Araara ċ — Apeundeku

This is the tenth in a series of expository blog posts on Araara. The preceding post introduced Asoho: reciprocals expressed with uppercase letters. We will now combine Asoho with Akurto.

We can already write three as ba, and its reciprocal as BA. But what if we would like to write two thirds?

Uppercase, then lowercase

Our two alphabets are the uppercase and lowercase Aglobasa alphabets, in their established order:

O AE BP CJ DT FV GK HX IU LR MN SZ WY

o ae bp cj dt fv gk hx iu lr mn sz wy

Consider a pair consisting of an Asoho term followed by an Akurto term. We interpret the pair by multiplying their values.

For example, BA represents one third, while b represents two. Together:

BAb = ⅓ × 2 = ⅔

Uppercase letters come first, followed by lowercase letters. The change of case tells us where the two components meet. Here are a few more examples:

Ba = ½ × 1 = ½

Pa = −½ × 1 = −½

Ac = 1 × 5 = 5

The uppercase term gives us a reciprocal; the lowercase term scales it.

Apeundeku

We will call this type Apeundeku, after the projective line P¹ over ℚ: pe un de ku.

Its points include all fractions of integers, together with one additional point: infinity.

Equivalently, we can read a pair as a numerator over a denominator: the lowercase component supplies the numerator, and the uppercase component supplies the reciprocal of the denominator.

We already have a way to express that point. Recall that O represents reciprocal zero in Asoho. We write:

Oa = ∞

There is just one infinity here. Oa and Oe represent the same point.

If the denominator is zero, the numerator must be nonzero. Thus Oo does not represent a point; neither does any pair whose numerator and denominator are both zero.

Combining parts

For finite values, we can concatenate these pairs to add their contributions. Each new uppercase run begins another pair:

AcBAb = 5 + ⅔

AbPa = 2 − ½ = 1½

The multiplication happens within each pair; the addition happens between pairs.

As before, different expressions can represent the same value. For example:

Ba = AaPa = ½

The first expression gives one half directly. The second gives one minus one half. We would like an agreed normal form.

Normal form

For a finite point, first take its whole-number part towards zero. Write A followed by that integer’s Akurto normal form.

Then write the remaining fractional part. Reduce it so that its numerator and denominator have no common factor greater than one. Its magnitude must be less than one. For example, CAbb is four sixths: dividing both numerator and denominator by two gives BAb, two thirds.

Write the numerator as a positive Akurto term. The uppercase component supplies the reciprocal of the denominator, including the negative sign when needed. Both components use their respective normal forms.

For a negative value, the whole-number and fractional contributions are both nonpositive. Omit either part if it is zero. If the entire value is zero, use Ao. Here are some normal forms:

For AcBAb, the first pair contributes five and the second contributes two thirds. For AjPEb, the first pair contributes negative five, while PE represents negative one third and b represents two.

When both parts are present, the integer part comes first and the reduced fractional part second. Integers and proper fractions need only one pair. Infinity has normal form Oa. This gives each point one agreed representation.

Exercises

1. What values do Ca, Bab, and AcPa represent?
2. Put Bb, AaPa, and AbBAb into normal form.
3. Write the normal forms of one third, negative two thirds, and seven halves.

We can now combine integer quantities with reciprocals, express every point of Apeundeku, and choose a normal form for each.

In the next post, we will extend the letter-value recurrence with Alungo.