# Reality, realizing, affinity, etc.: A terminology question

In Snapper & Troyer's book Metric Affine Geometry the word "reality" at one point occurs in the phrase (IIRC) "reality problems", meaning questions about whether the imaginary part of a complex number is indeed zero. And in a paper I had in the American Mathematical Monthly in 2003, I used the word "affinity" in some phrase like "the desired affinity follows", which the context made clear meant something like: "thus we have proved that this mapping is affine, as desired." No one complained about that, so I take it that I was "licensed" in the sense in which that word applies to poets and dramaturges.

So now some muse or demon rested on my shoulder and inspired me to write "realizing the denominator", meaning multiplying the numerator and denominator by the complex conjugate of the denominator. I've never seen it or heard it. Googling indicates that it is far from unprecedented, even though I've never come across it, but of course so are many offenses against good usage.

Since this forum is the priest to whom I confess my sins What normative advice can the sages here assembled offer me?

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"rationalizing the denominator" still applies. To make explicit that the process is using $k$ as its base field, "$k$-rationalizing the denominator" (or rationalizing "over $k$" / "relative to $k$"). The suggestion is therefore

$\mathbb{R}$-rationalizing the denominator

"Realize" and "decomplexify" have accepted meanings very different from this. "Realify" and "realification" share the same problem as decomplexification, that they read like synonyms for restriction of scalars from $\mathbb{C}$ to $\mathbb{R}$.

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What do you have in mind as the "accepted" meaning of "realize"? – Michael Hardy Mar 4 '14 at 18:29
To exhibit as, or by means of, an instance of a (contextually determined) preferred type of construction. – zyx Mar 4 '14 at 18:35
Similar to the nonmathematical idea of "actualize [some hypothetical thing] in concrete form". – zyx Mar 4 '14 at 18:39
Or simply, to "represent", in a sense close to (but more general than) that of representation theory, or "present" similar to the idea of a group presentation. There is a contextually determined type of preferred description and one asks if the abstract construct is presentable in that form. – zyx Mar 5 '14 at 17:18

"Decomplexizing the denominator" would not be as correct as "realizing the denominator", but it is less ambiguous and it sounds better for me.

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+1 for effort, but I wonder how "decomplexifying" can sound better. A real number is still a complex number, and a six-syllable word is cumbersome. – Michael Hardy Feb 28 '14 at 18:47