Talk:Casting out nines

Limitations to Casting out nines
While extremely useful, casting out nines does not "catch" all errors made while doing calculations. For example, 5x7 = 8. By the casting-out-nines method, it would appear that this multiplication is correct, which it is not. This would also happen with the result of 17, 26, et al. In other words there will be an "uncaught" mistake every nine numbers, this would yield a 90% effectiveness to the method.

This situation resembles a "typo" and thus is not a true or even a threadbare limitation of the Casting Out Nines method. 5x7 = 35. If and when 5x7 = 8 it is a case of a "typo" and therefore, the method is not applicable. — Preceding unsigned comment added by Mbidwai (talk • contribs) 01:03, 18 June 2014 (UTC)


 * I reverted your edits because you did not supply an explanation for section blanking. I think you have a good point, however, so I will leave the section alone for now.  If someone else has a good explanation of any "limitations", they will have to re-create the section and explain it better.  In the future, please leave a comment explaining any edit that you make. — Anita5192 (talk) 01:27, 18 June 2014 (UTC)


 * I'm afraid I don't understand this objection at all. If you make an error in a calculation and obtain a result whose digital root is the same as that of the correct answer, then the method of casting out nines will not flag it as an error.  Why on earth is it inappropriate to call that a "limitation" of the method?  I have now reworded the explanation to try and make it clearer.
 * David Wilson (talk · cont) 14:48, 11 July 2014 (UTC)

History
On checking out various references for the section on history, I have discovered there are several problems with it:
 * Equivocation on the expression "casting out nines". There are actually two closely related, but nevertheless different, senses in which this expression is used:
 * it is used to denote the procedure of merely calculating digital roots (or, equivalently, remainders mod 9) by some combination of computing digital sums and taking remainders on division by 9, irrespective of the purpose for which this is done; and
 * it is used to refer to the specific application of this procedure to the checking of arithmetical calculations.
 * The second of these is the only one currently acknowledged in the article, but it is only for the first that the procedure can reasonably be said to have been "known to the Roman bishop Hippolytus". In the passage from The Refutation of all Heresies where Hippolytus explained the form of casting out nines known to ancient Greek mathematicians, there's nothing at all to indicate any awareness of how it could be used to check the results of arithmetical calculations.  His only purpose for describing it was to explain how it was used by Greek practitioners of gematria.


 * Misleading citation of Latin expression. The use of the Latin expression "abjectio novenaria" in the first sentence seems gratuitous and is almost certainly anachronistic.  It certainly doesn't occur in Hippolytus's explanation of casting out nines, which was written in Greek.  The earliest known references to casting out nines in Latin appear to be those from the expositions and translations of Arabic arithmetical works in the high middle ages.  But neither John of Seville's Liber algorismi de pratica aritmetrice nor Fibonacci's Liber Abaci, two of the earliest such works, uses the term "abjectio novenaria".  Some of Leibniz's writings are the only places I have so far found where it has been used.  The editor responsible for  seems to have culled it from the reference he had just added with .   But the term "abjectio novenaria" is just one of two terms which the reference's editors merely imply were used by Leibniz to refer to casting out nines.  They do not imply that either term enjoyed any wider currency than that.
 * P.S: A previous comment on this talk page gives a quotation worded identically to the first sentence of the article's history section as it stood at the time, and says "This is a statement in Florian Cajori's history." I initially took this to mean that the quotation was alleged to have appeared verbatim in that work of Cajori's. I now suspect that I was mistaken, and that it was merely intended to assert that something roughly synonymous appears in Cajori's history.  At any rate, although Cajori does say, in the fifth edition of his history, that Hippolytus knew about casting out nines, he does not appear to have anywhere mentioned the expression "abjectio novenaria".


 * Irrelevant & misleading quotation. The article currently says:
 * "In the 17th century, Gottfried Wilhelm Leibniz not only used the method extensively, but presented it frequently as a model for rationality: "By means of this, once a reasoning in morality, physics, medicine or metaphysics is reduced to these terms or characters, one will be able to apply to it at any moment a numerical test, so that it will be impossible to be mistaken if one does not so desire..." "
 * But in the essay of Leibniz's from which the quoted passage has been taken, there's nothing at all to indicate that either the passage itself, or the essay as a whole, has anything whatever to do with casting out nines.  Casting out nines is mentioned only three times in the entire cited reference where the essay appears, all in footnotes by the reference's editors.  In my opinion none of those footnotes justifies the assertion that "Leibniz ... presented it [viz. casting out nines] frequently as a model for rationality".

David Wilson (talk · cont) 11:30, 18 July 2014 (UTC)

There is an error in the subtraction example.
To whomever produced the pictures of the math, there seems to be a mistake in the subtraction example. In the result of the subtraction (2752) the numbers 2, 5 and 2 are crossed out and the 7 remains, but according to the algorithm the first 2 and 7 add to 9 and should have been crossed out. The remaining 5 and 2 should then be added together giving (another) 7, which is the correct answer. 82.72.139.164 (talk) 23:43, 24 February 2018 (UTC)


 * It is no mistake, as the algorithm does not specify which digits to use: either approach yields the same result.  In fact, the result is what one would achieve by adding the digits, dividing by nine, and keeping only the remainder.—Anita5192 (talk) 23:53, 24 February 2018 (UTC)

"A variation on the explanation" needs clarification
Not really sure what the extended sequence of multiplication and digit sums is supposed to convey, and the formatting of equations and parentheses is very unclear. Also the language used is largely first- and second-person and should be revised. Not a mathematician so not sure how to fix. RookWeaver (talk) 09:50, 22 February 2022 (UTC)


 * I changed the second-person point of view in the Examples section to imperative mood to match the rest of the section. Other than that, I think the section is clear.—Anita5192 (talk) 17:05, 22 February 2022 (UTC)

Other bases
Does this also work in octal or hexadecimal, using the number one less than the base (7 or 0xF, respectively)? -- ~ SpareSimian (talk) 02:18, 6 April 2024 (UTC)
 * Sure does. It even works in binary i.e. you cast out all the 1 bits until either one 1 is left, or there is no 1 left. That's what we usually call the parity of the original binary representation, and parity is preserved under + - * and /.In fact, it even works in unary, but it's not very useful in that case, because the "digital root" is always 0. EEng 04:40, 6 April 2024 (UTC)