4 ms·

A Professor I TA'd for made a very good point: It's not so important if you think 0.9999... = 1 or not, the important thing is to have a (rigorous) definition f

A Professor I TA'd for made a very good point: It's not so important if you think 0.9999... = 1 or not, the important thing is to have a (rigorous) definition for what 0.999... means. So much time is lost to people arguing without first defining what they are talking about. Not to mention the exercised of writing down what a repeating decimal means is a great way to clarify your thinking.

Koshkin 4y ago
> a (rigorous) definition for what 0.999... means But 0.999... is not "defined" to be equal to 1, it is equal to 1 as a matter of fact. (In other words, both are just different spellings of the same number arising, automatically, from the construction of the decimal representation of the reals.)
- patrickthebold 4y ago
  I'm not familiar with the construction of the reals, but somewhere in there you'd have to say what 0.999... means for it to be equal to 1. I'd probably just define it as an infinite series, where it's straight forward to show it is 1. But on second thought, kids run into decimals before constructing the reals, I'd imagine you could define rational numbers and repeating decimals and end up with a simpler argument.
  - deleted 4y ago
    [deleted]
- pdpi 4y ago
  0.99… doesn’t equal 1 in, say, base 16. 0.11… can be 1/9 in decimal, or 1 in binary. Those equalities aren’t a “matter of fact”, they’re provable artefacts of positional number systems.
- macrolocal 4y ago
  See ~ page 16 here for the (historic) definition: https://personal.math.ubc.ca/~cass/courses/m446-05b/dedekind-book.pdf https://personal.math.ubc.ca/~cass/courses/m446-05b/dedekind... It’s possible to work with both {x : x < r} and {x : x <= r} and maintain the distinction between 0.999… and 1, but then eg. 1 = 0.999… + x has no solution.
constantcrying 4y ago
>the important thing is to have a (rigorous) definition for what 0.999... means And a rigorous definition of what "=" means too, which is another important point.

Koshkin 4y ago

> a (rigorous) definition for what 0.999... means But 0.999... is not "defined" to be equal to 1, it is equal to 1 as a matter of fact. (In other words, both are just different spellings of the same number arising, automatically, from the construction of the decimal representation of the reals.)

patrickthebold 4y ago
I'm not familiar with the construction of the reals, but somewhere in there you'd have to say what 0.999... means for it to be equal to 1. I'd probably just define it as an infinite series, where it's straight forward to show it is 1. But on second thought, kids run into decimals before constructing the reals, I'd imagine you could define rational numbers and repeating decimals and end up with a simpler argument.
- deleted 4y ago
  [deleted]
pdpi 4y ago
0.99… doesn’t equal 1 in, say, base 16. 0.11… can be 1/9 in decimal, or 1 in binary. Those equalities aren’t a “matter of fact”, they’re provable artefacts of positional number systems.
macrolocal 4y ago
See ~ page 16 here for the (historic) definition: https://personal.math.ubc.ca/~cass/courses/m446-05b/dedekind-book.pdf https://personal.math.ubc.ca/~cass/courses/m446-05b/dedekind... It’s possible to work with both {x : x < r} and {x : x <= r} and maintain the distinction between 0.999… and 1, but then eg. 1 = 0.999… + x has no solution.

constantcrying 4y ago

>the important thing is to have a (rigorous) definition for what 0.999... means And a rigorous definition of what "=" means too, which is another important point.