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## Main Question or Discussion Point

As a mathematical novice I have been following with great interest the recent debates (in more than one forum!) as to whether 0.999... = 1.

The question is whether two ostensibly different natural numbers are equivalent, equal and synonymous. (I quote the three terms because mathematicians and philosophers tend to attach different meanings to each, so I am just trying to cover all the bases). Now, there are few things in mathematics more fundamental than the series of the natural numbers, so, if we want to prove that two natural numbers are equal, the proof is unlikely to reside in a complex equation (unless we were resorting to a

This means we are effectively reduced to arguing in terms of the axioms of arithmetic. Within that context, the proposition that 0.999... = 1 obviously has difficult implications (especially to a layman like myself!). I'd like to ask some perhaps naive and obvious questions:

(1) Since we cannot use 0.999..., what is the correct method of representing a fraction which approaches infinitely close to 1?

(2) What is the product of 0.999... x 0.999...? Putting it another way, if 0.999...9 = 1, does that imply that 0.999...8 = 0.999...9?

(3) The number 0.999... consists of a decimal point followed (in this case) by an infinite series composed of the digit 9. Since this can be more economically represented - in fact, can ONLY be accurately represented - by the digit "1", would I be right to assume that the number 0.999... is logically redundant and meaningless, and should be replaced by "1" in every case? (vide Occam's Razor). Is there any conceivable situation, outside of this discussion, where 0.999... would be logically required?

(4) Can the principle be extended to every fraction, the final term of which is a single recurring digit (eg 0.1999... = 0.12)?

(5) How is the principle to be generalised to the case where the recurrent is not a single digit, but a group of digits (eg 0.13131313...)?

(6) '1' is a member of the set of whole numbers. How should we define 'whole number' in such a way that it includes some numbers which begin with '0.9'?

(7) The proposition entails, prima facie, the axiom that two different natural numbers may have the same value; is this axiom generally accepted?

These questions are posted in good faith, by one who has no mathematical aptitude, but an intense interest in mathematical philosophy... so if you could try to express your replies in lay terms, as far as is reasonable? I mean the kind of terms that Bertrand Russell would have used for a lay reader.

The question is whether two ostensibly different natural numbers are equivalent, equal and synonymous. (I quote the three terms because mathematicians and philosophers tend to attach different meanings to each, so I am just trying to cover all the bases). Now, there are few things in mathematics more fundamental than the series of the natural numbers, so, if we want to prove that two natural numbers are equal, the proof is unlikely to reside in a complex equation (unless we were resorting to a

*reductio ad absurdam*, which is not the case which has been argued here). Our proof has to be expressed in terms which are logically antecedent to the terms in question. Since the series of natural numbers is established by definition, the only available logical antecedents are the primitive propositions and ideas (as presented, eg, by Peano).This means we are effectively reduced to arguing in terms of the axioms of arithmetic. Within that context, the proposition that 0.999... = 1 obviously has difficult implications (especially to a layman like myself!). I'd like to ask some perhaps naive and obvious questions:

(1) Since we cannot use 0.999..., what is the correct method of representing a fraction which approaches infinitely close to 1?

(2) What is the product of 0.999... x 0.999...? Putting it another way, if 0.999...9 = 1, does that imply that 0.999...8 = 0.999...9?

(3) The number 0.999... consists of a decimal point followed (in this case) by an infinite series composed of the digit 9. Since this can be more economically represented - in fact, can ONLY be accurately represented - by the digit "1", would I be right to assume that the number 0.999... is logically redundant and meaningless, and should be replaced by "1" in every case? (vide Occam's Razor). Is there any conceivable situation, outside of this discussion, where 0.999... would be logically required?

(4) Can the principle be extended to every fraction, the final term of which is a single recurring digit (eg 0.1999... = 0.12)?

(5) How is the principle to be generalised to the case where the recurrent is not a single digit, but a group of digits (eg 0.13131313...)?

(6) '1' is a member of the set of whole numbers. How should we define 'whole number' in such a way that it includes some numbers which begin with '0.9'?

(7) The proposition entails, prima facie, the axiom that two different natural numbers may have the same value; is this axiom generally accepted?

These questions are posted in good faith, by one who has no mathematical aptitude, but an intense interest in mathematical philosophy... so if you could try to express your replies in lay terms, as far as is reasonable? I mean the kind of terms that Bertrand Russell would have used for a lay reader.