Talk:Engel expansion

Uniqueness

The example

1.175=\{1,6,20\}=\{1,6,21,21,21,\dots \}

given in the article shows that the non-decreasing sequence of positive integers which is an Engel expansion for a fixed positive number is not unique for rational numbers (if one allows finite sequences). --91.32.82.49 (talk) 19:13, 24 March 2011 (UTC)[reply]

As the article says, every positive rational number has a unique finite Engel expansion and a unique infinite Engel expansion. The example shows how the finite and infinite Engel expansions are related. Gandalf61 (talk) 10:25, 25 March 2011 (UTC)[reply]

But then it shows the Engel expansion is NOT, as the article says, unique. I guess, but I can not immediately see if it's true, the article should mention, a rational number has a unique finite and a unique infinite expansion. Madyno (talk) 08:22, 28 September 2022 (UTC)[reply]

The article already says: "Every positive rational number has a unique finite Engel expansion. ... Every rational number also has a unique infinite Engel expansion" What part of this do you think is ambiguous or missing? —David Eppstein (talk) 17:05, 28 September 2022 (UTC)[reply]

Well, the first sentence is not correct: The Engel expansion of a positive real number x is the unique non-decreasing sequence of positive integers.

Some numbers have more than one, in fact two, expansions. Madyno (talk) 15:43, 1 October 2022 (UTC)[reply]

Negative numbers in Engel expansion?

Are there ever negative integers in these expansions, especially in alternation with positive integers? For example, the Julian year's length of 365.2425 days is implemented in leap days via the expansion $365 + .mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}⁠1/4⁠ − ⁠1/100⁠ + ⁠1/400⁠$ . 𝕃eegrc (talk) 17:40, 14 July 2016 (UTC)[reply]

As it says fairly prominently in the article, Engel expansions themselves are all positive, but there is a version with alternating signs, called the Pierce expansion. —David Eppstein (talk) 18:33, 14 July 2016 (UTC)[reply]

Thank you 𝕃eegrc (talk) 14:28, 18 July 2016 (UTC)[reply]

Additive version

does a additive version exists? like how for for example 2 is

x={\frac {1}{1}}+{\frac {1}{1+2}}+{\frac {1}{1+2+3}}+\cdots

Does such a thing exist as well for sqrt(2) --KlokkoVanDenBerg (talk) 02:36, 8 September 2023 (UTC)[reply]