They&#x27;re quite as real as nonnegative numbers, which are also abstract things. It&#x27;s ironic that in your last line you use the terms &quot;real&quot; and &quot;imaginary&quot;, in their colloquial meanings, as these have long since been adopted and coopted to denote... the real and complex numbers respectively :) which, some of us contend, are no less real than the integers. Face it: groups abound in physics. A group has a binary operation such that for every element has a (unique) inverse element w.r.t. that operation. When the group is Abelian we write the operation as &quot;+&quot;, and then the &quot;additive&quot; inverse of an element x is -x, also an element of the group. It&#x27;s silly to claim that, for example, rotation through an angle $\theta &gt; 0$ should be considered more &quot;real&quot; than rotation through $-\theta$. One is counterclockwise, the other, clockwise, by the same amount.