The largest number less than one

I got into an argument with a friend about this.

I asked him what is the largest number less than 1, he said 0.999 (recurring) then I pointed out that 0.999… is equal to 1 because 1/3=0.333… and 0.333…*3=0.999…

Then he argued that 0.333…*3 is 1 and we started arguing until we came to the conclusion that because of the fact that decimals are uncountably infinite there is no answer.

Now I’m asking .

Reply 1

Labradoodle1

0.999... is equal to 1

Reply 2

mqb2766

Original post by kella3

The decimal representation isnt unique, so as you say 1 and 0.999... both represent the same number. So for any number with a finite decimal representation so say
1.23456
then its the same as
1.234559999....

Two numbers are different if there is a third number between them. Obviously thats not the case here. Also if you did assume there was a (different) number x closest to 1, then the number (1+x)/2 is different to both 1 and x and closer to 1 than x is, so contradiction. Uncountably infinite isnt really a proof.

(edited 1 year ago)

Reply 3

DFranklin

Original post by kella3

I got into an argument with a friend about this.

I asked him what is the largest number less than 1
~snip~
we started arguing until we came to the conclusion that because of the fact that decimals are uncountably infinite there is no answer.

If you are talking about the real numbers or the rationals, then there is no "largest number less than 1" (since if x was supposedly such a number, then (x+1)/2 is both larger than x and less than 1).

Note that the rationals are countable, so you do not need your number system to be countably infinite for there to be no answer.

Note also that if you were talking about the integers, then 0 would work.