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According to Wikipedia, there are two notable ways to render large numbers, reaching up to hundreds of thousands and higher: apostrophus and vinculum. The first uses a system of expanding rings, so that 100,000 looks something like CCC|ƆƆƆ. The second uses lines above and on the side as multipliers, so Ī might mean 1,000 and |X̄| might be 1,000,000.

Unfortunately, Wikipedia doesn't provide classical examples of the use of these, so I'm wondering: were both of these systems for rendering large numbers in use during the classical period (75 BC – AD 300)? Which system came first? Which was more commonly used?

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Well, there's the story from chapter five of Suetonius' life of Galba about how Tiberius cheated Galba out of his inheritance:

Observavit ante omnis Liviam Augustam, cujus et vivae gratia plurimum valuit et mortuae testamento paene ditatus est; sestertium namque quingenties praecipuum inter legatarios habuit, sed quia notata, non perscripta, erat summa, herede Tiberio legatum ad quingenta revocante, ne haec quidem accepit.

He showed marked respect to Livia Augusta, to whose favour he owed great influence during her lifetime and by whose last will he almost became a rich man; for he had the largest bequest among her legatees, one of fifty million sesterces. But because the sum was designated in figures and not written out in words, Tiberius, who was her heir, reduced the bequest to five hundred thousand, and Galba never received even that amount.

There's a pretty extensive discussion of large numbers here. It mentions the passage from Galba and offers the following illustration:

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