*L*_{q}

{**q**_{i}}:
All of the square roots of 1 such that i,j,
q_{i} + q_{j} 0.

Thus this is all the square roots of 1 leaving out the additive inverses.
For k=2, 1 is the only q_{i}.
For k=10, there would be two q_{i}, 1 and ...18751.
(Note that you could actually choose any of the square roots of 1 to be your
q_{i}'s as long as none of them are additive inverses.)
For k=30, there are four q_{i}.

*L*_{q}:
{ r_{1}*q_{1} + r_{2}*q*_{2} + ^{...} +
r_{n}*q_{n} | r_{i}
*L*_{r} }

For any number base that's a prime or power of a prime,
*L*_{q} = *L*_{r} .
This is because 1 is the only q_{i}.

Addition and multiplication are easy to define for *L*_{q} .
We will look at *L*^{10}_{q} as an example.

(a + bq) + (c + dq) = (a + c) + (b + d)q

(a + bq) * (c + dq) = (ac+bd) + (ad+bc)q

Clearly *L*_{q} is closed under addition and multiplication.
Thus *L*_{q} is a *commutative ring*.
*L*_{r} is a subring of *L*_{q} .

Interesting properties of 1+q:

(1+q)^{2} has only 2 square roots

(1+q)x = 1+q doesn't imply x=1 (for example x=q works)

(1+q)x = 0 doesn't imply x=0 (for example x=1-q works)

Base 30, we can choose q_{1}=1, q_{2}=...t1q7f1,
q_{3}=...dcm58b and q_{4}=...eaprnb. In the following list,
we will refer to these as 1, i, j and k.

i^{2} = j^{2} = k^{2} = 1

ij = i/j = j/i = k

ik = i/k = k/i = j

jk = j/k = k/j = i

1 has 8 square roots

r*L*_{r}
has either 0 or 8 square roots

(1+i)^{2} has only 4 square roots

r*(1+i) has either 0 or 4 square roots

1+i+j+k has only 2 square roots

r*(1+i+j+k) has either 0 or 2 square roots

#### Important Properties of *L*_{q}

**LQ1:**
If *L*_{q} *L*_{r}
, then m*L*_{q}
such that 1/m*L*_{f} .

Thus for number bases that aren't prime or a power of a prime,
*L*_{f} is not a *field*.

**LQ2:**
If *L*_{q} *L*_{r}
, then m,n*L*_{q} ,
m0 and n0 such that mn = 0.

Thus for number bases that aren't prime or a power of a prime,
*L*_{f} is not an *integral domain*.

One conclusion from this is that division is not necessarily unique
for these number bases within *L*_{q} .

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© Scott Sherman 1999