Complex Evens and Odds

 I have sometimes found myself enjoying experimental math with questionable practicality.  For example, fractional bases can be fun to play with, and negative bases can be really confusing.  The practical value of fractional bases probably does not exist, and negative bases may have some practical uses, but the complexity is high enough that there probably aren't many practical uses for them.  In writing the previous article on the terms "even" and "odd", I ended up being forced to consider the implications of those terms in complex numbers, and I have discovered some interesting things.  So, I am going to write about it!

Initially, I assumed even and odd cannot apply to complex numbers, however, it turns out I was wrong, and I even managed to prove my wrongness with an example.  There are a few problems that make this hard to understand.  First, complex numbers aren't just two independent numbers.  A complex number is essentially a 2D number, represented as a real and an imaginary component.  It's still one number though.  Now, the magnitude of a complex number isn't just its component added together.  It's more complicated than that.  It's actually the length hypotenuse of the right triangle that would be made, if the other two sides were the length of the two components.  This means that a complex number with magnitude 2 might be represented as sqrt(2) + sqrt(2)i.  The components don't necessarily have to be integers for the actual magnitude to be an integer value, and that really complicates things.  Another problem is that even and odd only make sense in the context of discrete math, thus we can't call sqrt(2) + sqrt(2)i even, because its components are not integers and thus don't fall into a domain governed by discrete math.  This is severely limiting, because it means that complex numbers that fall into the integer domain must have integer components and must have integer magnitude.  That excludes a ton of complex numbers that have integer components, because most don't have integer magnitude.

I am not sure how useful this is.  I think it clearly falls into the realm of experimental math, with questionable practicality, but I find this interesting, and if it even might have practical value, it is worth putting some time into exploring.  The rest of this will be some examples and exploration, to get a better feel for what we are looking at, and to see if anything interesting arises from it.

So first, 1 + 1i doesn't have an integer magnitude, so it isn't a valid complex integer.  Now, on a graph, complex integers can only exist on grid vertices.  This will help narrow down the possible options in a way that can be understood visually.  Since we know 1 + 1i isn't an integer (it's magnitude is 1.41), we know that not all vertices are complex integers.  I am thinking we could take a numbered grid, and then put circles on it at integer distances from the origin, and anywhere a circle precisely intersects a grid vertex, there is a complex integer.  We would, of course, want to verify any integers we find, as we may see an apparent intersection that is merely extremely close and only appears to intersect due to resolution limitations.

There are a few intersection points I know off the top of my head.  For example, the last article used 3 + 4i as an example.  This is the common Pythagoras theorem example, with a magnitude of 5.  We can also swap components, for 4 + 3i = 5, and we can swap signs as well.  This means we have rotational symmetry at 90 degree (quadrant) intervals, and within each quadrant we have reflective symmetry across the 45 degree quadrant bisector.  This is similar to algebraic rules in real math, like the Commutative property, except that because magnitudes are always positive, negative and positive values can be switched without changing the magnitude.  (This does bring up the question of if and how angle plays into this.  Not sure I want to deal with that right now, and I suspect that adding angles will break the whole integer thing, because I seriously doubt any/many complex integers will land on integer angles.  That said, the scale for angles is arbitrary, so "integer" really doesn't apply in any real way...unless maybe this exercise reveals a discrete scale of angles that has thus far remained undiscovered...  Maybe I need to print off this graph with circles and draw lines from complex integers to the origin, and see if that yields any interesting patterns...)  Any integer multiple of either of these is also going to have an integer magnitude.  And that leads to the realization that we are actually looking at vector math here.  We can also represent complex numbers as vectors, <3, 4> or coordinates, (3, 4), and realizing this, it is now obvious to me that the complex domain uses vector algebra for its operations, and this makes me wonder how vector algebra would work in the complex integer domain.  Honestly, I am beginning to think I may have just gotten myself way deeper than I did with negative bases.  I am hoping this isn't more complex than non-reciprocal fractional bases...  (Reciprocal bases, ie., fractional bases where the numerator is always 1, are trivially easy.  The one time I tried a fractional base with a numerator higher than 1, however, was a disaster.  That's one of the most complicated mathematical things I have tried and failed to do.  And I should note, I got an A in multivariable calculus.)

So, I think I am going to make the graph with circles at integer distances from the origin and see if I can find more complex integers.  From there, I can look for patterns.  I am sure I will find some, because that's just what happens with math, but the question now is whether I will find anything significant or not!

I guess I know where I am going from here.  I have convinced myself, through this line of thought, that this might have some practical value after all.  It might only serve to move math theory forward, but that is practical value, as most elements of math theory seem to eventually lead to something of significant value being discovered or created.  I am seeing a lot of potential in discovering interesting and perhaps undiscovered patterns from this, so maybe...