Lq

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: { r₁*q₁ + r₂*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 L¹⁰_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 q₁=1, q₂=...t1q7f1, q₃=...dcm58b and q₄=...eaprnb. In the following list, we will refer to these as 1, i, j and k.
    i² = j² = k² = 1
    ij = i/j = j/i = k
    ik = i/k = k/i = j
    jk = j/k = k/j = i
    1 has 8 square roots
    rL_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 mL_q such that 1/mL_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,nL_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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Lq

Important Properties of Lq

L_q

Important Properties of L_q