a few complex multiplication tables

Someone putting together a multiplication table of real integers has very few important decisions to make: Will it have as many rows as columns? The answer is usually yes. What range will be covered? Usually 1 to 10, or 1 to 12 in the old days, for both rows and columns. After making those decisions, all the other decisions are probably purely cosmetic: What font to use? What base? Usually base 10. What multiplication operator? The tacit multiplication operator is sometimes used, but the multiplication cross $\times$ is probably preferred.

But for someone wanting to put together a multiplication of complex integers there is suddenly an embarrasse de choix, and the two-dimensional table structure appears inadequate. It would be unfair to expect schoolchildren to memorize complex multiplication tables, and perhaps it might be better to just teach the identity $(a+bi)\times (x+yi)=(ax-by)+(ay+bx)i$. On the other hand, just showing a few complex multiplication tables might help make the subject more “real” (atrocious pun fully intended).

To help us orient ourselves, we could imagine our beloved real integer multiplication table as hanging on a wall, with each entry having an additional “$+0i$” which for convenience we normally leave out. On the wall behind that table there might be a table multiplying numbers of the form $a+i$. Here we’ll limit ourselves to five rows and five columns of results to avoid horizontal scrollbars or print-outs chopped off on the right edge. Another convenience we’ll avail ourselves to is using $r$ as the row number variable and $c$ as the column number variable.

On a perpendicular wall we might have a table in which the operand column is $ci$ and operand row is $1+ri$:

The real parts are what we would get in our “normal” table, though multiplied by $-1$. The imaginary parts are consistently $ri$. Plugging $b=y=1$ into our identity stated above, it reduces to $(0-y)+(0+x)i=-y+xi$ or $-c+ri$, which explains the observation.

Perhaps on the floor we might have a table in which both the rows and columns are of the form $0+mi$:

This looks an awful lot like the usual multiplication table. Our diagonal for $r=c$ has numbers of the form $-(n^2)$, which with a little reflection is an obvious consequence of the fact that $\sqrt{-1}=i$.

Things get more interesting when we go down to the basement, where upon the wall is a table in which the operand row has $r+i$ and the operand column has $c-i$:

There is an interplay of arithmetic progressions in both the real and imaginary parts. Observartion shows that the real parts follow the pattern $c+cr$ and the imaginary parts follow the pattern $ci-ri$. Plugging $b=1$ and $y=-1$ into our identity stated above reduces it to $(ax+1)+(-a+x)i$. A result of this interplay is that our $r=c$ diagonal has numbers with no imaginary parts, some of them we normally consider prime. But since they are on yield cells of a Gaussian integer multiplication table, they are not Gaussian primes.