Number systems, counting and numerals

A number system needs three separate tools, and history shows they arrived one at a time, not together.

• One-to-one correspondence (tallying). Match each object in a group with exactly one mark, notch, pebble, or knot. Two herders can compare flock sizes without a single number word, just by lining up marks against animals. This is the oldest and simplest counting method — it answers "how many" purely by matching, with no words at all.
• Grouping. Once tally marks get long and hard to read at a glance, people started bundling them — for example, crossing every fifth stroke, as in $\cancel{||||}$. A bundle of five (or ten, or twenty, depending on the culture) turns a long unreadable row of strokes into a small number of readable chunks. Grouping is the conceptual seed that later grows into place value, though it is not place value yet.
• Spoken number sequence. An ordered list of words — "one, two, three, …" — that a community agrees on and recites the same way every time. Spoken sequences generally appeared before any matching written numeral, since speech needs no tools.
• Written numeral. A symbol standing for a count, so information can be stored and shared without anyone reciting words aloud. Early numerals were often repeated tally-like strokes (Egyptian $|$, $||$, $|||$ for 1, 2, 3); later cultures moved to distinct, compact symbols — the digits $0$–$9$ used today are the endpoint of that long journey.

Grouping is worth making precise, because it is the one idea in this list that behaves like arithmetic. If a count of $N$ objects is bundled into groups of a fixed size $b$, then

where: $N$ is the total count, $b$ is the bundle size (how many strokes make one group — the "grouping base"), $q$ is the number of complete bundles, and $r$ is the number of leftover, ungrouped strokes. This is exactly the arithmetic a trader performs when bundling tally marks in fives: $q$ full bundles plus $r$ loose strokes reconstruct the original count. Equation (1) does not by itself give any symbol a position-dependent value — that further leap, where the position of a digit changes what it means, is place value, and it is a separate, later idea covered elsewhere in this chapter. The rough historical order this section describes is: objects → tally marks → grouped tallies → spoken words → written numerals → place-value numerals, with different cultures reaching different stages at very different times.

Problem. A shepherd with no number words checks every evening that all 18 sheep in her flock have returned, using only a pouch of pebbles for one-to-one correspondence. On one evening, only 15 sheep return. Show how she detects the shortfall, and state it using the grouping idea from equation (1) with a bundle size of $b = 5$.

Step 1. As each of the 18 sheep leaves the pen in the morning, the shepherd drops one pebble into her pouch — one pebble per sheep, so the pouch holds exactly $18$ pebbles by the time the flock has left, without her ever saying a number.

Step 2. In the evening, she removes one pebble for every sheep that returns. After the 15th returning sheep, she has removed 15 pebbles, but the pouch is not empty: $3$ pebbles remain unmatched.

Step 3. Each leftover pebble stands for one sheep still out grazing, so the pouch tells her directly, through matching alone, that $18 - 15 = 3$ sheep are missing — she never needs the words "fifteen" or "eighteen" to know this.

Step 4. Regroup the missing count using equation (1) with bundle size $b = 5$: for $N = 3$, dividing by $5$ gives zero complete bundles, so $q = 0$ and $r = 3$, i.e. $3 = (0 \times 5) + 3$. Three is too small to form even one bundle, so it would simply be drawn as three loose tally strokes, $|||$, with no crossed bundle at all.

Result. One-to-one correspondence alone reveals a shortfall of 3 sheep, and grouping that shortfall in fives confirms it stays as three loose strokes since $3 < 5$.

• Thinking counting, number words, and written numerals all appeared together historically — they developed as separate, staggered inventions, often centuries apart.
• Assuming every early numeral system used place value; most (tally marks, Egyptian hieroglyphic numerals) simply repeated symbols and had none.
• Confusing "grouping in fives or tens" (organising tally marks with equation (1), $N = qb+r$, purely for readability) with "place value" (where a digit's value depends on its position) — grouping is a stepping stone, not the same idea.
• Believing a system needs an explicit symbol for zero to count things — tallying and grouped tallies work fine with no zero, since there are simply no marks when there is nothing to count.

Ask the tutor: want to see how grouped tallies eventually turned into the ten place-value digits we use today, or get quizzed with more matching and bundling examples?