Talk:Quater-imaginary base

Can someone explain why this is spelled "quater-imaginary" instead of "quarter-imaginary"? What does "quater" mean?

It's a Latin adverb meaning "four times" (i.e. once beyond thrice). I would guess that it refers to the fact that the system can cover all four quadrants of the complex number system with a single numerical representation. AnonMoos 15:42, 23 February 2006 (UTC)

Actually, it's because the system uses four different digits. -- Milo

Because the system is equivalent to negaquanternary. –Orienomesh-w (talk) 08:23, 25 November 2012 (UTC)

*negaquaternary. Quanternary is not a word. — Preceding unsigned comment added by 77.169.168.165 (talk) 22:52, 7 December 2012 (UTC)

Non-standard positional numeral systems

I have addressed certain issues by creating the article Non-standard positional numeral systems, and making related changes to Unary numeral system, Golden ratio base, Quater-imaginary base, Positional notation, Base (mathematics), and Category:Positional numeral systems. I suggest further discussion of these issues takes place at talk:Non-standard positional numeral systems.--Niels Ø 14:32, 26 February 2006 (UTC)

Unique?

Spending just a minute looking at this, it appears that this system uniquely represents any complex number (no number has more than one representation). Is there a proof of this around? I think it would be good to include. If it doesn't, a counterexample would be nice too. Hjfreyer 12:52, 9 February 2007 (UTC)

Sadly, no. 1.0300(0300)... = 1/5 = 0.0003(0003)... Note that, as with 1.000... and 0.999... in base 10, these are respectively the smallest and largest numbers representable with integral part 1 and 0 respectively. –EdC 21:37, 19 April 2007 (UTC)

New section

This article really should have a “Quater-imaginary to decimal” section. — Ti89TProgrammer 04:28, 1 August 2007 (UTC)

Added Examples etc.

I added two sections on converting to and from the quater-imaginary system. I also added sections on adding, subtracting and multiplication, with examples. I haven't done division, because I don't (yet) know how this works (anyone?). I also changed the "powers of 2i" table to a horizontal table for readability. -GjjvdBurg (talk) 15:38, 21 December 2007 (UTC)

I think some of the rows got messed up. 2i^-1 should be 1/2i, not -1/2i, 2i^-3 should be -1/8i, not 1/8i, and 2i^-5 should be 1/32i, not -1/32i. Equivalently, since 1/i = -i, they could be presented as -i/2, i/8, and -i/32, respectively.131.107.0.94 (talk) 02:10, 17 March 2012 (UTC)

10i-base equivalent

Is it worth mentioning that a similar system can be made using 10i as the base, which is similarly able to represent every complex number using only the digits 0 through 9 without a sign (but which may be slightly more comprehensible to novices)? — Loadmaster (talk) 05:06, 24 December 2007 (UTC)

Sure, it would be alright to add some links to other numeral systems but I think a system with base 10i should have a separate article (if it isn't already there). -GjjvdBurg (talk) 11:31, 25 December 2007 (UTC)

This doesn't actually work; for base ni you need n² digits, not n, which makes the whole thing impractical (if it wasn't already) for n much beyond 4. For n = 3 there are a couple of nice balanced systems with digit values of −4, −3, ..., 3, 4 and {0,±1} + {0,±1} i. Actually for the latter you don't need base 3i; base 3 works just as well. -- BenRG (talk) 21:07, 4 February 2008 (UTC)

difference in notation

hello, It's a mistake to use an 'x' as the multiplication sign in the first two formulas and a dot in the rest, there should be only one way to express this instead of two because it might lead to think that they have different meanings and (afaik) it's not the case. Samus_ (talk) 20:25, 20 January 2008 (UTC)

It seemed to be the only instance of that in the article. I think they have indentical meaning so I changed it on the spot. Johan G (talk) 04:18, 1 March 2009 (UTC)

A new representation(Squary?) of 2i base numeral system

We may use the {0,1,i,1+i} instead of {0,1,2,3}. The system will be more clear(no radix point for Gaussian integer), and the multiplication will be more simple. in the sample, h means 1+i

	-4i	-3i	-2i	-i	0	+i	+2i	+3i
-6	hi0	hii	hh0	hhi	1i0	1ii	1h0	1hi
-5	hi1	hih	hh1	hhh	1i1	1ih	1h1	1hh
-4	h00	h0i	h10	h1i	100	10h	110	11h
-3	h01	h0h	h11	h1h	101	10h	111	11h
-2	ii0	iii	ih0	ihi	i0	ii	h0	hi
-1	ii1	iih	ih1	ihh	i1	ih	h1	hh
0	i00	i0i	i10	i1i	0	i	10	1i
1	i01	i0h	i11	i1h	1	h	11	1h

addition	multiplication
+ 0 1 i h 0 0 1 i h 1 1 i1i0 h i1i1 i i h 10 11 h h i1i1 11 i1h0	× 0 1 i h 0 0 0 0 0 1 0 1 i h i 0 i i1 ih h 0 h ih 10

–Orienomesh-w (talk) 13:39, 24 November 2012 (UTC)

Though original research (?) I'm glad you put this here. I presented this idea to my local college's math faculty in 1988. (I used "I" as the 1+i digit, but I prefer your choice "h".) Hooray Gaussian integers! Johntobey (talk) 15:58, 26 July 2023 (UTC)

Mistake in "Decomposing the quater-imaginary" section

It seems there is a mistake in the section about converting from quater-imaginary to decimal (or at least unclear phrasing). It says that the imaginary part of a quater-imaginary number can be represented in base -4 as $\ldots d_{5}d_{3}d_{1}.d_{-1}d_{-3}\ldots$. However, the preceding line gives the imaginary part as $2i\cdot [...+d_{5}\cdot (-4)^{2}+d_{3}\cdot (-4)^{1}+d_{1}+d_{-1}\cdot (-4)^{-1}+d_{-3}\cdot (-4)^{-2}+\ldots ]$, which is twice the value of the other expression. The later example of $1101_{2i}$ has imaginary part $-8i$, but using the base -4 representation gives $10_{-4}=-4$. Am I missing something here or does this need clarification? Exodus6395 (talk) 12:26, 11 October 2014 (UTC)

Simplest self-contained numeral system for complex numbers?

Anyone know if this numeral system (Quater-imaginary base) is the simplest possible self-contained numeral system for complex numbers, analogous to signed ternary for integers?

2806:264:4403:84F:9C6B:850A:557A:EDCF (talk) 17:21, 24 October 2020 (UTC)

It depends on what you mean by simple. A base ${\sqrt {-2}}$ or base ${\sqrt {-3}}$ system could represent all complex numbers with arbitrary precision but might be harder to work with. For example, $i=10.2_{2i}=10.101010100010001\dots _{\sqrt {-2}}$ (non-repeating). Johntobey (talk) 04:11, 27 July 2023 (UTC)

Explanation revamping

In my most recent edit, I (quoted from my edit description)

"Rewrote the short description to be slightly less technical, and rewrote the entire section "Decomposing the quater-imaginary" to clarify the meaning of the equations and explain them in more-accessible and less-ambiguous text."

The actual meaning of this may seem slightly confusing, so I decided to explain my reasoning better here (especially figuring this was a large edit:

1. The short description at the top really felt too much like a jump into the history before actually explaining what it is, so I replaced the opening historical statement, "The quater-imaginary numeral system was first proposed by..." with a descriptive statement, "The quater-imaginary numeral system is a numeral system, first proposed by..." as it is the same statement, but worded in what seems to me to be a more explanatory way.
2. Instead of just linking to "non-standard positional numeral system", I edited the passage to explain the difference between the quater-imaginary system and other numeral systems: "Unlike standard numeral systems, which use an integer (such as 10 in decimal, or 2 in binary) as their bases, it uses the imaginary number 2i (equivalent to √-4) as its base."
3. The section header "Decompose the quater-imaginary" (which is an imperative, or command statement) with "Decomposing the quater-imaginary", since that describes what is being done.
4. Under that same header was a section of almost pure math. I liked it, but pure math doesn't have an encyclopedic tone, and fails to get the point across to non-specialists without them having to tediously follow strings of multiplication and addition. Because of this, I added a section of text which explained more-or-less what the math was doing, and which provided a little more context. I also removed the slightly POV 1st-person statement "as we know". It also makes the equations themselves fit next to the template.

If anyone has any complaints with this edit, please tell me. UnbiasedBrigade (talk) 17:43, 19 January 2023 (UTC)

Alignment

@Yet Another Internet User: Did you really match the alignment requirements which correspond to the well-known place-value notation? In my opinion they contain the following:

1. All (low level) arithmetic starts from the right and works to the left.
2. Thereby, so-called carries may be transported to the left.

From this it would follow that the digits (if aligned according to their value) of a number start in a certain right column and advance to the left column by column (according to their value). If I understand your example correctly the 2i in the last (= 6th) line would have to be placed exactly below the 8i in the 5th line; and the 20 in the 6th line would have to be placed exactly below the 01 in the 5th line. Nomen4Omen (talk) 20:41, 15 March 2023 (UTC)