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One way to introduce the idea that a number represented as decimal digits can have multiple representations is to talk about the numbers 1 and 1.0 being exactly

by adunk 6y ago

One way to introduce the idea that a number represented as decimal digits can have multiple representations is to talk about the numbers 1 and 1.0 being exactly the same. And that 1.00 is the same as 1. Just like 0, 0.0, and 0.00 are the same number. Most people would agree at this point. Then keep stretching the number of zeroes to 0.000... - which, again, is exactly the same as 0. From there, it is not a huge stretch to be able to go from that 0.000... is another way to write 0, then 0.999... is another way to write 1.

andrewprock 6y ago
The real question is what do you get if you add: 0.999 ... infinite number of 9s ... 9 and 0.000 ... infinite number of 0s ... 1
- aw1621107 6y ago
  > 0.999 ... infinite number of 9s ... 9 > 0.000 ... infinite number of 0s ... 1 Wouldn't these be ill-defined? You can't say "infinite number of 9s" then have the numerical representation terminate with a 9. If the decimal representation terminates, then by definition it isn't infinite.
  - andrewprock 6y ago
    Whether or not a specific construction is well defined depends on the system you are using. This representation is certainly well definable. I think your objection here is in the same vein as those who object to the notion that 0.999... = 1.0 For most people, the concept of an infinite representation is not well defined.
    - aw1621107 6y ago
      > This representation is certainly well definable. Do you mind defining it more formally for me then or pointing me to explanations/systems that would permit such a definition? I'm not really sure what such a formulation might look like, but then again I'm not nearly well-acquainted enough with mathematical topics past what is commonly taught. Does it involve the hyperreals, surreals, or one of the other systems beyond the "standard" reals as mentioned by other comments in the thread? > I think your objection here is in the same vein as those who object to the notion that 0.999... = 1.0 > For most people, the concept of an infinite representation is not well defined. I think (or at least hope) it's a bit more nuanced than that. I understand that some reals with infinite decimal representations can be well defined as an infinite series, and that defining 0.999... using such a series allows other manipulations to be done to complete the proof. However, adding the concept of an "end" to said infinite series kind of breaks that understanding. The translation to an infinite sum no longer seems to hold, so I'm at a bit of a loss. It's also somewhat counterintuitive to have an "end" to infinity, but as the rest of the thread shows intuition isn't always reliable for this kind of thing, especially for those who aren't particularly familiar with more detailed bits of math.

by adunk 6y ago

andrewprock 6y ago

The real question is what do you get if you add: 0.999 ... infinite number of 9s ... 9 and 0.000 ... infinite number of 0s ... 1

aw1621107 6y ago
> 0.999 ... infinite number of 9s ... 9 > 0.000 ... infinite number of 0s ... 1 Wouldn't these be ill-defined? You can't say "infinite number of 9s" then have the numerical representation terminate with a 9. If the decimal representation terminates, then by definition it isn't infinite.
- andrewprock 6y ago
  Whether or not a specific construction is well defined depends on the system you are using. This representation is certainly well definable. I think your objection here is in the same vein as those who object to the notion that 0.999... = 1.0 For most people, the concept of an infinite representation is not well defined.
  - aw1621107 6y ago
    > This representation is certainly well definable. Do you mind defining it more formally for me then or pointing me to explanations/systems that would permit such a definition? I'm not really sure what such a formulation might look like, but then again I'm not nearly well-acquainted enough with mathematical topics past what is commonly taught. Does it involve the hyperreals, surreals, or one of the other systems beyond the "standard" reals as mentioned by other comments in the thread? > I think your objection here is in the same vein as those who object to the notion that 0.999... = 1.0 > For most people, the concept of an infinite representation is not well defined. I think (or at least hope) it's a bit more nuanced than that. I understand that some reals with infinite decimal representations can be well defined as an infinite series, and that defining 0.999... using such a series allows other manipulations to be done to complete the proof. However, adding the concept of an "end" to said infinite series kind of breaks that understanding. The translation to an infinite sum no longer seems to hold, so I'm at a bit of a loss. It's also somewhat counterintuitive to have an "end" to infinity, but as the rest of the thread shows intuition isn't always reliable for this kind of thing, especially for those who aren't particularly familiar with more detailed bits of math.