Concepts: A Story of Numbers (part 1)

1.1 A slip of paper with strange marks

Reema finds an old slip of paper covered with strange symbols. Her father explains that it is a Mesopotamian numeral, about 4000 years old, from the region that is now Iraq. This starts a journey through the history of number representation.

People counted long before they wrote. They needed numbers for food, livestock, trade, offerings and the calendar: new moon, full moon, seasons. The first number systems used objects (sticks, pebbles), then names, and finally written symbols.

1.2 Number systems and numerals

A number system is a standard sequence of objects, names or symbols with a fixed order. Counting pairs every object with one member of the sequence, in order. For example, each cow is paired with one stick, so the number of sticks equals the number of cows. The written signs of a number system are its numerals.

A sequence must be endless to count every collection. Sticks give an endless sequence, but a large number needs a very long row of sticks. Letter names are easier to say but a language has only a fixed number of letters. English gives a to z for 1 to 26, so single letters cannot count a group of 30. Written symbols face the same problem unless the system is designed carefully.

1.3 Counting in groups

The Gumulgal people of Australia counted in twos: urapon is 1, ukasar is 2, ukasar-urapon is 3, ukasar-ukasar is 4, ukasar-ukasar-urapon is 5, and ukasar-ukasar-ukasar is 6. Any number greater than 6 was called ras. The same pattern appears in the Bakairi and Bushmen systems; historians think the three groups may share ancestors. Grouping by 2 is more efficient than one mark per object, because each word stands for several objects.

History used many group sizes: 2, 5, 10 and 20. The Roman system groups by 5 and 10, the Egyptian and Hindu systems by 10, the Mesopotamian system by 60 and the Mayan system by 20 (with an exception at the third place).

1.4 Tally marks and landmarks

Tallies are notches cut on bone or stone. The Lebombo bone, around 44,000 years old, has 29 notches; the Ishango bone, 20,000 to 35,000 years old, has notches in columns and may be a calendar. Tally marks are the written form of counting one by one.

A landmark number is a reference number with its own symbol or word. The Roman landmarks are I for one, V for five, X for ten, L for fifty, C for a hundred, D for five hundred and M for a thousand. A number is written by grouping it from the largest landmark downwards. For 2367 we take two 1000s, three 100s, one 50, one 10, one 5 and two 1s, giving MMCCCLXII.

1.5 Worked examples

Example 1. Write 27 in Roman numerals. 27 = 10 + 10 + 5 + 1 + 1, so the numeral is XXVII.

Example 2. Write 40 in Roman numerals. Since 40 is 10 less than 50, it is written XL; the older form XXXX also appears in historical writing.

Example 3. Read CCXXXII + CCCCXIII without converting to Hindu numerals. Count C's, X's and I's: there are 6 Cs, 4 Xs and 5 Is. Five Cs regroup to D, so the sum is DCXXXV.

Example 4. Convert 1222 to Roman form. 1222 = 1000 + 100 + 100 + 10 + 10 + 1 + 1, so it is MCCXXII.

A useful check when writing Roman numerals is to start from the largest landmark smaller than the number and keep subtracting. For 1984, the largest landmark is M (1000), leaving 984. The next useful landmark is CM (900), leaving 84. Then L (50) leaves 34, XXX leaves 4, and IV finishes the numeral: MCMLXXXIV.

Reading in the other direction also works. A smaller symbol before a larger one is subtracted: IV is 4 and IX is 9. The same symbol after a larger one is added: VI is 6 and XI is 11. This rule explains XL (40) and XC (90) as well.

1.6 Why number systems matter

A number system does more than record quantity. It shapes what can be calculated easily. A system with fixed-value symbols, like Roman numerals, can record a number but makes multiplication and division slow. A system with a base and place value turns those operations into the arithmetic of digits, which is why it became the standard.

The history of numbers is therefore a history of inventions: tally marks, landmark numbers, bases, place values and zero. Each solved a limitation of the previous stage, and none of them appeared all at once. A good system needs to be endless, unambiguous and easy to compute with; these three demands explain the order in which the ideas appeared.