How Mathematics Began: Counting, Tally Marks and the First Numbers

Long before anyone wrote an equation, people needed to answer a simple question: how many? How many sheep left the pen this morning, how many days until the next full moon, how many baskets of grain are owed? The story of how humans learned to answer that question is the story of how mathematics began.

Counting Without Numbers

The oldest idea in mathematics is not a number at all. It is one-to-one correspondence: matching each object with a mark, a pebble or a finger. A shepherd who drops one pebble into a bag for every sheep that goes out to graze can check that every animal has come home, simply by removing one pebble per returning sheep. If pebbles are left over, sheep are missing. No counting words are needed.

This idea is so natural that young children still use it. When a child holds up three fingers to show "three", they are doing exactly what our ancestors did tens of thousands of years ago.

Notched bones

The earliest physical evidence of counting comes from animal bones with deliberate cuts.

• The Lebombo bone, found in a cave in the Lebombo Mountains between present-day South Africa and Eswatini, is a baboon fibula with 29 notches. It is dated to around 43,000 years ago. Some researchers suggest it tracked a lunar cycle, though this cannot be proved.
• The Ishango bone was discovered in 1950 by the Belgian geologist Jean de Heinzelin near Lake Edward, in what is now the Democratic Republic of the Congo. It is roughly 20,000 years old. Its notches are grouped into columns, and one column contains groups of 11, 13, 17 and 19, which are exactly the prime numbers between 10 and 20. Whether this was intentional is debated, but two of its columns each total 60, which suggests careful grouping.

These bones show that people were recording quantities long before writing existed.

Tally Marks: Grouping Makes Counting Easier

A long row of identical scratches is hard to read. Try counting 37 vertical lines quickly, and you will probably lose your place. The solution, discovered independently in many cultures, was grouping. The familiar modern tally groups marks in fives, with the fifth mark drawn across the first four. Chinese shopkeepers traditionally use the five strokes of the character 正 for the same purpose.

Worked Example 1: Reading and writing a tally

Suppose a teacher records attendance with tally marks and ends the week with 7 complete bundles of five plus 2 extra strokes.

Step 1: Each bundle represents 5, so the bundles give

$$ 7 \times 5 = 35 $$

Step 2: Add the loose strokes:

$$ 35 + 2 = 37 $$

So the tally shows 37 attendances. Notice that grouping turned a long, error-prone count into a multiplication and an addition. This is the first step towards the idea of a base: counting in groups of a fixed size.

Body Counting and Number Words

Many societies counted using parts of the body. The Oksapmin people of Papua New Guinea traditionally count by touching 27 body positions in order, starting at the thumb of one hand, moving up the arm, over the head and down the other side. The word for a number is simply the name of the body part.

Fingers gave most cultures their base. Ten fingers led to base 10, still used worldwide. Fingers and toes together led to base 20, which survives in French quatre-vingts ("four twenties" = 80) and in the old English word score, as in Lincoln's "four score and seven years" (87). The Maya of Central America used a full base-20 system.

From Clay Tokens to Written Numerals

In Mesopotamia (modern Iraq), from around 8000 BCE, farmers used small clay tokens of different shapes to represent goods: a cone for a small measure of grain, a sphere for a larger one, a cylinder for an animal. The archaeologist Denise Schmandt-Besserat argued that these tokens were sealed inside clay envelopes, and that people began pressing the tokens onto the outside of the envelope so the contents could be checked without breaking it. Eventually, the impressions alone were enough, and the tokens were dropped.

By around 3200 BCE, in cities such as Uruk, scribes were drawing number signs on clay tablets with a reed stylus. This was one of the first moments in history where a number was separated from the thing being counted. "Five" could now mean five sheep, five jars or five workers. That abstraction is the real birth of arithmetic.

Early Number Systems

Once numbers could be written, different civilisations designed different systems.

• Egyptian hieroglyphs (from around 3000 BCE) were additive: a stroke for 1, a heel bone for 10, a coil of rope for 100, a lotus flower for 1,000, and so on. You repeated each symbol as many times as needed.
• Roman numerals were also mostly additive, with a subtractive shortcut: I = 1, V = 5, X = 10, L = 50, C = 100, D = 500, M = 1000, and a smaller symbol before a larger one is subtracted.
• Babylonian numerals used base 60 with place value, a revolutionary idea where the position of a symbol changes its value.
• Maya numerals used dots (1), bars (5) and a shell symbol for zero, stacked vertically in base 20.

Worked Example 2: Converting ancient numerals

(a) Write 2026 in Roman numerals.

Break the number into thousands, hundreds, tens and units:

$$ 2026 = 2000 + 20 + 6 $$

• \( 2000 = \text{MM} \)
• \( 20 = \text{XX} \)
• \( 6 = \text{VI} \)

So 2026 = MMXXVI.

(b) Write 1994 in Roman numerals.

$$ 1994 = 1000 + 900 + 90 + 4 $$

• \( 1000 = \text{M} \)
• \( 900 = 1000 - 100 = \text{CM} \)
• \( 90 = 100 - 10 = \text{XC} \)
• \( 4 = 5 - 1 = \text{IV} \)

So 1994 = MCMXCIV.

(c) Read a Maya number. A Maya number has two levels. The upper level (the "twenties" place) shows two dots. The lower level (the units place) shows one bar and two dots.

The upper level is worth \( 2 \times 20 = 40 \). The lower level is a bar (5) plus two dots (2), giving 7. Therefore

$$ 2 \times 20 + 7 = 47 $$

Compare how much work the Roman version needs: 47 = XLVII. A place-value system scales far better, which is why our modern Hindu-Arabic numerals, also place-value, eventually conquered the world.

Why Additive Systems Were Not Enough

Additive systems are easy to read but terrible for calculation. Try multiplying MCMXCIV by XLVII without converting to modern numerals. Merchants in Rome and medieval Europe used counting boards and the abacus precisely because the written numerals were not designed for arithmetic. The great leap came with place value plus a symbol for zero, which was perfected in India and later spread through the Islamic world to Europe.

Did You Know?

• The word "calculate" comes from the Latin calculus, meaning "small pebble", a reminder of pebble counting.
• The English word "tally" comes from the Latin talea, a stick. Medieval England used notched wooden tally sticks as receipts for taxes until 1826, and burning the old sticks in 1834 accidentally started the fire that destroyed much of the Houses of Parliament.
• Some languages, such as that of the Pirahã people of the Amazon, appear to have no exact words for numbers beyond about two, showing that number words are a cultural invention.

Key Takeaways

• Counting began with one-to-one correspondence: matching objects with fingers, pebbles or marks.
• Notched bones such as the Lebombo and Ishango bones are the oldest evidence of recorded quantities, tens of thousands of years old.
• Grouping marks into bundles led naturally to the idea of a base, such as 10, 20 or 60.
• Clay tokens in Mesopotamia evolved into written numerals around 3200 BCE, separating numbers from objects.
• Place-value systems with a zero are far more powerful for calculation than additive systems like Roman numerals.

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