How are quaternions a finite set?

Question

I'm having trouble understanding how Quaternions are a finite set when you can express a quaternion as Q = a + ib + jc+ kd, because a, b, c, d are $\in$ of $\Re$ would this not mean that the set is infinite?

Answer 1

The quaternions, as a set, is infinite.

However, the quaternions, seen as a vector space over the reals, has finite dimension, e.g. the set $\{1,i,j,k\}$ form a basis, so the dimension is 4. This means that any quaternion $x$ can be written uniquely on the form: $$x = \alpha_11 + \alpha_2i + \alpha_3j + \alpha_4k$$ for $\alpha_1,\alpha_2,\alpha_3,\alpha_4 \in \mathbb R$.

The quaternion group $$Q = \langle -1, i, j, k \mid (-1)^2 = 1, i^2 = j^2 = k^2 = ijk = -1 \rangle$$ is a group of finite order, the order being 8.

Answer 2

The set of all quaternions is not finite. However, the basis quaternions (1,i,j,k) generate a finite group under quaternionic multiplication.