How the Ancient Greeks Did Math With Letters, Not Numbers

The ancient Greeks were incredibly talented mathematicians—but they rarely used numbers in their math. Their particular specialty, geometry, dances around actual quantities, focusing on higher-level logic and constant relationships. Even Pythagoras, whose triangles we navigate with easy examples like “3, 4, 5,” or “5, 12, 13,” was way more interested in diagrams than in specific situations.

But the ancient Greeks certainly had numbers. In fact, they had what was in some ways a much more elegant system than the clunky Roman numerals—like I, II, III—that we still occasionally adopt today. Like Roman numerals, their system borrowed letters; like the Arabic numerals we still use, it only needed one symbol for each decimal place.

In the 6th century BCE, the Greek alphabet used 24 letters. To make numbers, the Greeks added three more symbols (accounts differ as to whether these were resurrected older letters or newly designed characters), then doled them out nine at a time to account for the ones, tens, and hundreds columns. Hence:

1–9: Α, Β, Γ, Δ, Ε, [digamma], Ζ, Η, Θ

10–90: Ι, Κ, Λ, Μ, Ν, Ξ, Ο, Π, [qoppa]

100–900: Ρ, Σ, Τ, Υ, Φ, Χ, Ψ, Ω, [sampi]

These were combined to create individual numbers—for example, MA meant 41, and PNE meant 155. After they ran out of alphabet, a small comma-like tail at the lower left of a letter between Α and Θ signified thousands—so ,BTKZ meant 2327. An M with a small letter or set of letters above it stood for tens of thousands and higher. (Confusingly, in this instance the M represented the Greek ancestor of “myriad,” not the number 40.)

All together, that got a Greek up to 99,999,999—a much bigger number than they ever really needed, either for daily life or complex mathematics. They could accommodate fractions, too: A tick mark to the upper right of a number meant one divided by that number, and they developed special symbols for common fractions like 1/2.

Like any system, it had its pros and cons. Using different symbols for different magnitudes tidily got around the Greeks’ lack of a symbol for zero. Where we use 0 as a placeholder in writing numbers of different magnitudes (1 versus 10, for example), they used different symbols entirely—but that meant they needed more symbols. In certain cases, like abbreviation-prone inscriptions, figuring out whether a character was a number or a letter might have taken a second glance, but numerology gave a deeper meaning to letters and words [PDF].

This alphabetic counting system first arose as a local phenomenon on Greek islands in the Aegean during the 6th century BCE, likely adopted from the Egyptians through trade. At about the same time, Pythagoras was revolutionizing math with his theorem just a few islands away, on Samos (although we don’t have any clear evidence he used alphabetic numerals). It was one of many local number systems that varied across the Greek-speaking world and presaged the Roman system of focusing on repeating ones and fives.

Even locally, alphabetic numerals fell out of popularity for about 150 years, only to come rushing back on the scene during the late 4th century BCE—just in time for Archimedes to discover the joy of applied mathematics and for Ptolemy to calculate the latitude of thousands of places.

The timing isn’t a coincidence: the system benefited from Alexander the Great’s contemporary imperial conquests. Greek-speaking territories were suddenly more politically unified than ever before, and using similar but not quite identical systems across the land was simply not going to work. 

The alphabetic system shone in comparison because it didn’t cause miscommunication, and that geographic mobility gave it domination over the other systems. It also gave it staying power: the system remained in use in Greek texts and Greek-speaking territories—even as the Roman numerals still seen today were born—into the 15th century CE, when Greek letters finally lost out to Arabic numerals.

Arabic numerals aren’t going anywhere, but if you’re a little bored with balancing your checkbook or tallying your fantasy baseball score with twos, sevens, and fives, brush off those Bs and Zs and imagine yourself in ancient Greece. 

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