The Roman Hand-Abacus

by Steve Stephenson

In the history of mathematics, the contributions of the Roman Empire are sometimes overlooked. Roman Numerals are considered cumbersome and the Roman's lack of contributions to mathematics, and the lack of the Zero, are held in low esteem.

And yet, the Roman Empire was likely the largest when viewed as a percent of world population. Their empire consistently built engineering marvels: roads that survive and are used to this day, homes and bath houses with indirect heating emulated today, plumbed sewer and water lines in and out of homes and public buildings, indoor toilets, aquaducts that included long tunnels and bridges, and huge, beautiful buildings. Their engineers and architects designed and built these using counting boards and hand abaci; using Roman Numerals only to record the results.

roman hand-abacus

Roman Hand-Abacus: A photocopy of a photograph of the Roman hand-abacus; the top image is the front and the bottom is the back. Image is from Museo Nazionale Ramano at Piazzi delle Terme, Rome. (Click to download PDF document.)

The longevity of their empire was due to their commercial trade-- they were businessmen. The intricate, complex, and extensive accounting of their trade was conducted with counting boards and hand-abaci; again using Roman Numerals only to record the results.

And as anyone knows who has used a counting board or abacus, your rows or columns often represent nothing, or zero. Since the Romans used Roman Numerals to record results, and since Roman Numerals were positively definitive, there was no need for a zero notation. But the Romans certainly knew the concept of zero occuring in any place value, row or column.

One could also infer that they were aware of the concept of a negative number. How else would Roman merchants understand and manipulate liabilities against assets and loans versus investments?

The Romans developed their hand-abacus as a portable counting board-- the first portable calculating device for both engineers and businessmen.

Layout of the Roman hand-abacus

Here's the London Science Museum's Roman hand-abacus layout, where the ~3 was actually a symbol that looked like a 3 that was flattened on the top then flipped top to bottom and right to left, or rotated 180 degrees:

  | |    | |    | |    | |    | |    | |    | |    | |
  | |    | |    | |    | |    | |    | |    | |    | |
  |*|    |*|    |*|    |*|    |*|    |*|    |*|    |*|
  ___
  |X|  (((|))) ((|))   (|)     C      X      I      0     ~3  

  | |    | |    | |    | |    | |    | |    | |    | |    | |
  | |    | |    | |    | |    | |    | |    | |    | |    | | )
  |*|    |*|    |*|    |*|    |*|    |*|    |*|    |*|    | |
  |*|    |*|    |*|    |*|    |*|    |*|    |*|    |*|    | |
  |*|    |*|    |*|    |*|    |*|    |*|    |*|    |*|    | |
  |*|    |*|    |*|    |*|    |*|    |*|    |*|    |*|    |*| 2
                                                   |*|    |*| 
roman hand abacus

Roman "pocket abacus": (in bronze), beginning of Common Era (Cabinet des Médailles, Bibliothèque nationale, Paris). Note that fig. 16.94 has beads missing from most of the slots. The drawing on the bottom has incorrectly labeled the right-most slot. This abacus is similar to the one being described in this article. Image and caption from, The Universal History of Numbers, Georges Ifrah, Wiley Press 2000. (Click to enlarge.)

The abacus was made of a metal plate where the beads ran in slots. The size was such that the abacus could fit in a modern shirt pocket. The upper slots contained a single bead while the lower slots contained 4 beads, the only exceptions being the two right-most columns, marked 0 and ~3.

Note the longer slots below the 0 and ~3 positions, the 5 beads in the lower slot of the 0 position, the 2 beads in the lower slot of the ~3 position, and the absence of an upper slot in the ~3 position. I wonder what the ')' and '2' symbols along the right side of the ~3 slot meant?

Obviously the units in the 0 position were 1/12 of the I position, and the units in the ~3 position were 1/3 of the 0 position. So the upside down reversed 3 character seems appropriate to represent 1/3; or, more likely, our symbol for 3 came from the Roman symbol for 1/3.

It is also worth noting that:

More information and conjectures on counting boards and Roman Hand Abaci can be found on Mr. Stephenson's web site.

A Possible Time Abacus

The last point, above, might lead one to develop a Time abacus1.


                         |*|       |*|       |*|
|*|                      |*|  |*|  |*|  |*|  |*|  |*|
| |                      | |  | |  | |  | |  | |  | |
| |                      | |  | |  | |  | |  | |  | |
 y    q    w    d   6h    h   6m    m   6s    s   s/10
| |  | |  | |  | |  | |  | |  | |  | |  | |  | |  | |
| |  | |  | |  | |  | |  | |  | |  | |  | |  | |  | |
|*|  |*|  |*|  |*|  |*|  |*|  |*|  |*|  |*|  |*|  |*|
|*|  |*|  |*|  |*|  |*|       |*|       |*|       |*|
|*|  |*|  |*|  |*|  |*|       |*|       |*|       |*|
|*|       |*|  |*|            |*|       |*|       |*|
          |*|  |*|
          |*|  |*|
          |*|
          |*|
          |*|
          |*|
          |*|
          |*|

Column Description Base
y year 10
q quarter 4
w week 13
d day 7
6h six hours = 1/4 day 4
h hour 6
6m six minutes = 1/10 hour 10
m minute 6
6s six seconds = 1/10 minute 10
s second 6
s/10 1/10 second 10
1The quarters and years calculate an exact 364 day year, not the (currently) correct 365.242190 day year. For additional details about calendars, refer to Calendars Through the Ages..

Further reading: Number Words and Number Symbols: A Cultural History of Numbers by Karl Menninger (1977).

Last modified: Thu Nov 27 16:07:59 2003

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