Araara c· — Attic
This is the eighth in a series of expository blog posts on Araara. The preceding post explored references to books, atoms, galaxies, and colours. The series begins here.
Let us introduce a new alphabet: Attic. We will use the Greek zero sign 𐆊 as its zero element, followed by ten Attic numeral symbols.
This is a gapped alphabet. Each nonzero symbol is followed by a gap, so none of these symbols has an inverse in the alphabet. As before, we write each gap as ui.
uissllggfddc uihhgfdd ui uihhgfddcb ui uihhgfdcbb ui uissllggbb ui uihhgfdccb ui uissllggc ui uihhgffa ui uissllggca ui uihhgfddba ui uissllggcb ui
𐆊 Ι □ Π □ Δ □ 𐅄 □ Η □ 𐅅 □ Χ □ 𐅆 □ Μ □ 𐅇 □
Decoded: Greek zero first, then each Attic symbol followed by a gap. There are twenty-one positions in total.
The value of each symbol
We give the ten nonzero symbols their usual Attic numeral values. The positions organise the alphabet; the values follow Attic notation rather than our earlier recurrence:
| Symbol | Value |
|---|---|
| 𐆊 | 0 |
| Ι | 1 |
| Π | 5 |
| Δ | 10 |
| 𐅄 | 50 |
| Η | 100 |
| 𐅅 | 500 |
| Χ | 1,000 |
| 𐅆 | 5,000 |
| Μ | 10,000 |
| 𐅇 | 50,000 |
The Greek zero sign is our added zero element. The gaps remain empty positions; they are not numeral symbols. With no negative symbols, Attic terms represent nonnegative integers. Attic numeral reference
Adding the symbols
In Attic, just like in Akurto, we add the values of all the letters in a term. The order does not matter!
For example, we can write 2026 as:
ΧΧΔΔΠΙ = 1000 + 1000 + 10 + 10 + 5 + 1 = 2026
Rearranging the symbols leaves the total unchanged:
ΙΠΔΧΔΧ = ΧΧΔΔΠΙ = 2026
Normal form
We choose a normal form by working from big to small: write the largest numeral that fits the remaining total, subtract its value, and repeat until nothing remains. Our first expression for 2026 is already in normal form!
This chooses both the symbols and their order: five ones become a five, and five tens become a fifty. The result runs from largest value to smallest. For zero, use the Greek zero sign.
Exercises
- Write 26 and 365 in Attic normal form.
- Write 2026 using only the symbols for one, ten, and a thousand. How many symbols do you need?
- Rearrange one of your terms. Check that its value stays the same, then put it back into normal form.
A new alphabet gives us a new way to write a sum. The characters have changed, but addition is still familiar.
In the next post, we will consider reciprocals.
