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So I recently got into an argument with someone about this. If $x\in\mathbb{Z}$ satisfies $|x|<10$ in base ten, how many digits does it have? My position is that we can't say for sure, it can have one or more, because for example $x=one=1=1.0=1.00=0.999...$ is an integer and it satisfies $|x|<10$. The other person says that in mathematics, insignificant digits are ignored. So $x$ is just 1, which is a single digit. Is he/she correct?

Kashados
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    As we're dealing only with integers, yes, one. – DanLewis3264 Oct 15 '20 at 21:17
  • You can express 1 as 0.999... recurring, that being said it depends on what you mean by "how many digits", let's say $ S:={ x\in\mathbb{Z}: |x|<10 }={-9,-8,...,-1,0,1,...,8,9} $ which are indeed all one digit numbers, and they are all and only the ones satisfying the condition we are imposing. I would say your friend it's correct, still it depends on what you take as definition of "number of digits" – Spasoje Durovic Oct 15 '20 at 21:22
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    "$x\in\mathbb{Z}$" just means that $x$ is an integer. Integers don't have digits -- it's their digital representations that have digits (and these representations, even in a given base, need not be unique). – r.e.s. Oct 15 '20 at 21:31
  • It is worth also clarifying what definition you are using with regards to how many digits the number $0$ has. In some definitions one might say it has one digit, zero digits, or maybe even negative infinitely many digits. See here. – JMoravitz Oct 15 '20 at 21:43
  • I think, everyone would say that "$4$" has one digit , if nothing else is specified. By default, we assume that we have the decimal expansion and the we neither have trailing zeros like $1.00$ nor leading zeros like $001$ nor something like $3.\bar 9$. So, I would agree the other person. The answer is $1$. – Peter Jan 21 '21 at 07:58
  • If we speak of a "$799$-digit prime number" , I think everyone knows how this is meant without cumbersome definitions. – Peter Jan 21 '21 at 08:02

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