Talk:Signed-digit representation

Informal illustration for simple integer case?

This page reads quite formally and feels a little inaccessible. Is it worth adding an informal example to motivate the discussion? For example, use balanced quinary with the digits -2,-1,0,1,2 show how you can convert an integer N to its unique representation by repeatedly casting out remainders mod 5? e.g d_0 = N mod 5 from the digit set and then N is replaced by (N - d_0)/5 and we repeat.~~User:Patricksurry~~ — Preceding undated comment added 14:46, 10 February 2023 (UTC)

Balanced forms guarantee unique representation?

The article states "The non-adjacent form does guarantee a unique representation for every integer value, as do balanced forms." This is right after an example of four different balanced ternary representations for the same integer. Am I missing something?--Wikimedes (talk) 02:12, 5 May 2013 (UTC)

Yes, you have found a contradiction. The number 7 is written four ways using balanced ternary. For unique representation, more conditions must be placed on the digits. Excluding the 8 cuts out some alternatives, so requiring minimal word length helps. Also, 7 is positive, so starting with a negative digit introduces an alternative. One could specify that the first digit shares the same sign as the total to be represented.Rgdboer (talk) 21:04, 5 May 2013 (UTC)

Minimal wordlength gives unique representations when finite. The popular article 0.999 shows up an ambiguity in ordinary decimals. Similar non-unique representations occur for infinite words of signed-digits.Rgdboer (talk) 21:15, 5 May 2013 (UTC)

Thanks. I removed from the article the part about balanced representation guaranteeing uniqueness. I'll leave it to someone more knowledgeable than I to add other uniqueness criteria.--Wikimedes (talk) 23:16, 5 May 2013 (UTC)

The above examples are not balanced (look at the powers), they are powers of two but allowing positive and negative. Balanced forms do guarantee uniqueness, 7 in balanced twrnary is 1,-1,1 = 9-3+1 which cannot be represented in any other way. -- Q Chris (talk) 07:57, 6 May 2013 (UTC)

Ah yes, signed binary is different from balanced ternary. Always good to ask before editing on a topic I've just learned about. Sorry to have jumped the gun in editing.--Wikimedes (talk) 16:29, 6 May 2013 (UTC)

The observation of contradiction remains. Minimal word length must be required for unique expressions, and even then the infinite word can produce two expressions for the same value as in 0.999. Ideally a source would provide the algorithm for conversion, then conclusions could be drawn. Conversion is an interesting exercise.Rgdboer (talk) 20:21, 6 May 2013 (UTC)

It explains further on that recurring fractions can represent the same number even in balanced systems with recurring numbers (I think this is true of all positional based number systems). For integers the form is unique in balanced bases. -- Q Chris (talk) 08:33, 7 May 2013 (UTC)

The use of "signed-binary numbers" should be removed. In binary there are just two digits and one of them is zero. Balanced ternary is more appropriate for examples. Multiple representations with digits beyond the minimum is unnecessary distraction from the economy of signed-digit arithmetic.Rgdboer (talk) 20:33, 6 May 2013 (UTC)

I disagree, the article is about signed-digit representation ... representations that allow signed values of digits. Why would you miss out signed-digit binary? Should you also exclude balanced ternary because it is not ternary (having digits 0, 1, and 2) or signed-digit decimal because its not decimal? You'd end up with an article about nothing! -- Q Chris (talk) 08:33, 7 May 2013 (UTC)

My mistake. Of course, signed-binary is essential, perhaps a special section.Rgdboer (talk) 22:17, 9 May 2013 (UTC)

When the number of digits is equal to the radix, it is unique, when superior there are several writings, when inferior some values cannot be written. Radix 8 (octal) with 10 digits 0 to 9 is not unique. Decimal with 8 digits 0 to 7, you cannot write 8. Binary with 3 digits 0, 1, 2 is not unique. Binary with 3 digits -1, 0, 1: it is not unique. Balanced or not is not the question. — Preceding unsigned comment added by Tacob~frwiki (talk • contribs) 12:04, 17 October 2016 (UTC)

Sourcing

The following was removed:

the ancient Etruscan language, concept regarding numerals was also quite similar. As of being that time the most advanced way of expressing positive integers (in that region), the concept had quickly found its way to the newly established city of Rome, and had also appeared in the Roman numerals, loosely written in a manner how it was generally said in the spoken language. The majority of people collective founding Rome and brought it to rise were mainly Estruscans. Aside the that for some non-trivial letters contained of Roman numerals took up to a century or more to evolve to its final form which allowed the old Etruscan symbols to be discarded, the concept, rising out of the spoken language, more or less also had left a permanent footprint.<ref from my own studies regarding Ancient Rome's Numerals and Calendar issues ref>

Editors may not cite themselves. See WP:RS for guidelines. Rgdboer (talk) 04:35, 26 April 2021 (UTC)

However, WP:Suggestions for COI compliance allows some participation. Rgdboer (talk) 04:38, 18 May 2021 (UTC)

For example, a blog post from 2012 Complementary Numerals at iMechanica is on the topic of this article. It uses a for minus one and refers to a "rule of signs" in arithmetic with sign-digit notation. A base ten system might use 5 or 1e where e is minus five. The two systems are compared for their rounding by truncation. Another editor may find this blog post useful in improvements to this article. Rgdboer (talk) 17:28, 18 May 2021 (UTC)

Extension to Complex Numbers?

Is there an extension of signed digit representation to complex numbers? 130.76.25.51 (talk) 22:26, 6 May 2023 (UTC)