Wikipedia:Reference desk/Archives/Mathematics/2022 December 29

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December 29

Mathematical constants that are not real numbers

Where integers count as a subset of the reals. By mathematical constants I mean numbers with names or symbols connected to them, like π, e, the Hardy-Ramanujan number 1729, and so on. Among constants that are not real-valued, I can think of the imaginary unit i, and some transfinite ordinals and cardinals like ω or $\aleph _{0}$. Are there any more, particularly complex-valued ones? I keep expecting a few numbers like ${1+i{\sqrt {3}}} \over 2$ to have "names", but none come to mind. The constants should be numbers, which I suppose is a loose term in this context, but let's not count matrices, graphs, etc. Thanks. 2601:648:8200:990:0:0:0:B9C2 (talk) 21:44, 29 December 2022 (UTC)

There is a positive infinitesimal surreal number denoted by $\varepsilon .$  --Lambiam 00:04, 30 December 2022 (UTC)

The basic quaternions are commonly denoted by $\mathbf {i} ,$ $\mathbf {j}$ and $\mathbf {k} .$  --Lambiam 00:08, 30 December 2022 (UTC)

Our article Greek letters used in mathematics, science, and engineering states that $\omega$ denotes a complex cube root of unity, a claim also found in the The Concise Oxford Dictionary of Mathematics.^[1] I think, however, that $\omega$ more generally denotes a primitive $n$th root of unity, not a specific constant.  --Lambiam 00:46, 30 December 2022 (UTC)

Wikipedia's list of mathematical constants doesn't have any complex constants, but WolframAlpha apparently does. It seems to be somewhat obscure, but they call the complex half-period of the equianharmonic case of the Weierstrass elliptic function the $\omega _{1}$ constant. You can see MathWorld's page on the corresponding real half-period, the $\omega _{2}$ constant, for more info. GalacticShoe (talk) 01:02, 30 December 2022 (UTC)

There are a number of named ordinals, e.g., $\epsilon _{0}$, $\omega _{1}^{ck}$, and other large countable ordinals.--2600:4041:5640:8400:255F:1894:B6EA:66BB (talk) 15:38, 30 December 2022 (UTC)

Thanks everyone. 2601:648:8200:990:0:0:0:B9C2 (talk) 01:32, 3 January 2023 (UTC)