Hi. How does it work when the base of your number system is not natural, or even rational? You can't have phi different digits?
Hey, thanks for asking about mysymbiote number system headcanon!
The full (or full-ish) answer for how phinary took me a while tofigure out how to explain. I tried to write this as clearly and succinctly aspossible, so please tell me if anything is still confusing. I usedphinary through out this to refer to base phi/the golden ratio base,because I also talk about base pi a bit, and I wanted to minimizeconfusion. Also, phinary is just catchier-sounding than “base phi.”
The first part is relatively simple: phinary uses 0and 1 as its digits, same as binary (base 2), but with onespecial restriction: you can’t have multiple 1s in a row, like in“11″ or “111″. (More on this later.) In the original post,that’s why I suggested using both binary and phinary, since you canrepresent them with the same set of characters.
As it turns out, all number systems with non-integer basesuse “integerdigits up to (but not including) the base”. Therefore, sincephi is roughly 1.618, we can use 0 and 1 in phinary. In base e, wecan use 0, 1, and 2. In base pi, we can use 0, 1, 2 and 3. I amnot a mathematician, and I cannot give the deets on why it works likethis, but I can tell you that it does. ¯\_(ツ)_/¯
This can make representing integers difficult. For example, inbase pi, “10″ represents the value 3.14159…. But how do yourepresent the value 4? Because pi is irrational, writing the value of4 in base pi requiresinfinite digits! Obviously, this is impractical.
However, in phinary, we can sidestep this using a funproperty of the golden ratio: every integer can be written in termsof sums of powers of phi! For example:
phi ≈ 1.618, and phi^-1 ≈ 0.618, so phi - phi^-1 = 1
(1 can also be written as phi^0, since any number to the zeroth power equals 1.)
phi^-2 ≈ 0.382, so phi + phi^-2 = 2
phi^2 ≈ 2.618, so phi^2 + phi^-2 = 3
phi^2 + phi^0 + phi^-2 = 4
In base ten, we write the powers of ten like so:
Becausewe’re using phinary, we know that the powers ofphi will follow the same pattern:
Thenby combining this with the list we just made, we can see that 4 inphinary would be written like this:
phi^2 + phi^0 + phi^-2 = 100+ 1 + 0.01= 101.01.
So unlike base pi, in phinary we canuse a finite number of digits to write 4. (The golden ratiobase/phinary is actually the only irrational base where it’spossible to write out every integer using a finite number ofcharacters.)
Thenumbers from 1 – 10 are written out in phinaryin theexamples section of Wikipedia page for the golden ratio base, soI won’t calculate them all here. But hopefully that makes itclearer how they’re getting there.
Now, you may notice that there’s multiple ways to sum up powersof phi to get a given integer. For example, you could represent 5 as:
phi^2+ phi^0 + phi^0 + phi^-2 = 100 + 1 + 1 + 0.01 = 110.01.
But the correct (or “standardform”) way to write 5 is as:
phi^3 + phi^-1 + phi^-4 = 1000 + 0.1 +0.0001 = 1000.101.
So there’s two ways to write one number? Whatgives?
Well, yeah. There are. That’s why the restriction on consecutive1s comes into play. It turns out, numbers written in phinary are onlydefinitely finite and unique if they’re written without consecutiveones. This is called standard form.
But writing all the numbers without “11” is easy, if you useanother fun property of phi:
The actualmath is simple:
But we already know that in phinary, we write phi^0 as 1, phi as10 and phi ^2 as 100. So we can rewrite phi^0 + phi as 1 + 10, and see that in phinary …11 = 100.
(Yes, this is DEEPLY WEIRD.)
So for larger numbers in phinary, you actually have to shiftnumbers around a bit to get a number that works. Fortunately, a fewkind souls have built decimal-to-phinarynumber converters, including one that can do rangesof numbers so I will never ever have to do this math.
As for what practical applications phinary has? …The only thing I was able to find was some research into image enhancement/decomposition methods. But the golden ratio famously appears in a bunch of place in nature, so there might be cool stuff you can do with phinary that we just do a different way because it’s hard to think about. I mean, with symbiotes, they’re already so comic-book-sciencey I’m perfectly fine with making up any explanation I need to make the real science concepts I think are cool work. :P But the recent association of symbiotes with spirals is definitely fun fuel for this headcanon!
I genuinely enjoyed reading through all the math pages to be ableto explain this. Please reblog or comment with any additionalquestions you have if I didn’t explain something clearly enough. :D
