Post #3832635

## True Infinitesimals To proceed to have infinitesimals we need to accept one of two things: 1. Much more complicated mathematics than I want to explain here 2. Limitations on what you can do with the number system being developed (e.g. a partial order, or arithmetic that isn’t always defined) The first option is taken by the development of the Levi-Civita field. This is formed not of sequences but of functions with *rational* number inputs, so is quite a bit more complicated. More complicated still is the mathematics of *Non-standard Analysis* which uses ordinary sequences but to create the notion of equivalence requires an object called an *ultrafilter*, which is quite advanced to explain. The second option is taken if you examine the structures I mentioned above, as well as others, such as the *dual numbers*, but none really captures the desire to say that 0.999… is infinitesimally less than 1: depending on the exact structure, either decimal expansions no longer make sense at all, or the statement is not true, or the two things are incomparable, or arithmetic doesn’t work so it is too far from the real numbers that 0.999… and 1 are clearly members of. ### Back to Decimals It’s worth thinking about where we came to this discussion from - decimal expansions. When someone unschooled in higher mathematics doubts that 0.999… = 1, they typically want to say that 1 - 0.999… = 0.000…1, i.e. “zero, followed by a decimal point, followed by infinitely many zeroes, *followed by a one*”. This thing is not a decimal expansion, at least, not the decimal expansion normal mathematicians use. Those decimal expansions do not have any digit which has infinitely many digits before it, whereas this one does. If we decide to define this kind of expansion as an infinitesimal, we ought still to be able to do arithmetic with it: if we call it ε, we should be able to work with 10ε and ε/10. If the expansions are to be worth having, they should obey the normal rules of place value, meaning that 10ε should shift the 1 one decimal place to the left, and ε/10 should shift it 1 decimal place to the right. But if we describe those in words they would still be “zero, followed by a decimal point, followed by infinitely many zeroes, followed by a one”, exactly the description of ε itself. In order to not have arithmetic collapse and have all numbers equal to all others, so the description was incomplete. There must be an identified spot - an infinite place value - where this 1 is inserted, so that we can move the 1 to the left when we multiply ε by 10. In particular, there must be zeroes to the left (and right) of that spot, so what we have is not 0.000…1, but rather 0.000…;…0001000…! The “;” here represents an imaginary boundary between infinitely many zeroes stretching to the right in the conventional decimal part, and infinitely many zeroes stretching to the left in the unconventional part, to indicate that the dots hide an infinite, rather than finite, number of zeroes. The problem with this though is that the idea of describing ε in this way was that by adding this one digit to 0.999…, it would be added to the “last” 9 digit, and hence cause infinitely many carries, resulting in 1 exactly. This does not happen if there are infinitely many zeroes for the 1 to move left into, instead 0.999…;...000… + 0.000…;...0001000 would have to be 0.999…;...0001000… So this only works if we conceive of 0.999… as being a shorthand for 0.999…;...9999000… And this genuinely *does* work in an appropriate development of Non-standard Analysis, except for one issue I believe to be fatal: using 0.999… as a shorthand for something whose decimal expansion actually *stops* having nines and from that point is all zeroes is nonsense. If the notation “0.999…” means that the nines never stop, as is the case in standard analysis, then it only makes sense to have this be another way of writing the number 1. Nevertheless, subject to arguments of this kind, analysis with infinitesimals *can* be carried out in this way. There is a simple way of thinking about it: in infinitesimal analysis you generally have a way of chopping off the infinitesimal part of a number, leaving only its “standard” part. The rule is that if you want a structure in which you can perform ordinary arithmetic with infinitesimals, then you can never add an infinitesimal number to another and change the standard part. The notation 0.999… suggests a number whose standard part is less than 1 (hence the confusion), but the desire to see ε as an infinitesimal is a desire for 0.999… to be only infinitesimally less than 1, hence at the very least its standard part must be 1. ## Conclusion This has been a very brief look at a very big topic (Fréchet filters didn’t even make the cut!) I hope it encourages you to think and read about infinitesimal analysis, and where the boundary between figures of speech and mathematical practice lies. As before, the inspiration for this has been one annoying charlatan, but the ongoing motivation is a love of mathematics for its own sake. As always, if you have any questions I’m happy to try and answer (though I know very little about the Levi-Civita field). Until next time! 2/2