A calculator made of sticks
For many centuries, Chinese mathematicians and officials did arithmetic with small rods, usually called counting rods. They were thin sticks, commonly of bamboo, though other materials such as wood, bone or ivory were also used. The rods were laid out on a flat surface, or on a ruled board, in rows and columns. Each rod arrangement stood for a digit, and the position of the arrangement told the reader its place value.
Counting rods are often mentioned in sources from the Han dynasty onward, and written descriptions show the practice had probably developed earlier. Exact dates for its beginnings are uncertain, because wood and bamboo rarely survive and early texts describe the practice only in passing.
Vertical and horizontal forms
The most distinctive feature of rod numerals is the way digits were drawn. There were two forms.
- Vertical form: the digits 1 to 5 were shown as that many upright rods side by side. For 6 to 9, one horizontal rod laid across the top counted as five, with upright rods beneath it for the rest.
- Horizontal form: the digits 1 to 5 were shown as that many horizontal rods stacked one above another. For 6 to 9, one upright rod on top counted as five, with horizontal rods underneath.
The two forms were used in alternating places. Units, hundreds and ten-thousands used the vertical form, while tens and thousands used the horizontal form. The reason is practical: if two neighbouring digits both used the same form, their rods would merge into one confusing bundle. Alternating shapes kept each digit visually separate.
Reading a number
Numbers were read from the highest place on the left to the lowest on the right, and the system was decimal. Consider 4,637. The thousands digit, 4, is in the horizontal form. The hundreds digit, 6, is vertical, with a horizontal rod across the top for five and one upright below. The tens digit, 3, is horizontal. The units digit, 7, is vertical, with the five-rod across the top and two uprights below. Reading the result as a row of alternating shapes gives 4,637.
When a place was empty, the calculator simply left a blank square on the board. This avoided the ambiguity of an unmarked gap in written text, since the board itself made the empty position obvious. A written symbol for zero, drawn as a circle, appears in Chinese mathematical texts in a later period, around the Song and Yuan dynasties, though scholars still discuss its sources and its possible links to Indian influence. The wider story is part of the history of zero.
Positive, negative and the counting board
Some sources describe using rods of two colours, red for positive amounts and black for negative ones, though later texts also reverse or vary this. Chinese mathematics of the Han and later periods dealt with deficit and surplus quantities in this way, which is part of the history of negative numbers. The details of when and how this convention began are not always certain.
The surface used for calculation was often a board or even a table marked with squares, but ordinary flat ground or a cloth could serve. Multiplication, division, and even the extraction of roots and the solving of systems of equations were carried out by moving rods from place to place. Texts such as the Nine Chapters on the Mathematical Art set out problems whose methods are generally understood as procedures on the counting board, though the book itself describes the steps in words.
From rods to written signs
Rod shapes were also written down. Mathematical texts used brush-drawn marks resembling the rod patterns to record numbers on paper, so the written form copied the arrangement on the board. These written signs were used alongside the ordinary Chinese characters for numbers, which were common in everyday text. Later, a commercial style of numeral characters with more complex forms, used to prevent forgery, was developed for financial documents, a separate matter.
The rod notation also survived in a few unexpected places. In some regions, the rod-derived numerals, sometimes called Suzhou numerals, continued in use in markets and in handwritten bills well into modern times.
The shift to the abacus
Over time, the counting board and rods were replaced in everyday work by the bead abacus, the suanpan. The abacus appears in sources of the Ming period as the standard instrument, and some scholars believe the change was gradual and began earlier. The key difference is that the beads stay attached to rods in a frame, so the instrument is easier to carry and quicker to use than loose rods. The wider abacus history reaches well beyond China.
What rod calculation shows
Chinese rod numerals were a place-value system long before many other societies had one. They also show a close tie between a calculating tool and a notation: the written digits were pictures of physical rods. That link helps explain why the system worked well for calculation but was seldom used to write very large or unusual numbers in literary text. A distant parallel is the Inca quipu, which also stored decimal numbers by the position of knots on cords. Rod numerals remain one of the clearest examples of a society doing decimal positional arithmetic without a written zero.