Any literate society can record numbers by writing them down as words – assuming they have words for numbers, beyond “one, two . . . many.” But literate societies have separate systems for writing down numbers; e.g., “144” or “CXLIV” instead of “one hundred forty-four.” These come in three broad types:
Tallies (TL0) involve making a mark, commonly a line, for each object in a collection or each event in a series. Objects dating to the Paleolithic carry such marks, possibly tracking phases of the moon. A number larger than seven or eight can’t be seen at a glance, but must be counted off line-by-line. Tallies can also be matched up with objects, such as a herd of sheep, even by people who can’t count.
Nonpositional numerals (TL1) – such as Roman numerals – use signs for various numbers, which are added up (or sometimes subtracted) to get the total number that they stand for. These may be letters of the alphabet (as in the Roman, Greek, and Hebrew systems) or signs for number words (as in the Chinese system). Nonpositional numerals are useful for recording numbers, but awkward for calculation; roll vs. Accounting or Mathematics (Applied or Statistics) to solve any problem more complex than adding on your fingers.
Positional numerals (TL3) – such as Arabic numerals (actually invented in India), Babylonian cuneiform numerals, and Mayan numerals – use the same symbol for different numbers (1, 10, 100, . . ., or 1, 60, 3,600, . . .) depending on where it’s placed; such systems have a symbol for zero as a “placeholder.” This notation makes arithmetic relatively straightforward, so that it doesn’t require use of a computational skill.
It isn’t historically accurate to treat positional numerals as “more advanced” (higher TL) than nonpositional ones; both the Maya and the Babylonians went straight to positional notation at TL1. But historical GURPS campaigns will be set in Western civilization more than any other – and the West acquired positional numerals only at TL3. Treat civilizations that went straight to positional numerals as “advanced in mathematics.”
For a more-detailed treatment of computation, see GURPS Low-Tech Companion 1.
Many societies have tools for making arithmetic faster or more accurate. Using these devices requires a computational skill and familiarity with the specific device.
Abacus (TL2). A frame, usually wood, that holds beads strung on wires. The beads are moved back and forth to represent calculations. Positional relationships are built into it. People who use nonpositional systems, such as Roman numerals, can calculate on an abacus without a skill roll if they have points in Accounting or Mathematics (Applied). Anyone experienced with an abacus can calculate faster than with pencil and paper, in any notation. $50, 2 lbs. A collection of pebbles laid out on a flat surface can be used as an improvised abacus at no cost, but is much slower to use due the care needed to avoid mixing up pebbles.
Cube Root Extractor (TL2). Ancient Greek engineers designing catapults (see Mechanical Artillery, pp. 78-83) needed to take the cube root of a stone’s weight to determine the engine’s dimensions. The cube root extractor is a mechanical device invented in the third or fourth century B.C. to solve this problem. It has several rods set at angles to each other, one of which slides back and forth. By making a rough initial guess and positioning the rods accordingly, the exact cube root could be measured. Using it requires Mathematics (Applied). $25, 2 lbs.
Napier’s Bones (TL4). This is a set of numbered rods, one for each digit from 0 to 9, showing the multiplication table for that digit, plus an 11th rod for the multiplier. By arranging the rods, it’s possible to read off the product of a multidigit number by any single-digit number. Several sets may be needed if the multi-digit number has repeated digits! Not as fast as a slide rule (GURPS High-Tech, p. 18), but able to produce results with any desired number of digits. Only works with positional numerals. Using it requires Mathematics (Applied or Statistics). One set, with a wooden box (5”x2.5”x1”): $25, 0.5 lb.