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Quick Question: A complex number $z$ with real part $Re(z) = 0$, i.e. something like $-17i$ -- would you call it "pure imaginary", or "purely imaginary"?

I'm not a native English speaker. My language sense says it should be "purely", but I have come across various sources that say "pure imaginary".

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    $\begingroup$ I would know what you meant either way. I suspect "purely" is technically correct, but they both seem to be in usage. In particular, check out the Google Ngram comparison for these two phrases. $\endgroup$ Commented Apr 11 at 17:21
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    $\begingroup$ Your language sense is right, but many people, including me, sometimes repress their language sense. For example, one also hears "This mapping is onto." $\endgroup$ Commented Apr 11 at 18:41
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    $\begingroup$ Following Wikipedia (a good source for non natives): purely imaginary. $\endgroup$
    – Pedro
    Commented Apr 11 at 19:06
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    $\begingroup$ The waters are perhaps muddied by "imaginary numbers" also being an old term for all complex numbers. Then "pure imaginary" is a "pure" complex number with real part equal to zero. $\endgroup$
    – J W
    Commented Apr 12 at 5:53
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    $\begingroup$ Do you have the complete sentence? In isolation it should be "purely imaginary". But then maybe one can think of a "pure imaginary number" $\endgroup$
    – qrsngky
    Commented Apr 12 at 9:56

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checking the most popular textbooks:

  • Papa Rudin has 3 occurrences of 'pure imaginary' vs. 0 of 'purely'

  • Ahlfors has it 0 vs. 14

  • Stein-Shakarchi: 0 vs. 9

  • Conway: 0 vs. 4

  • Gamelin: 2 vs. 1

  • Needham: 0 vs. 4

  • Ablowitz-Fokas: 6 vs. 3

  • Remmert: 0 vs. 5

so it seems there's no issue in using either form, or even both, it's really up to the writer

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    $\begingroup$ I'm certainly more familiar with "pure imaginary" as the term of art. E.g., Lang's Complex Analysis has it 9-0. $\endgroup$ Commented Apr 12 at 13:26
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    $\begingroup$ +1 for the time you took to provide a definitive answer with examples. $\endgroup$ Commented Apr 14 at 17:58
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I guess you could get away with either, but strictly speaking, grammatically, the correct term is "purely imaginary."

  • "pure imaginary number" means that the number is being described as both imaginary and pure. But what is a "pure number"? That doesn't make sense.

  • "purely imaginary number" means that the number is being described as imaginary. How imaginary? Purely imaginary.

More technically: the "-ly" suffix creates an adverb, which indicates that it describes the adjective rather than the noun.

  • In the phrase "pure imaginary number," there are no adverbs. Both "pure" and "imaginary" are adjectives that describe the noun "number."

  • In the phrase "purely imaginary number," the adverb "purely" describes the adjective "imaginary" which in turn describes the noun "number."

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    $\begingroup$ I would suggest that the kind of analysis you are suggesting is, perhaps, not entirely correct. "Pure imaginary" is a kind of fixed phrase. I don't think that you should try to parse the individual words. $\endgroup$
    – Xander Henderson
    Commented Apr 12 at 13:17
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    $\begingroup$ @ac15 It should be hyphenated "finite-dimensional," but like "-ly", hyphens are disappearing, too. $\endgroup$
    – user12357
    Commented Apr 12 at 13:52
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    $\begingroup$ @user12357 Adverbs without -ly, called flat adverbs, (where an -ly form is available) have a long history in English. It seems that they were actually much more common in 18th century literature than they are today: merriam-webster.com/grammar/drive-safe-or-safely $\endgroup$ Commented Apr 12 at 16:22
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    $\begingroup$ As I interpret "pure imaginary number", "pure" modifies all of "imaginary number", not just "number". What kind of imaginary number is it? It's a pure one: one uncontaminated by any real part. (I would still use "purely imaginary", personally; I just don't entirely agree with the analysis.) $\endgroup$ Commented Apr 13 at 0:54
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    $\begingroup$ I think this answer is reading "pure imaginary number" as "pure, imaginary number", which is possible but not the only possibility, as @MishaLavrov points out. $\endgroup$ Commented Apr 13 at 6:42
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I generally take a more descriptivist (rather than prescriptivist) view of language. If a particular phrase is commonly used by competent native speakers of a language, then the phrase is correct, and you should have no worries about using it. A useful tool for looking at how words and phrases are used is the Google Ngram viewer.

With respect to "pure imaginary" vs "purely imaginary", it would appear that both phrases appear in the corpus that Google uses, and that "purely imaginary" occurs only a bit more than twice as frequently as "pure imaginary". In the grand scheme of the English language, this seems to indicate that either is fine, from the point of view of English speakers. The distinction is even less pronounced, and the ordering of the relative frequencies changes, when comparing "pure imaginary number" to "purely imaginary number".

Note that the corpus Google uses is not specifically a mathematical corpus, so the frequencies represented are across a wide range of texts written in English. I would not be surprised to find that, in a corpus of mathematical texts, there are relatively more occurrences of "pure imaginary" (this is a bit of a fixed phrase in mathematics, and may be something of a shibboleth).

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    $\begingroup$ Adding "number" to both strings in the ngram will probably get a more accurate picture of mathematical usage. (It reverses the relative prevalence.) $\endgroup$
    – Adam
    Commented Apr 12 at 13:43
  • $\begingroup$ @Adam Oh, indeed! Good call! $\endgroup$
    – Xander Henderson
    Commented Apr 12 at 13:51
  • $\begingroup$ Also, "pure imaginary numbers" is prescriptively correct, while "pure imaginary" is not. So it makes sense to have more "pure imaginary numbers" relative to "purely imaginary numbers" $\endgroup$
    – justhalf
    Commented Apr 14 at 18:37
  • $\begingroup$ @justhalf Maybe? The real problem is disaggregating the use of these phrases in mathematical argot from use in vernacular English. But, descriptively, I think that the take home should be "both are fine". $\endgroup$
    – Xander Henderson
    Commented Apr 14 at 18:40
  • $\begingroup$ yep. I'm just adding on to your comment 😃 $\endgroup$
    – justhalf
    Commented Apr 14 at 18:43
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It depends whether you think of "imaginary" as an adjective (a property of a subset of numbers) or a noun (a shortened name for the set of imaginary numbers). "Purely imaginary" treats it as an adjective; a description. "Pure imaginary" treats it as a noun; a category of number, a name for this set of numbers. Either is acceptable, and for most purposes it makes no difference, but they have subtly different meanings.

A lot of mathematical terms for sets of objects can be interpreted as either nouns or adjectives, because we commonly specify the set (the noun) by giving the property they satisfy (the adjective).

It ought to be noted that the mathematical names for the sets of numbers are "real numbers", "imaginary numbers", and "complex numbers". Adding "pure" or "purely" doesn't change the definition. "purely imaginary numbers" are exactly the same as "imaginary numbers". I've found people sometimes get confused about that, thinking there is a distinction. (Using "imaginary numbers" to refer to the non-real complex numbers is even worse, as phrases like "the imaginary part" then make no sense at all.) I prefer to deliberately avoid using "pure" or "purely" until the basic definitions are firmly established.

Real numbers are laid out on the x-axis, and square to a non-negative real number. Imaginary numbers are laid out on the y-axis, and square to a non-positive real number. Complex numbers are laid out on the whole xy plane, and each have a real and an imaginary component, added together. Then you can talk about a complex number with only an imaginary component (like 0+5i) as "a purely imaginary complex number", and the corresponding imaginary number (i.e. 5i) as "a pure imaginary number".

That's not the only way to do it. I'm sure other ways are equally valid.

I might add, I like motivating the topic by talking about the matrices of the form $\pmatrix{a & -b \\ b & a}$ first. You have to talk about the Argand plane at some point because that's how everybody else does it. But it's arguably not the best place to start.

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Following Wikipedia (a good source for non natives): purely imaginary.

Remark: Originally, this answer was a comment. However, I accepted Taladris' suggestion to include it as an answer. In the words of the said user, this answer is simple but important: a significant part of discussions on Wikipedia is about how to name pages and terminology, so people there gave a lot of thoughts about this kind of question.

Addendum: Following Encyclopedia of Mathematics (a good source for math in general), we would also use purely imaginary.

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This is really a question for English Learners but let’s go:

The word imaginary is an adjective; that is, it modifies a noun. You might have an imaginary friend, an imaginary ailment, or (relevantly) an imaginary number.

But English grammar allows you to promote adjectives to nouns, to mean people or things that have the indicated qualify. That’s why you can say “the British”, “the quick and the dead”, and “the bold and the beautiful”.

Are you using “imaginary” as a noun?

For example, can you pluralize it? English, unlike most Western languages, does not pluralize ordinary adjectives, but it does nouns, even nouns that used to be adjectives. We say “shorts”, meaning “short pants” or (in finance) investors in short positions. And we have reals (members of ℝ) and imaginaries (members of 𝕀).

You can say “2i and 3i are pure imaginaries” (used as a noun) or “2i and 3i are purely imaginary” (as an adjective).

Also, singular nouns typically take articles (“a”, “an”, and “the”). “2i is a pure imaginary” or “2i is purely imaginary”.

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  • $\begingroup$ Your last sentence captures what immediately came to mind for me. $\endgroup$
    – Mark S.
    Commented Apr 14 at 1:47
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I have come across the term purely imaginary in Text Books when the real part is missing in a complex number. But there may be no issue if you are consistent in using one term. This can be somewhat similar to asking Pythagorean triples or Pythagorean triplets is correct. This issue I have already raised in Mathematics Educators.

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  • I am familiar with the term "pure imaginary" from mathematics, as in $5i$ is a pure imaginary number, but $i+10^{-100}$ is not.

  • OTOH, if I were to debunk some proposal, I might say that its benefits are "purely imaginary", i.e. fictitious.

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  • $\begingroup$ This seems like an analysis saying "I have never used this term, therefore it doesn't exist". $\endgroup$ Commented Apr 13 at 0:55
  • $\begingroup$ @MishaLavrov Maybe it really seems that way to you: "pure" is an adjective, "purely" an adverb, however, and the words have different functions (in English, but maybe other languages are different). I suggest that you read Strunk & White, especially the section on the sparing use of adverbs. I know that adverbs have a extra syllable, but that isn't a mark of good style. BTW, do you think Hardy should have called his textbook "Pure Mathematics" or "Purely Mathematics"? The meanings are different, you know. $\endgroup$ Commented Apr 13 at 1:51
  • $\begingroup$ See, I disagree with your comment as well, but at least your comment actually tries to make an argument, unlike your answer. Consider that Strunk & White's adage is "omit needless words", not "omit all words". $\endgroup$ Commented Apr 13 at 2:27

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