculation. He just needed to record the final results. The Romans even had a small hand abacus or as we would say a pocket calculator. The words to calculate, calculus and calculator all come from the Latin calculi, which were the small pebbles used as counters on the counting board. In antiquity it was also common practice to create a counting-board in a sand tray by simply making parallel groves in the sand with ones fingers.

A reconstruction of a Roman hand abacus, made by the RGZ Museum in Mainz, 1977. The original is bronze and is held by the Bibliothèque nationale de France, in Paris. This example is, confusingly, missing many counter beads. Source: Wikimedia Commons

A reconstruction of a Roman hand abacus, made by the RGZ Museum in Mainz, 1977. The original is bronze and is held by the Bibliothèque nationale de France, in Paris. This example is, confusingly, missing many counter beads.
Source: Wikimedia Commons

Moving away from the counting-board to written calculations it would at first appear that Professor Evans is correct and that multiplication and division are both much simpler with our Hindu-Arabic number system than with the Roman one but this is because we are guilty of presentism. In order to do long multiplication or long division we use algorithms that most of us spent a long time learning, often rather painfully, in primary school and we assume that one would use the same algorithms to carry out the same tasks with Roman numerals, one wouldn’t. The algorithms that we use are by no means the only ones for use with the Hindu-Arabic number system and I wrote a blog post long ago explaining one that was in use in the early modern period. The post also contains links to the original post at Ptak Science books that provoked my post and to a blog with lots of different arithmetical algorithms. My friend Pat Belew also has an old blog post on the topic.

I’m now going to give a couple of simple examples of long multiplication and long division both in the Hindu-Arabic number system using algorithms I learnt I school and them the same examples using the correct algorithms for Roman numerals. You might be surprised at which is actually easier.

 

My example is 125×37

125

   37

875 Here we have multiplied the top row by 7

3750 Here we have multiplied the top row by 3 and 10

4625 We now add our two partial results together to obtain our final result.

To carry out this multiplication we need to know our times table up to nine times nine.

Now we divide 4625 : 125

4625 : 125 = 37

375

875

  875

  000

First we guestimate how often 125 goes into 462 and guess three times and write down our three. We then multiply 125 by three and subtract the result from 462 giving us 87. We then “bring down” the 5 giving us 875 and once again guestimate how oft 125 goes into this, we guess seven times, write down our seven, multiply 125 by 7 and subtract the result from our 875 leaving zero. Thus our answer is, as we already knew 37. Not exactly the simplest process in the world.

 

How do we do the same with CXXV times XXXVII? The algorithm we use comes from the Papyrus Rhind an ancient Egyptian maths textbook dating from around 1650 BCE and is now known as halving and doubling because that is literally all one does. The Egyptian number system is basically the same as the Roman one, strokes and bundles, with different symbols. We set up our numbers in two columns. The left hand number is continually halved down to one, simple ignoring remainders of one and the right hand is continually doubled.

1 XXXVII CXXV
2 XVIII CCXXXXVV=CCL
3 VIIII CCCCLL=CCCCC=D
4 IIII DD=M
5 II MM
6 I MMMM

You now add the results from the right hand column leaving out those where the number on the left is even i.e. rows 2, 4 and 5. So we have CXXV + D + MMMM = MMMMDCXXV. All we need to carry out the multiplication is the ability to multiply and divide by two! Somewhat simpler than the same operation in the Hindu-Arabic number system!

Division works by an analogous algorithm. So now to divide 4625 by 125 or MMMMDCXXV by CXXV

1 I CXXV
2 II CCXXXXVV=CCL
3 IIII CCCCLL=CCCCC=D
4 IIIIIIII=VIII DD=M
5 VVIIIIII=XVI MM
6 XXVVII=XXXII MMMM

We start with 1 on the left and 125 on the right and keep doubling both until we reach a number on the right that when doubled would be greater than MMMMDCXXV. We then add up those numbers on the left whose sum on the right equal MMMMDCXXV, i.e. rows 1, 3 and 6, giving us I+IIII+XXXII = XXXIIIIIII = XXXVII or 37.

Having explained the method we will now approach Professor Evan’s challenge

1 I XXXIX=XXXVIIII
2 II XXXXXXVVIIIIIIII=LXXVIII
3 IIII LLXXXXVVIIIIII=CLVI
4 IIIIIIII=VIII CCLLVVII=CCCXII
5 VVIIIIII=XVI CCCCCCXXII=DCXXIIII
6 XXVVII=XXXII DDCCXXXXIIIIIIII=MCCXLVIII

Adding rows 6, 3 and 2 on the right we get MCCXLVIII+CLVI+LXXVIII=MCML i.e. MCMLXVI less XVI so our result is XXXII+XVI+II = L remainder XVI

6 + 5 + 2 = MCCXLVIII+DCXXIIII+LXXVIII = 1950 + 16(reminder) is the correct value for the given example (MCMLXVI) Thanks to Lucas (see Comments!)

Now that wasn’t that hard was it?

Interestingly the ancient Egyptian halving and doubling algorithms for multiplication and division are, in somewhat modified form, how modern computers carry out these arithmetical operations.

* Added 13 February 2017: I have been criticised on Twitter, certainly correctly, by Eleanor Robson, a leading expert on Cuneiform mathematics, for what she calls a sloppy and outdated account of the sexagesimal number system.  For those who would like a more up to date and anything but sloppy account then I suggest they read Eleanor Robson’s (not cheap) Mathematics in Ancient IraqA Social History, Princeton University Press, 2008

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41 Comments

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41 responses to “The widespread and persistent myth that it is easier to multiply and divide with Hindu-Arabic numerals than with Roman ones.

  1. Definitely one of the perpetual questions one gets as a math historian (not quite as popular as ‘who invented the zero?’) that become a lesson about presentism and path dependence. I quite like your way of explaining it; thanks for writing it up! You might have seen philosopher Dirk Schlimm’s work on numeral systems and arithmetic,… apparently one can extract a lot of interesting philosophy from this very question.

  2. I think this is a little unfair on the ‘placeholders are useful’ argument. Your Roman multiplication did appear to have twice as many operations. And now lets see the Roman multiplication of 1.532×10^23 by 4.247×10^44…

    … And I HATE WordPress. It keeps trying to get me to sign into an old WordPress account I don’t use anymore. Stop it, WordPress!

    • Ah, but the operations in the Egyptian algorithm are much more simple 😉

      • Several comments:
        Your Egyptian algorithm is also known as Russian Peasant multiplication:
        http://www.wikihow.com/Multiply-Using-the-Russian-Peasant-Method

        If you are calculating using an abacus the representation you use for the numbers is only needed at the start and finish. Skilled abacus calculators are (were?) very fast:
        http://www.ee.ryerson.ca/~elf/abacus/abacus-contest.html

        There is a trick that you can use for division: write down the multiples of the divisor below each other e.g. 125/250/375/500/…/1000/1125 before starting. This is a straightforward addition of 125 to the number above in the list and removes the need to guess. You don’t need it for simple division like this, but it becomes more useful the more digits that the divisor has.

        @Philip Helbig
        I don’t think that IC is a valid Roman numeral; 99 should be XCIX.
        https://www.math.nmsu.edu/~pmorandi/math111f01/RomanNumerals.html

      • You of course can’t do decimal fractions with any of the Greek, Roman or Egyptian number systems, which is one reason why astronomers retained the Babylonian sexagesimal number system right down to the early modern period. However it should be noted that the Europeans didn’t really work out how to do decimal fractions until the early seventeenth century, Simon Stevin De Thinned, 1585. Multiplying by ten is of course trivial whether you have one or forty-four zeros!

  3. This raises the question: Why, then, did Hindu-Arabic numerals overtake the older systems in Europe, in a culture very well versed in abacus methods? Surely there must have been some advantage to the switch: people would not have undergone such an elaborate retraining for nothing. I agree that in basic arithmetic there is no advantage to speak of, but I believe the Hindu-Arabic numerals have advantages for more advanced arithmetic——not least for square root extraction, which was also a rather important standard operation back in the day.

    • “Why, then, did Hindu-Arabic numerals overtake the older systems in Europe?”

      The answer is in the post if somewhat hidden. It’s when people started writing numbers extensively i.e. with the introduction of double
      -entry bookkeeping. If we take a relatively simply number like 1988 in Roman numerals it becomes MDCCCLXXXVIII. The first advantage of Hindu-Arabic numerals become immediately obvious.

      Also as indirectly pointed out by Brian Clegg in his snarky comment, you can’t do decimal fractions in Roman numerals, which is why my version of Richards Evans’ challenge ends with ‘remainder 16’. This is also the reason why astronomers retained the Babylonian sexagesimal number system up to the sixteenth/seventeenth century.

      Even with the Hindu-Arabic number system the Europeans didn’t really work out how to do decimal fractions until the early seventeenth century.

      • Indeed the economy of Hindu-Arabic numerals for writing big numbers has much to commend it. Fibonacci introduced his famous series to make precisely this point, if I’m not mistaken.

  4. You’ve kinda left out a step of explanation, which is how you halve e.g. “XXXVII”. I would do it by converting it to decimal and halving that and then reconverting, but is the algorithm something like “convert all double symbols into singles, then take left-over ones (like the third X. or the V) and split them into things that are smaller and then combine with others?” That doesn’t seem terribly simple.

    • Remember not to confuse numbers with notation! You can pronounce XXVII as thirty seven or septem et trigenta or trente-sept or šalašā u sebû (all of which mean 30 + 7), it does not matter. To do any division and multiplication, you need to memorize a table of conversions: either half 37 is 18 remainder one, or VII is twice III remainder one. You can write the results down in any notation you know, but you carry out the operation on abstract numbers in your head.

    • It’s really easy to halve Roman numerals

      To halve XXXVII you simply have the number of symbols you don’t even have to know what the number actually is.

      Half of XXX is X with one X left over equals twoVs

      Half of VVV is V with one V left over equals fine Is

      Have of IIIIIII is III with I left over for this algorithm simply discarded

      So half of XXXVII is XVIII

      Doubling is equally trivial. You simply double the number of each symbol and regroup as necessary.

      So XXXVII doubled is XXXXXXVVIIII equals LXXIIII

  5. “three thousand and sixtieths”

    3060 or 3600?

  6. “he X halved horizontally through the middle gives a V for a bundle of five.”

    We might never know, of course, but I always thought that V is a hand with five digits: a thumb on one side and the other four on the other (like in a mitten). And X represents two arms (of course each with a hand with five fingers). But halving the X also makes sense. Or perhaps both ideas contributed.

    “A lower symbol placed before a higher one reduces it by one, so LX is sixty but XL is forty. Of interest is the well-known IV instead of IIII for four was first introduced in the Middle Ages.”

    Any truth that this was only for IV the case, while XL, IC, etc were standard, and it was because IV is like JU, i.e. the begining of Jupiter?

    • The derivation of the Roman V for five from the X for ten is historically fairly well established. Don’t forget the X isn’t really an X but is a crossed I.

      On the use or non use of IV for four I have really no idea why the Romans didn’t use it but it’s an interesting question.

  7. Reblogged this on Kirstie Earlene Ink and commented:
    This is a fascinating look at the original mathematics systems and a very thorough explanation of how to do multiplication and division in Roman Numerals. I highly recommend reading this post.

  8. Baerista

    Great post. But I suspect that in “Various medieval scholars such as Robert Grosseteste wrote elementary textbooks explaining the Hindu-Arabic system and how to use it,” you meant to write John of Sacrobosco instead of Robert Grosseteste.

  9. Jacob Rus

    Check out retired engineer Steve Stephenson’s intriguing (if not necessarily historically accurate) speculations about possible uses of counting boards to do various kinds of computations in base ten, sixty, and twelve (Romans and others used base twelve fractions): http://ethw.org/Ancient_Computers

    I really like his two-column signed-pebbles methods. I plan to teach something similar to my own children in a few years as the basis of their study of arithmetic (with Hindu–Arabic style pen-and-paper methods layered on top).

  10. Pingback: The widespread and persistent myth that it is easier to multiply and divide with Hindu-Arabic numerals than with Roman ones. | The Renaissance Mathematicus – N S Gill – Ancient/Classical History

  11. David McDonough

    I don’t understand how this is a vindication of Roman numerals since the algorithm can be accomplished with Hind-Arabic numerals. The algorithm is virtuous without regard to the system used. Disappointing. I was hoping for an actual demonstration of the usefulness of the Roman system.

    • Jacob Rus

      David: the key to understanding Roman numerals and similar systems is to realize that they weren’t a tool for performing arithmetic, but only a tool for writing down finished results. They’re just a written version of tally marks scored on a stick, or flashed hand-gesture counts.

      Actual calculations were done on a counting board. When someone wanted to some bit of complicated arithmetic, they would “take it to the pebbles”, pulling out a physical counting board or drawing one on a flat surface, and then moving pebbles or coins around to perform the calculation (“calculate” comes from the Latin word for pebble).

      Someone used to that system would directly relate written Roman numerals to their physical representation on a counting board, much the same way a musician can relate music notation to an imagined sequence of sounds, or the way some modern East Asians can visualize arithmetic on a mental image of a counting frame (soroban) when they see Hindu–Arabic numbers written.

      So the goal of the Roman numeral is to just be as easy and direct a representation of the counting board configuration as possible. Nobody is going to write down a long list of Roman numerals representing every intermediate computation in the middle of a division problem. The materials were too expensive (parchment, papyrus) or cumbersome (wax tablets) for that.

      When trying to analyze the relative efficiency of Roman numerals vs. Hindu–Arabic numerals for doing some concrete computation, the fair comparison is to do the Roman numeral computation on a counting board. e.g. http://www.drthomasoshea.com/uploads/8/4/0/2/84021994/algorist-vs-abacist_orig.jpg?450

      We have something similar today. All of our actual calculation nowadays is done using binary arithmetic in some complicated network of transistors in an electronic calculator somewhere. The Hindu–Arabic numerals are becoming less a practical tool for arithmetic, and more a notation system for written final answers.

      • If you’re only going to judge the Roman system on how easy it is to write down the final answer, then a fair comparison would be to how easy it is to write the final answer in Arabic numerals. Comparing the Roman ease of writing down the answer to the Arabic ease of running the whole algorithm is comparing apples and oranges.

        If you’re going to say that Roman numerals don’t work for pen-and-paper algorithms, and then complain about the Arabic pen-and-paper algorithm, the obvious retort is “At least Arabic *has* a pen-and-paper algorithm.”

      • Jacob Rus

        beleester: You’re missing my point.

        The Hindu–Arabic numerals are clearly a beneficial invention overall, which is why they’ve spread throughout the world. Pen and paper arithmetic, and especially written algebraic equations are incredibly powerful and flexible tools. Being able to write 3888 rather than MMMDCCCLXXXVIII saves a whole lot of space on the page. Easy extensions to decimal fractions, floating point numbers (“scientific notation”), written representations of logarithms, etc. are essential for modern science and engineering.

        The problem I have is that typical discussions dismiss the Roman numerals as cumbersome and stupid, without considering the historical / cultural context. The criticisms are simplistic and ignorant, usually pure ridicule – with the Roman numeral system treated as a foil for modern arithmetic – rather than on trying to understand *why* the number system worked the way it did.

        There are some notable advantages to the Roman system: (1) It only requires learning a small number of easy-to-read symbols, and several of the symbols are clearly related to each-other or are directly based on the spoken words. (2) Once the symbols are known, arithmetic relations between written numbers are immediately obvious even to illiterates. This makes numbers easy to compare and easy for non-experts to verify. Addition and subtraction in particular are essentially trivial. (3) Basic arithmetic is easier to teach because the representation is more direct, e.g. it is much easier to teach grouping/“carrying” using Roman numerals vs. Hindu–Arabic numerals. (4) The context of a written symbol doesn’t need to be carefully considered in order to understand its meaning, as happens with a written place-value system. (5) The written numbers have a very direct and obvious relationship to the configuration of pebbles or coins on a counting board. (6) Simple large numbers don’t require any more symbols to express than simple small numbers, and arithmetic can be done pretty well the same on any scale (within 4 orders of magnitude) without carrying around confusing extra zeros; this aspect matches the spoken form of numbers. (7) Arithmetic algorithms designed for Roman numerals often don’t require any memorization of tables, which is helpful in a society where not every person spends several years of schooling memorizing basic arithmetic facts.

        In a context where writing is expensive or cumbersome, you don’t need to do lots of written scratch work because everything is worked out using physical manipulation of counters, most people are expected to be unschooled, and the vast majority of calculations are relatively straight-forward addition, subtraction, and multiplication problems related to trade, accounting, taxation, simple measurement, and the like, then many of the modern advantages of the Hindu–Arabic numbers are much less relevant than say in a 20th century context.

        Cheers.

  12. Just a wee typo, looks like. “algorithms I learnt I school and them the same examples”. Cheers – going to read the rest, now that’s not bothering me any more. 😉

  13. Pingback: The widespread and persistent myth that it is easier to multiply and divide with Hindu-Arabic numerals than with Roman ones. | ExtendTree

  14. Lucas

    There’s a little mistake on the last paragraph
    “Adding rows 6, 5 and 2 on the right we get MCCXLVIII+CLVI+LXXVIII=MCML”, but you’re adding 6, 3 and 2

  15. sarahemilybond

    Reblogged this on History From Below .

  16. Reblogged this on Talmidimblogging and commented:
    HT Sarah E. Bond

  17. Pingback: XXX – MeasurementDataBases for Industry & Science

  18. Pingback: XXX | A bunch of data

  19. Max

    This misconception pops up (more or less) as a plot point in the Ursula Le Guin short story “The Masters”. I still quite like the story, but I think it’s very interesting how the “Roman numerals are bad” thing has penetrated, and I hadn’t had any reason to doubt its truth. Cool!

  20. Pingback: 11 Things We Learned This Week | Nat Geo Education Blog

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