3.2 Some Early Number Systems

I. Counting with body parts

Plenty of peoples counted on their hands and bodies. A community in Papua New Guinea ran through a fixed run of body parts, fingers, then wrist, elbow, shoulder, and onward, as their standard counting sequence, touching each part in a set order.

II. Tally marks on bones

Among the oldest methods is the tally mark, a notch sliced into bone, wood, or a cave wall, one notch per object. It mirrors the pebble method exactly, except the marks are carved rather than gathered.

The Oldest Counting Bones Archaeologists have turned up bones more than 20,000 years old scored with what resemble tallies. The Ishango bone (Democratic Republic of Congo, 20,000 to 35,000 years old) carries notches set in columns, perhaps a calendar. The Lebombo bone (South Africa) is older still, roughly 44,000 years old, bearing 29 notches, possibly a lunar calendar. These rank among humanity’s oldest mathematical objects.

III. Counting in twos

The Gumulgal people of Australia counted in 2s:

Number Gumulgal name Meaning
1 urapon 1
2 ukasar 2
3 ukasar-urapon 2+1
4 ukasar-ukasar 2+2
5 ukasar-ukasar-urapon 2+2+1
6 ukasar-ukasar-ukasar 2+2+2

Anything above 6 was simply called ras (“many”). Strikingly, the Bakairi of South America and the Bushmen of South Africa, oceans apart, with no known contact, arrived at the same counting-in-twos scheme. One suggestion is that all three trace back to shared ancestors who carried the idea along as they spread out.

The First Big Idea: Grouping Counting in 2s beats a plain tally, because a single word (“ukasar”) now stands for a whole group. This is the first great idea: count in groups of a fixed size, and give the group a name. Through history the favourite group sizes have been 2, 5, 10, and 20. (You can spot grouping by 5 sitting inside the Roman system: V = 5.)

Why did grouping appear at all? Partly because of a limit built into our own eyes. Glance at a scatter of dots: most of us read off “1, 2, 3, 4” instantly, but a clump of 5 or more has to be counted one at a time. So swapping each group of 5 tallies for one fresh symbol made numbers legible at a glance.

Math Talk What breaks down if you group using only one size, say only 5s? How would you write a number like 1345 that way? (You would need 269 groups of 5, still a mouthful! The next idea, several group sizes, repairs this.)

IV. The Roman numerals

The Romans bettered single-size grouping by deploying a sequence of special numbers. We will call any number that earns its own brand-new symbol a landmark number.

Symbol I V X L C D M
Value 1 5 10 50 100 500 1000

To write a number, break it into landmark numbers, taking as many of the largest as you can, then the next, and so on. For instance:

1872 = 1000 + 500 + 100 + 100 + 100 + 50 + 10 + 10 + 1 + 1 \;\Rightarrow\; \text{MDCCCLXXII}.

A smaller symbol placed before a larger one means “subtract”: 4 = \text{IV} (one short of five), 40 = \text{XL}, 9 = \text{IX}. (The Romans were not always tidy, sometimes 40 came out as XXXX.)

The Second Big Idea: Landmark Numbers The Roman scheme runs well ahead of a tally because it groups with a whole sequence of landmark numbers (1, 5, 10, 50, …) rather than one fixed size. This is the second great leap in the story of numbers.

But the Roman scheme has a real weakness: arithmetic is painful. Because the landmark numbers do not grow by one steady factor (1 to 5 is \times 5, yet 5 to 10 is \times 2), adding and especially multiplying call for fussy, uneven regrouping. To calculate, Romans leaned on a device called the abacus, and only trained specialists handled it well.

Worked Example

Add CCXXXI + CCCXXIV in Roman numerals, without converting to Hindu numerals.

Gather like symbols: \text{I}: 1 + 4 = 5, \text{X}: 3 + 2 = 5, \text{C}: 2 + 3 = 5.

Now regroup with the landmark rules. Five Is make a V; five Xs make an L; five Cs make a D. The total is

\text{D} + \text{L} + \text{V} = \text{DLV} \quad (= 555).

Notice that the regrouping factor here is 5 each time, but elsewhere in the system it is 2, and that unevenness is exactly why Roman arithmetic feels awkward.

Worked Example

Write 409 in Roman numerals.

Break 409 into landmark pieces, largest first. The hundreds part is 400, which is written as “one less than 500”: \text{CD}. The remaining 9 is “one less than 10”: \text{IX}.

409 = 400 + 9 = \text{CD} + \text{IX} = \textbf{CDIX}.

Figure it Out

  1. Write in Roman numerals: (i) 1444   (ii) 2848   (iii) 409   (iv) 936.
  2. Multiply these landmark-number pairs (give the Roman answer): \text{VI} \times \text{L}, \;\text{VI} \times \text{D}, \;\text{VIII} \times \text{VII}.
  3. Take the Gumulgal “counting by 2s” scheme extended past 6. With urapon =1 and ukasar =2, evaluate (write the answer in Gumulgal form):
    1. (uk-uk-uk-uk-uk-ur) + (uk-uk-uk-uk-ur);
    2. (uk-uk-uk-uk-uk-ur) - (uk-uk-uk);
    3. (uk-uk-uk-uk-uk-ur) \times (uk-uk);
    4. (uk-uk-uk-uk-uk-uk) \div (uk-uk).
  4. List the features of the Hindu system that make it more efficient than the Roman system.
    1. $1444 = $ MCDXLIV   (ii) $2848 = $ MMDCCCXLVIII   (iii) $409 = $ CDIX   (iv) $936 = $ CMXXXVI.
  2. $ = 6 = 300 = $ CCC.   $ = 6 = 3000 = $ MMM.   $ = 8 = 56 = $ LVI.
  3. In Gumulgal, urapon =1, ukasar =2, so uk-uk-uk-uk-uk-ur = 11 and uk-uk-uk-uk-ur = 9, etc.
    1. $11 + 9 = 20 = $ uk-uk-uk-uk-uk-uk-uk-uk-uk-uk (ten ukasars).
    2. $11 - 6 = 5 = $ uk-uk-ur.
    3. $11 = 44 = $ uk-…-uk (twenty-two ukasars).
    4. $12 = 3 = $ uk-ur.
  4. The Hindu system uses place value (a symbol’s worth depends on its position), it has a symbol for 0, and it makes all four operations easy. The Roman system has fixed-value symbols, no zero, and is hard to calculate with.