Number systems across civilisations

Long before "0, 1, 2, 3..." became the world's shared language for counting, every great civilisation had already solved the same problem in its own way — how do you write down "how many"? The Romans stacked letters, the Egyptians repeated pictures, the Babylonians grouped wedges in clusters of sixty, the Maya stacked dots and bars around a shell for zero, and the Chinese combined digit-symbols with power-of-ten characters. Each system answers two quiet questions differently: what base does it count in, and does the same symbol mean different amounts depending on where it sits? This article compares five historical numeral systems side by side, shows exactly how a place-value system turns a row of symbols into a single number, and reworks the same two examples from the syllabus — 1994 in Roman numerals and a Babylonian base-60 number — in more depth, so you can see precisely why place value quietly won.

The core insight

A numeral system is fixed by two choices — its base (the group size before starting over) and whether it uses place value (same symbol, different value by position) or is purely additive — and switching to a base-10 place-value system with a zero is what makes arithmetic on very large or very small numbers manageable.

The core idea

Base: the size of one "group." Our decimal system regroups every $10$ units into a new place (units, tens, hundreds, ...). Ancient Babylon regrouped every $60$ ; that choice still survives in $60$ minutes to an hour and $360° = 6\times 60°$ in a full turn. The Maya regrouped every $20$ (vigesimal), likely from counting fingers and toes.

Place value versus purely additive. In a place-value system such as ours, Babylonian, or Mayan numerals, a symbol's value depends on its position — the two 2's in $222$ mean $200$ , $20$ and $2$ . In a purely additive (repetitive) system such as Roman or Egyptian numerals, a symbol keeps the same value no matter where it appears, and the total is just the sum of the symbols shown (Roman numerals add a small subtraction trick on top of this, explained below).

Quick tour of five systems:

Roman: $I=1,\ V=5,\ X=10,\ L=50,\ C=100,\ D=500,\ M=1000$ . Additive ( $VI=6$ ), with a subtraction rule when a smaller symbol precedes a larger one ( $IV=4$ , $IX=9$ ). No place value, no zero.
Egyptian: distinct hieroglyphs for $1, 10, 100, 1000,\dots$ , each repeated as many times as needed and simply added. Purely additive, no place value, no zero.
Babylonian: base- $60$ , written in cuneiform wedges, using just two basic marks (a "one" wedge, a "ten" wedge) added together within a place, but with true place value across places — for most of its history it had no symbol for zero, which sometimes made a number ambiguous.
Mayan: base- $20$ , using a dot (value $1$ ), a bar (value $5$ ), and a shell for zero — and, crucially, arranged vertically with real place value, zero symbol included.
Chinese: symbols for $1$ – $9$ paired with characters for $10, 100, 1000,\dots$ — compact and partly positional in spirit, but historically written with explicit "place" characters rather than pure position.

Figure

Comparing how the decimal number two hundred twenty two gets its place value from position while the Roman numeral for the same idea repeats symbols instead

Derivation APlace value: turning digit groups into a number

A place-value system's real engine is one short rule, and the goal of this derivation is to state that rule in general form for any base $b$ and then use it to convert both a familiar decimal number and an unfamiliar Babylonian base-60 number into an ordinary counted quantity, showing that decimal place value is only the special case $b=10$ of a much more general idea.

Step 1. In any place-value system with base $b$ , a number is written as a row of digit-groups $d_{n-1}\, d_{n-2}\,\dots\, d_1\, d_0$ , where each $d_i$ is a value from $0$ up to $b-1$ (for Babylonian base- $60$ , each "digit" is itself a cluster of wedges standing for a value from $0$ to $59$ ).

Step 2. Each position, counting from the right starting at $0$ , stands for a power of the base: the rightmost position is worth $b^0$ , the next is worth $b^1$ , the next $b^2$ , and so on.

Step 3. The number's total value is the sum of every digit multiplied by the power of the base at its position:

N = d_{n-1} b^{\,n-1} + d_{n-2} b^{\,n-2} + \dots + d_1 b^1 + d_0 b^0.

where: $N$ is the number's value in ordinary counting, $b$ is the base of the system, $d_i$ is the digit (or digit-group) sitting in position $i$ , and $n$ is the number of digit-groups.

Step 4. Check the rule on decimal $222$ with $b=10$ : $N = 2\times10^2 + 2\times10^1 + 2\times10^0 = 200+20+2=222$ , exactly the everyday meaning of "two hundred twenty-two."

Step 5. Apply the same rule with $b=60$ to the Babylonian pair of digit-groups $\{2\}\{35\}$ (leftmost group first): $N = 2\times60^1 + 35\times60^0 = 120+35 = 155$ . The identical rule, only the base changed, which is exactly why place value is a portable idea and not a decimal accident.

Worked example

Problem. Write $1994$ in Roman numerals using the addition and subtraction rules, then convert the Babylonian base-60 pair $\{2\}\{35\}$ into base-10 using the place-value formula above.

Step 1. Split $1994$ into parts that match Roman values: $1994 = 1000+900+90+4$ .

Step 2. Convert each part: $1000=M$ ; $900=CM$ (since $100$ before $1000$ means subtract, $1000-100$ ); $90=XC$ ( $100-10$ ); $4=IV$ ( $5-1$ ).

Step 3. Concatenate largest to smallest: $M\,CM\,XC\,IV = \text{MCMXCIV}$ . Check by adding back: $1000+900+90+4=1994$ . ✓

Step 4. Now convert the Babylonian number using $N=d_1\times60^1+d_0\times60^0$ from the derivation: $N=2\times60+35\times1=120+35=155$ .

Result. $1994$ is MCMXCIV in Roman numerals — nine symbols and two subtraction tricks for a four-digit number — while the Babylonian pair $\{2\}\{35\}$ is simply 155 in base-10, reached with one multiplication and one addition. The contrast is the whole lesson: additive systems grow long as numbers grow large, but place-value systems stay short because each position silently carries a power of the base.

Competitive stretch

Try converting $\{1\}\{0\}\{25\}$ (three Babylonian digit-groups, leftmost first) using the same rule: $N = 1\times60^2 + 0\times60^1 + 25\times60^0 = 3600+0+25=3625$ . Even without a true zero symbol in ancient practice, the idea of an empty middle place — worth zero — was already essential to reading the number correctly, foreshadowing why the Mayan shell-zero and, later, the Hindu-Arabic zero mattered so much.

Common traps

Calling every ancient system "primitive" — Babylonian and Mayan numerals already had genuine place value, centuries before the Hindu-Arabic system spread.
Mixing up additive systems (Roman, Egyptian) with place-value systems (Babylonian, Mayan) — additive systems need more symbols as numbers grow, since no symbol's meaning changes with position.
Applying positional reasoning to Roman numerals — $IV$ and $IX$ come from a specific subtraction rule, not from any place value, since Roman numerals have none.
Forgetting that a missing "zero" digit in a place-value system can make a number ambiguous — early Babylonian notation lacked a zero symbol for exactly this reason.

Ask the tutor: Want more practice converting numbers between Roman, Egyptian, Babylonian, Mayan and decimal notation, or a side-by-side chart comparing all five systems for the same number?

Why did place-value systems eventually replace purely additive ones like Roman numerals

Because place value lets a handful of symbols represent any size of number by reusing them in different positions, while additive systems need longer and longer strings of repeated symbols as numbers grow. Arithmetic — especially multiplication and division — is also far easier to carry out with place value and a zero, which is a big reason Hindu-Arabic numerals spread worldwide for calculation while Roman numerals stayed mostly for labelling (clock faces, chapter numbers).

Did the Babylonians really not have a zero at all

For most of their history, no dedicated zero symbol existed, so an empty place inside a number could sometimes be genuinely ambiguous — readers had to judge from context whether a place was empty or simply not written. Later Babylonian texts introduced a placeholder mark for an empty middle place, but it still was not used at the very end of a number, so some ambiguity remained until other civilisations, especially in India, developed a true positional zero.

Is the Mayan base-20 system the same idea as our base-10 system, just with a different base

Yes, in structure — the Maya used place value and even a true zero symbol (the shell), exactly the two features that make a system efficient. The base was $20$ instead of $10$ , likely from counting fingers and toes, and their vertical arrangement (rather than left-to-right) is the main visual difference from what you are used to, but the underlying place-value logic, digit-times-power-of-base-and-add, is identical to the formula used above.

Number systems across civilisations

The core idea

Worked example

Competitive stretch

Common traps

Before this

Where this leads