The Romans developed their hand-abacus as a portable counting board—the first portable calculating device for engineers, merchants and presumably tax collectors. It greatly reduced the time needed to perform the basic operations of Roman arithmetic using Roman numerals.

The engine of the Empire was Roman commerce. 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.

When using a counting board or abacus the rows or columns often represent nothing, or zero. Since the Romans used Roman numerals to record results, and since Roman numerals were all positive, there was no need for a zero notation. But the Romans clearly knew the concept of zero occurring in any place value, row or column.

It is also possible to infer that they were familiar with the concept of a negative number as Roman merchants needed to understand and manipulate liabilities against assets and loans versus investments.

The Late Roman abacus shown here as a reconstruction contains seven longer and seven shorter grooves used for whole number counting, the former having up to four beads in each, and the latter having just one. The rightmost two groves were for fractional counting. 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.

 | |   | |   | |   | |   | |   | |   | |   | |
 | |   | |   | |   | |   | |   | |   | |   | |
 |O|   |O|   |O|   |O|   |O|   |O|   |O|   |O|
 
 MM    CM    XM     M     C     X     I     0    ~3  
 ---   ---   ---   ---   ---   ---   ---   ---   --- 
 | |   | |   | |   | |   | |   | |   | |   | |   | |
 | |   | |   | |   | |   | |   | |   | |   | |   | | )
 |O|   |O|   |O|   |O|   |O|   |O|   |O|   |O|   | |
 |O|   |O|   |O|   |O|   |O|   |O|   |O|   |O|   | |
 |O|   |O|   |O|   |O|   |O|   |O|   |O|   |O|   | |
 |O|   |O|   |O|   |O|   |O|   |O|   |O|   |O|   |O| 2
                                           |0|   |O|

This diagram is based on the Roman abacus at the 
London Science Museum.

Here is the basic Roman hand-abacus layout: The leftmost seven rows are used in whole item arithmetic and the abacus can count up to ten million. The two rightmost slots where used to count fractions - 1/12ths and 1/3rds.

The lower groove marked I indicates units, X tens, and so on up to millions. The beads in the upper shorter grooves denote fives—five units, five tens, etc., essentially in a bi-quinary coded decimal system.

Computations are made by means of beads which would probably have been slid up and down the grooves to indicate the value of each column.

The upper slots contained a single bead while the lower slots contained four beads, the only exceptions being the two rightmost columns, marked 0 and ~3.

The longer slot with five beads below the 0 position allowed for the counting of 1/12th of a whole unit making the abacus clearly suited for use with Roman measures and Roman currency.

The Roman pound, libra, consisted of 12 unicae (28 grams) and a measure of volume, congius, consisted of 12 heminae (0.273 litres). Also the Roman foot, pes, was 12 unciae (2.43 cm), and the actus, the standard furrow length when plowing, was 120 peds.

The as, the principal copper coin in Roman currency, was divided into 12 unicae. Again, the abacus was ideally suited for counting currency.

The rightmost position, the ~3, with only two beads allowed the counting of 1/3rds of a whole unit. The use of this position is not clear but could be used to count partially full containers.

A few interesting facts about the Roman abacus:

• The Roman abacus predates the Chinese "invention" of the Suan Pan by centuries;
• The Romans traded with the Chinese over the Silk Road and the similarity of the two abaci suggest that Mongol traders transferred the technology to the Chinese before invention of the Suan Pan in the 13th century;
• The Roman hand-abacus has the refinements attributed to the modern Japanese Soroban; i.e. one bead above and four beads below the bar; and
• The Roman hand-abacus incorporates mixed-base arithmetic (in the two rightmost columns), another original enhancement by the Romans that is not present in any other abacus.