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science:mathematics

# Digital Mixed-Power non Binary Coding

Higher order symbolics instead of binary (-2) [vote for, against]

I could envision circuitry for new type of numbering:

Mixed powers:

Let's say we want to use 2 and 3:
The order is: (I wrote these one under the other...)
0 = 0000,
1= 0001,
2 = 0010 (the second place is for 2's),
3 = 0100 (the third place is for 3's),
4 = 0101 (3 and 1),
5 = 0110,
6 = 1000 (that's 2x3),
7 = 1001 (2x3+1),
8 = 1010 (2x3+2),
9 = 1100 (2x3+3),
10= 1101 (2x3 + 4),
11= 1110 (+5),
12=10000 (2x6)

It creates a lot of interesing mathematics and in particular eases even/odd number arithmetic. Could cause quite a stir in computer devices.

 — pashute, Oct 30 2002

What is 0,011? 0,111? 1,111?
 — phoenix, Oct 30 2002

Thanks jutta
 — pashute, Oct 31 2002

 The rule is: use 3 places each time (you don't have 011, and you don't have 111). Uses an extra "bit" but makes math faster.

You also don't have 11000 which instead is 100000
 — pashute, Oct 31 2002

 Erm, why? In your example, you counted up to the number 12, and already you're using five bits. Traditional binary only has to use five bits once you get to 16, so already your system is significantly less efficient - it uses a lot more memory space for the same information.

 As for making math faster, how can it possibly do that? At best it will be the same speed as normal binary, and at worst you will have to have insanely complex addition and subtraction circuitry - I don't even want to contemplate implementing multiply!

So it seems you have an idea that's slower, less space efficient and more complex to implement than what we have already. Or am I misunderstanding something?
 — pmillerchip, Oct 31 2002

 No, I think you've got it, pmillerchip. It's a number system with redundancy and complicated rules. I can't see an application, just lots of downside.

Someone tell me the point.
 — st3f, Oct 31 2002

There's a school called Downside. In fact there's a whole school of thought, in fact, it's the Halfbakery. I can see some interesting number theory coming out of this. An axiom of pure maths is you don't ask what it's for.
 — General Washington, Oct 31 2002

 There have historically been a few computers which have had registers with non-uniform number bases. For example, there was one in Britain which had mostly base-10 counters, but the three least-significant digits were a two-state tube, a ten-state tube, and a twelve-state tube. Can anyone in the UK (or elsewhere) figure out what such a machine might have been used for?

 Generally, though, there is little reason to use anything other than binary, and no reason to mix number basis within a storage register.

 BTW, the one reason one might conceivably want (or have wanted) to use a different number base would be that the cost of each unit of storage were proportional to the number of states, three-state storage devices would be more cost-efficient than two-state ones. For example, if one is/was constructing a computer out of bipolar transistors and they are/were expensive, a bit would require two transistors but a trit would require three. Nineteen bits [38 transistors] can hold a number up to 524,288. Twelve trits [36 transistors] can hold a number up to 531,441. Two fewer transistors thus hold a larger number.

 Although I may be mistaken, I believe the Russians developed some computers using 'balanced ternary' notation. In many ways, balanced ternary has a very nice sort of elegance to it. I don't know when it was last used.

BTW, it's possible to construct a full adder using only two transistors (and a lot of resistors) per bit. Modern LSI devices use many more transistors, but results are computed much more rapidly. Two two-transistor-per-bit adder was useful, though, in the days where transistors were expensive.
 — supercat, Oct 31 2002

 General Washington thanks. Now waugs please notice carefully: the Babylonians used a system of 12 for a reason. (Notice the words "dozen" and "twelve" in English). I know Sadam is not popular these days, but let's stick to math and computers. Take a paper and do a few additions and subtractions. Then do a few multiplications. This is definitely NOT "infinitely more complicated". You will notice that both doubling and times 3 is an almost simple shift (I call it a "ruled shift").

 Mathematically, there is a LOT of interesting behavior with these numbers.

About the non-binary: this was an example of mixed powers using binary numbers (0 and 1) but does not necessarily have to be so. (It would just take much more time to explain). A system with a mixed power of 2 and 3 using symbols (of 0 1 and 2) is Very interesting mathematically AND hardware-wise.
 — pashute, Nov 01 2002

 There is NOT 'a LOT of interesting behavior with these numbers', they are the same numbers we are all familiar with manipulating in the decimal NOTATION.

 I can accept that you may find the use of mixed powers in this notation to be mathermatically interesting, however I seriously doubt that any practical advance can come from this.

What we have here is a system which intruduces notational redundancy and conditional rule sets where none are required. Sure to cause quite a stir when the boys at Intel hear about it.
 — scarra, Nov 18 2002

 — egnor, Nov 19 2002

 "Ruled shift" is what the US Marine Corps did in Iraq, I think. No, wait, that's "shift rulers".

As for the math, it is interesting that in this system 100 = 11, while 1000 = 111. Isn't it? Come on guys!
 — gardnertoo, Dec 15 2004

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