Nice quaternions

I will call a quaternion nice if both the number and its multiplicative inverse have finite decimal representations. In other words, if $z=a+bi + cj + dk$, then the real and imaginary components of both $z$ and $z^{-1}$ have terminating decimal representations.

Always remember, mathematicians use $i$ for the complex imaginary unit, while engineers use $j$. Fortunately, however, $\textrm{span}\{ 1, i \}$ and $\textrm{span}\{ 1, j \}$ are both isometrically isomorphic sub-fields of the quaternions.

All complex numbers are quaternions, so please see this document to see all nice quaternions that have only two non-zero components.

We will first consider quaternions of the form $z=\alpha+\beta i + \gamma j$, where $\alpha$, $\beta$, and $\gamma$ are relatively prime integers. We need consider only one representative with $\alpha\geq\beta\geq\gamma \gt 0$, since swapping the various components or independently changing their signs produces other nice quaternions.

Now,

A rational number has a terminating decimal representation if and only if its denominator, after cancellation, has no prime factors other than $2$ and $5$. Since $\alpha$, $\beta$, $\gamma$ and $\delta$ are relatively prime, no prime can divide all $\alpha$, $\beta$, $\gamma$, or $\delta$ together with $\alpha^2+\beta^2 + \gamma^2 + \delta^2$, Consequently, no factors in the denominator can be cancelled from both components of $z^{-1}$. Therefore, $z^{-1}$ has finite decimal components precisely when

for some non-negative integers $r$ and $s$.

The following are all quaternions with at most three significant digits in three components with the last being zero:

The only quaternions with three non-zero components, including scalar multiples of the above, that have only one significant digit in all components are:

The following are all quaternions with at most three significant digits in four components with the last being zero, ordered by their absolute value:

The only quaternions with four non-zero components, including scalar multiples of the above, that have only one significant digit in all components are:

Nice quaternions on the unit hypersphere

A particularly useful subset of nice quaterions consists of those that can be scaled onto the unit hypersphere while retaining finite decimal representations for both components. For a primitive quaternion $z = \alpha + \beta i + \gamma j + \delta k$, this occurs when

The first such numbers are:

All components with one significant digit

Two components with two significant digits

Three components with two significant digits

Four components with two significant digits

Changing either sign or swapping the components produces other corresponding nice points elsewhere on the unit hypersphere.

Rotations

One significant application of quaternions is in the area of representing rotations in 3-dimensional space. If $\textbf{v}$ is a 3-dimensional vector represented by $v = v_x i + v_y j + v_z k$, then a rotation around $\hat{\textbf{u}}$, represented by $u = u_x i + u_y j + u_z k$ with $|u| = 1$, of $\theta$ radians is $$q = \cos\left(\frac{\theta}{2}\right) + \sin\left(\frac{\theta}{2}\right)u = \cos\left(\frac{\theta}{2}\right) + \sin\left(\frac{\theta}{2}\right)(u_x i + u_y j + u_z k),$$ and then performing this rotation is equivalent to calculating $qvq^{-1}$, so if $v = 6i + 2j - 3k$, so $|v| = 7$, and the rotation is represented by $q = 0.98 + 0.06i + 0.18j - 0.06k$, so we are rotating around the vector $\left(\begin{array}{r} 1 \\ 3 \\ -1\end{array}\right)$ with an angle of $\theta \approx 0.4007\textrm{ radians}$ or approximately $22.957^\circ$, then the result is $qvq^{-1}$ or $$(0.98 + 0.06i + 0.18j - 0.06k)(6i + 2j - 3k)(0.98 - 0.06i - 0.18j + 0.06k) = 4.8096i + 1.8128j − 4.752k,$$ which also has an absolute value equal to seven, as

$$\phantom{+}23.13225216 \\ \phantom{+0}3.28624384 \\ +\underline{22.581504\phantom{00}} \\ \phantom{+}49\phantom{.00000000}$$

For a rotation of approximately $0.4007\textrm{ radians}$ or approximately $22.957^\circ$, use any one of these with the three imaginary components swapped or negated: $0.98 + 0.14i + 0.14j + 0.02k$, $0.98 + 0.18i + 0.06j + 0.06k$, or $0.98 + 0.14i + 0.1j + 0.1k$.