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I don't recall the reference, but I read once in a work of Hardy's that "number-theoretic" should not be used when talking about number theory, however tempting it may be.

So what do we say? I read on Wikipedia that the word "arithmetical" is preferred.

Is "number-theoretical" wrong? Is "arithmetical" the correct word to use? Recently I've been looking into the Number Theoretical Transform (NNT). Ought this be called the "Arithmetical Transform" instead?

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    $\begingroup$ "Number-theoretical" was criticized by many, not only by Hardy because it just sounds awful. The only reason why some people prefer it, is that "Arithmetic" as a book title can be confused with an elementary school textbook. Otherwise, "arithmetic", "arithmetical" must be preferred. $\endgroup$ – Alexandre Eremenko Sep 16 '15 at 17:50
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    $\begingroup$ Greek arithmos means counting number, and arithmetic was historically associated with counting and other elementary uses of numbers. Modern number theory is much broader, including analytic number theory for example, so authors that favor "arithmetical" go against the colloquial usage. There is nothing wrong with "number theoretic" grammatically, "set theoretic" is used the same way, and it makes the meaning more precise. $\endgroup$ – Conifold Sep 16 '15 at 20:00
  • $\begingroup$ Someone elsewhere in SE was claiming an ending -ical is an abomination, combining ic from Greek -ikos with al from Latin -alis. $\endgroup$ – Gerald Edgar Apr 17 '16 at 23:08
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For the sake of taking this question off the unanswered list, these comments seem themselves to constitute answers:

"Number-theoretical" was criticized by many, not only by Hardy because it just sounds awful. The only reason why some people prefer it, is that "Arithmetic" as a book title can be confused with an elementary school textbook. Otherwise, "arithmetic", "arithmetical" must be preferred. – Alexandre Eremenko Sep 16 '15 at 17:50

 

Greek arithmos means counting number, and arithmetic was historically associated with counting and other elementary uses of numbers. Modern number theory is much broader, including analytic number theory for example, so authors that favor "arithmetical" go against the colloquial usage. There is nothing wrong with "number theoretic" grammatically, "set theoretic" is used the same way, and it makes the meaning more precise. – Conifold Sep 16 '15 at 20:00

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  • $\begingroup$ BTW, it needs an upvote to make it officially unanswered. Of course, if either commenter wishes to post their own I answer, I can delete this. $\endgroup$ – Michael E2 Apr 17 '16 at 15:20
  • $\begingroup$ Upvote now done! $\endgroup$ – Rory Daulton Apr 17 '16 at 16:22

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