Complex Bases

Bellow is Donald Knuth, his most famous work is probably The Art of Computer Programming for which the content won him a Turing Award in 1974. He also came up with the Up Arrow notation used in my posts on Large Numbers and God. Those posts were inspired by one of Knuth’s books, this post was inspired by his number system.

The Quater-Imaginary System
As a student at High School he entered into a science talent search and his submission was the Quater-Imaginary system, or Base 2i. It’s interesting because it uses the digits {0,1,2,3} for representation so it would seem at first glance to be quaternary (Base 4) – but it’s not!

It is actually an imaginary base, as the complex version is (0r±2i). Like the decimal system, quater-imaginary can finitely reperesent all positive real integers -but it can do more- it can finitely represent all positive AND negative real AND imaginary integers without signs (i.e., 3i, -8). Which can be seen in this table:

Base 10 Base 2i Base 10 Base 2i Base 10 Base 2i Base 10 Base 2i
1 1 -1 103 1i 10.2 -1i 0.2
2 2 -2 102 2i 10.0 -2i 1030.0
3 3 -3 101 3i 20.0 -3i 1030.2
4 10300 -4 100 4i 20.0 -4i 1020.0
5 10301 -5 203 5i 30.2 -5i 1020.2



The Dragon System

The Twindragon also known as the Davis-Knuth dragon!

It’s not offically called that, but the second most known complex system is Base (−1±i) which has an associated fractal shape (twindragon). It uses the numbers {0,1} for representation. It’s a very clean and eligant number system that was created by Walter F. Penney in 1965. As with quater-imaginary, this number system can be used to finitely represnt the Gaussian Integers.

Interestingly the radix starts with a negative number, but this isn’t a problem, negative bases work just aswell as positive ones. Infact in 1957 a Polish computer called BINEG was designed using negabinary!

Base 10 Base (-1±i) Base -2 Base 2
1 1 1 1
2 1100 110 10
3 1101 111 11
4 111010000 100 100
5 111010001 101 101

Base (-1±i√7)/2 and Others

Here the length of the numbers don’t increase monotonically, for example 12(11001100) and 13(11001101) are shorter than their predecessor 11(11100110011) which is the same length as 14(11100010110). It isn’t the only base to hold this attribute but it’s one of the quickest to show it. Source and explination here.

As well as whole-number radix systems it is possible to use fractions and even irrational numbers, one example I go over is Phinary (Base 1.61803…) also known as the Golden Ratio Base. On my Research Page I am looking at a series of systems, starting with the golden ratio, which I call the Metallic Series that can all be used under the same rules.

I find number systems quite interesting and I have started messing around modelling them in different ways, in a previous post (Numeral Automata) I look at them using cellular automata.


18 thoughts on “Complex Bases

  1. I was going to say that there is a typo in your base -2 column, but reading the rest I wasn’t so sure. As always, an interesting read, albeit above my head in most respects.

    • Thanks for once again delving into my non-political posts! It’s much appreciated 🙂

      If the misstype is that the binary and negabinary 4 and 5 are the same, it looks odd but it’s actually correct. A three digit number is in the form x * b^2 + y * b^1 + z * b^0 and if you punch in the numbers for 101_(-2) you get 1 * (-2)^2 + 0 * (-2)^1 + 1 * (-2)^0 = 5

      • No, I thought that since the second and third items in that column were higher numbers than the ones in the fourth and fifth ones, that it was a mistake, but then I read on. You’re up awful early. I didn’t even go to bed “last night” until about 5 am, my time of course.

    • My sleeping pattern fluctuates like madness! I was going to do some programming but first I’m going to read up on your third candidates post. I saw Larry King’s debate with Jill, Gary and two others.

    • Ah man, it’s partly my fault. There was so much I wanted to talk about that I tried to condence it, otherwise it would have been pages long. In the near future I think I’ll do a post on each system from theory to use.

      If you take anything away from it, take that Donald Knuth is a pioneer of theoretical computer science.

      • 😀
        Not to worry or stress, it’s not your fault, I’m just very curious and try and pick up things but have never really studied this level of math or anything like most of the stuff you share – I just like to dabble and try and comprehend what I can manage.

  2. Great stuff. Have you by any chance messed around with p-adic numbers? I’ve been recently working on hypercomplex extensions of the p-adic metric (i.e. split-quaternions using the p-adic metric). I have no idea how it’s all going to work out just yet, but that’s ok – uncertainty and curiosity is what science is all about!

    Really great post. Knuth is a genius; I’ve learned a ton from his work. 🙂

    • No there’s a couple I’ve been wanting to look into for some time, P-adic and Factorial number systems, they look very interesting – moreso the P-adics.

      If you do any later posts, link them in the comments here as I’d love to see.

    • The problem with e is that it is transcendental, so the units in Base e [0-2] would be finite but anything more would instantly be infinitely long.

      Essentially the rule for digits used in any general irrational system is {\lfloor}R{\rfloor} + 1 where R is the radix (so e in this case). We run into trouble trying to represent the rational number four, but not irrational 2e. This is a close-ish representation I came up with, you can go closer but you’ll never reach exactly 4:

      4_{10} \approx 11.02002_e = 4.00242828893044...

      The same applies for base pi and other transcendental numbers like (i^i).

      • Thanks, what about phi, I see there are minimal and maximal kind of representations. I am interested because I have seen some interesting patterns when you see the binary representations of prime numbers. Like reversed bits of many prime numbers are another prime number etc.. You seems to be interested in numbers quite well. 🙂

      • Well phi is interesting in that it has a field of Q[√5] (which is of the form Q + √5Q), meaning it can represent any non-negative rational integer and any element (rational or irrational) of that field.

        I would hazard a guess and say if you found something interesting it would be exclusively when Prime == 5 or when Prime != 5 but not both, because of the field relations. Although that’s just an off-the-top-of-my-head thought!

        Have you scrolled through my research page? There’s not much there and the metallic series might be a good one to look at because for instance the silver ratio has a field of Q[√2] which is another prime-root.

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