Positional Numerals

In a positional numeral system a few digits are enough to write any number, because each digit's value depends on the place it occupies. With a zero to mark an empty place, sums, products and divisions can be worked out in writing, column by column, instead of on a counting board.

What it is

In 707, the first seven counts hundreds and the last counts units. That is the whole idea of positional notation: a short set of signs whose value depends on where each one stands. The ten digits from 0 to 9 can write any whole number, however large, without a new sign.

Older systems added their signs together. Egyptian scribes had one sign for units, another for tens, another for hundreds, and repeated each as often as needed. Roman numerals work the same way with a few subtractions: 1888 is MDCCCLXXXVIII, thirteen signs. Such numbers record a total well but are awkward to calculate with, so Roman and medieval accountants worked on a counting board and wrote down only the result.

A place-value system turns the written number into a counting board on paper, so the calculation itself can be written down and checked. Zero makes this work: it first marked an empty place, then became a number in its own right.

How it works

A Babylonian number was pressed into wet clay one wedge at a time.

The base is the number of distinct digits, and each place is worth that many times the place to its right. In base ten, 2748 means 2 thousands, 7 hundreds, 4 tens and 8 units. In 1081 the zero says there are no hundreds; without it, 1081 and 181 would look alike.

Other bases follow the same rule. Babylonian scribes counted in sixties, with two marks: an upright wedge for one and a corner wedge for ten. One thousand is 16 sixties and 40, written 16,40. The Maya counted in twenties with a dot for one, a bar for five and a shell for zero, and stacked the places from the bottom up. One thousand is 2 four-hundreds, 10 twenties and no units: two dots over two bars over a shell. In calendar dates the third place counted 360 days instead of 400. Computers use base two, where 13 is written 1101: one eight, one four, no two and one unit.

When a column reaches ten or more, you write its units and carry its tens to the next column; when it runs short, you borrow one from the left. Moving a number one place to the left multiplies it by ten, so 47 × 23 is 47 × 3 plus 47 × 20: 141 + 940 = 1081. A table of products up to 9 × 9 is enough to multiply any two numbers. Long division works from the left, one place at a time: guess a digit of the quotient, multiply, subtract, bring down the next digit.

How to build

You need a flat board or a patch of smooth sand, chalk or a stick, about forty pebbles, paper and a pencil.

Draw the board. Draw four parallel lines a hand’s width apart: from the bottom, units, tens, hundreds and thousands. Once all carrying is done, no line holds more than nine pebbles.

Lay out a number. For 2748, put 2 pebbles on the thousands line, 7 on the hundreds, 4 on the tens and 8 on the units. A line left empty is a zero.

Carrying on the board: ten counters on one line are exchanged for a single counter on the line above.

Add with carrying. Add 1375 by putting 1, 3, 7 and 5 pebbles on the same lines. The units line now holds 13: take ten off, put one on the tens line, and 3 remain. The tens line then holds 12, so carry again, and do the same with the hundreds. The board reads 4123.

Multiply. Clear the board. For 47 × 23, lay out 47 three times, carrying as you go: 141. Then lay out 47 twice more with every pebble one line higher, which adds 940. Carry and read 1081.

Write it on paper. Do the same sums in columns, the carried figure written small above the next column. Each column is a line of the board, each digit a count of pebbles, each 0 an empty line.

Change base. To write 4000 seconds in base sixty, divide by 60: 66, remainder 40. Divide 66 by 60: 1, remainder 6. Read the last quotient, then the remainders backwards: 1,6,40, or 1 hour, 6 minutes and 40 seconds.

Safety. Small pebbles are a choking hazard for young children. Choose stones too large to swallow, or keep the board out of their reach.

History and evolution

The oldest firmly dated use of place value is a scratch tablet from Ur, of 2043 BCE, that adds up weights in base sixty. By about 1800 BCE, Babylonian school tablets use it routinely. There was no zero and no mark separating whole numbers from fractions, so the reader judged the size of a number from context. From around 700 BCE some scribes put a sign in an empty middle place, never at the end. Greek astronomers kept base sixty for fractions, which is why hours and degrees still have sixty minutes.

The Maya reached place value and zero on their own. Long Count dates appear in southern Mexico from the first century BCE, and stelae at Uaxactun dated 357 CE write empty places with a zero sign.

Across late-medieval Europe, accounts were often still cast on a ruled cloth with brass counters.

Our digits come from India. In 628 Brahmagupta defined zero as a number minus itself and gave rules for calculating with it, though his claim that zero divided by zero is zero is wrong. The Bakhshali manuscript, found near Peshawar in 1881, writes zero as a dot; radiocarbon dates obtained in 2017 for three of its leaves range from the third to the tenth century, and the date of the text is disputed. The oldest undisputed dated zero in India is at Gwalior, in 876.

In ninth-century Baghdad, al-Khwarizmi wrote a book on Indian reckoning, now known only in Latin; his name gave the word algorithm. Fibonacci learned the digits in North Africa and taught them in his Liber abaci of 1202. In 1299 the Florentine money changers’ guild barred them from its account books, and counting boards stayed in use in northern Europe into the sixteenth century.