- Thread Starter
- #21

So, agree with the gist of it, but there's an implicit use of limits.Let me explain it better.

1 - 0.9 = 0.1

1 - 0.99 = 0.01

1 - 0.9999 = 0.0001

1 - 0.9bar = 0.0bar (A never ending sequence of nines results in a never ending sequence of 0's)

So since 0.0bar clearly equals 0 then 1 must = 0.9bar.

EDIT: @BDWoody 's works also

In your sequence of 0.1, 0.01, 0.001 .... you will never get to 0.0bar. You can never get rid of that pesky "1", even if it is farther and farther to the right.

If you want to explain 0.9bar = 1 this way to someone who doesn't know limits, there's some hand-waving involved.

In contrast, BDWoody's proof requires no hand-waving.

So, interestingly, in analysis books you learn the construction of the real numbers with Dedekind cuts.

Dedekind cuts contain an infinity of numbers, but are easier to reason about than "a never-ending sequence of decimals".

The gist is to define a real number as the set of all rational numbers greater than it.

Having done that, you need to define all the arithmetic operations, and prove that indeed this set of numbers works as a number.

E.g. you define the square root of 2 as, the set of rational numbers such that their square is greater than 2.

So, you could think of defining 0 as the set of all rational numbers greater than 0.