Ancient Number Systems Converter

How to Use the Ancient Number Systems Converter

1. 1. Enter a whole number. Every system starts at 1. Each has its own top: Babylonian 999,999,999,999, Maya 999,999,999, Egyptian 9,999,999, Attic Greek 99,999 and Ionic Greek 9999.
2. 2. Choose the Babylonian empty place. Early scribes left a gap where a base-60 digit was missing; later ones marked it with a placeholder. Pick either to see how a number with an empty place was written.
3. 3. Choose the Maya place values. The Long Count calendar uses 1, 20, 360, 7200 and 144,000. Pure base 20 uses 1, 20, 400, 8000.
4. 4. Read each card. Every system shows the numeral, a table of what each sign contributes, and a note saying whether it is positional or additive. Copy any numeral into your notes.

Positional and Additive Systems

Two kinds of system appear here. In a positional system the same sign is worth more further left: the Babylonians (base 60) and the Maya (base 20) wrote numbers this way, as we do. In an additive system the signs are simply added up in any order, as with Roman numerals: the Egyptian and both Greek systems work like this. Additive systems need no zero. Positional ones do, or at least a way to show an empty place.

Babylonian: Base 60 and the Missing Zero

Babylonian scribes pressed two wedges into clay: a vertical wedge for 1 and a corner wedge for 10. A digit from 1 to 59 is built from up to five tens and nine ones, and the digits are placed left to right in powers of 60. Sixty has many divisors, which is a good reason for a base used in trade and astronomy, and the survival of 60 in minutes, seconds and degrees comes from this system.

Early scribes had no sign for an empty place and left a gap, so 61 (digits 1 and 1) and 3601 (digits 1, 0 and 1) could look alike unless the reader knew which was meant. Late in the Seleucid period, in the third century BCE, a placeholder of two slanted wedges appeared. Historians describe it as a separator and not a true zero, and in mathematical texts it was written inside a number but not at the end. That is why a trailing zero is not drawn here, and why 1 and 60 have the same cuneiform.

Maya: Base 20 and the 18×20 Exception

The Maya wrote the digits 0 to 19 with a shell for zero, dots for ones and bars for fives, and stacked the places vertically with the highest at the top. In a pure base-20 system the places would be worth 1, 20, 400 and 8000. The Long Count, which counts days, breaks the pattern once: 20 k'in (days) make a winal, but 18 winals, not 20, make a tun of 360 days, a number close to a solar year. After that the multiplier returns to 20: 20 tun make a k'atun (7200 days) and 20 k'atun a b'ak'tun (144,000 days). In Long Count mode the tool applies this rule, so the winal digit runs only from 0 to 17, and shows the dotted form 13.0.0.0.0 for 1,872,000 days.

Egyptian, Attic and Ionic Greek

Egyptian hieroglyphic numerals have a sign for each power of ten up to a million: a stroke (1), a heel bone (10), a coil of rope (100), a lotus (1000), a finger (10,000), a tadpole (100,000) and the god Heh (1,000,000). Each is repeated up to nine times. The writing can run in either direction, and the tool writes the largest sign first.

Attic Greek numerals, found in Athenian inscriptions, use the initial letters of the number words: Ι for one, Π (pente) for 5, Δ (deka) for 10, Η (hekaton) for 100, Χ (khilioi) for 1000 and Μ (myrioi) for 10,000. A Π with a smaller letter inside it makes 50, 500, 5000 and 50,000. The compound signs are in Unicode's Ancient Greek Numbers block, and a plain-letter fallback is shown for fonts that lack them. Ionic (Milesian) numerals use the whole alphabet, with three archaic letters, stigma (ϛ, 6), koppa (ϟ, 90) and sampi (ϡ, 900), to reach 1 to 9, 10 to 90 and 100 to 900. A raised keraia (ʹ) marks the numeral and a lower keraia (͵) marks thousands: ͵βκϛʹ is 2026. Modern Greek still uses these for chapter numbers and ordinals.

Frequently Asked Questions

Did the Babylonians have a zero?

Not as a number. Early texts left a gap, and from about the third century BCE a placeholder of two slanted wedges marked an empty place inside a number. It was a separator, not a value: nobody added or subtracted zero.

Why did the Babylonians use base 60?

Nobody knows for certain. A frequently given reason is practical: 60 divides evenly by 2, 3, 4, 5, 6, 10, 12, 15, 20 and 30, which suits sharing and measuring. Its legacy is the 60 minutes in an hour and 360 degrees in a circle.

How did the Maya write numbers?

With three marks: a shell for 0, a dot for 1 and a bar for 5, giving the digits 0 to 19. Places were stacked vertically in base 20, the units at the bottom. Unicode has a Maya numerals block, which this tool uses.

Why is the Maya Long Count 360 and not 400?

Because the third place counts 18 twenties. A tun is 18 winals, or 360 days, which is close to the length of a solar year. Every other step is a factor of 20, so the place values run 1, 20, 360, 7200 and 144,000.

In which direction do you read Egyptian numerals?

Either. Hieroglyphs can run left to right or right to left, and because the system is additive the order of the signs does not change the value. Conventionally the largest sign comes first, as in this tool.

What is the difference between Attic and Ionic Greek numerals?

Attic numerals are acrophonic: signs are the first letters of the number words, repeated to add up. Ionic numerals give every letter of the alphabet a value from 1 to 900, so a number such as 999 needs only three letters. The Ionic system won out and is the one still used in modern Greek.

Why do I see empty boxes instead of symbols?

Cuneiform, Egyptian hieroglyphs, Maya numerals and the Attic compound signs are rare Unicode characters, and your device needs a font that includes them, such as a Noto Sans font for that script. The plain-text forms next to each numeral work without one.