How Do You Do Arithmetic With Sticks? Lay Them Out in Columns

Short rods of bamboo, bone or ivory were laid on a board or a table ruled into columns, with each column a decimal place. Digits one to five were shown by that many rods laid side by side, and six to nine by one rod placed across at right angles, representing five, with the remainder beside it. Crucially, the orientation alternated between columns, with vertical rods in one and horizontal in the next, which removed any ambiguity about where one digit ended and the next began. An empty column meant zero and was simply left empty, so the system was positional in the modern sense.

Working with physical tokens on a grid supports operations that are awkward on paper:

• •Addition and subtraction by moving and exchanging rods
• •Multiplication and division by laid-out procedures on the board
• •Square and cube roots, by an algorithm still recognisable today
• •Negative numbers, shown with rods of a different colour or set aside
• •Fractions, shown as two rows of rods
• •Systems of linear equations, arranged as a rectangular array

The most striking achievement is a procedure for solving systems of several linear equations by arranging their coefficients as a rectangular array on the board and then manipulating columns to eliminate unknowns one at a time. That procedure appears in a mathematical text compiled by roughly the first century of the common era, and it is essentially identical to the elimination method taught in linear algebra today and attributed in the West to Gauss. Working with a physical array made the manipulation natural, since the operation is literally moving and combining columns of rods, and negative quantities arose unavoidably during it, which is why the system accommodated them.

The equipment was portable and the details of it appear in contemporary descriptions. A set consisted of a few hundred rods roughly ten centimetres long and a few millimetres across, carried in a bag or a bundle, with different colours or cross sections distinguishing positive from negative quantities in the systems that used that convention. The surface could be a purpose-made board ruled into squares, a table, or simply the ground. Officials and merchants carried sets as a matter of course, and their presence in tombs and in illustrations is how much of the practice is known, since the rods themselves survive far better than any record of how they were used.

The abacus displaced the rods in China from around the fourteenth century onwards, and the reasons are practical rather than mathematical. Beads on wires cannot be knocked out of place, which rods on a loose board can, and a portable frame suits a merchant where a ruled surface does not. Calculation on an abacus is considerably faster for routine arithmetic. Against that, the older system left a record of the working on the board that could be checked, and it handled the array procedures that beads do not, so it survived among mathematicians and astronomers for some time after commerce had moved on.

Rods laid in ruled columns represented digits positionally, with orientation alternating between columns to keep them apart and an empty column for zero. The arrangement supported negative numbers, fractions and a column elimination method for simultaneous equations recorded by the first century and equivalent to what is taught now. The abacus replaced it because beads cannot be knocked out of place.