1554
rating
15
debates
73.33%
won
Topic

# 0.999... = 1

Status
Finished

All stages have been completed. The voting points distribution and the result are presented below.

Arguments points
24
0
Sources points
16
12
Spelling and grammar points
8
6
Conduct points
8
0

Nemiroff
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Category
Education
Time for argument
Three days
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Open voting
Voting period
One month
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Four points
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1294
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Round 1
Pro
How to convert a repeating decimal into a fraction: https://studymaths.co.uk/topics/convertingRecurringDecimalsToFractions.php

X = 0.333...
10x = 3.333....
10x - x = 3.333... - 0.333...
9x = 3
X = 3/9 = 1/3
0.333...  = 1/3

X = 0.111...
10x = 1.111...
10x - x = 1.111... - 0.111...
9x = 1
X = 1/9
0.111... = 1/9

X = 0.999...
10x = 9.999...
10x - x = 9.999... - 0.999...
9x = 9
X = 9/9 = 1

Con
Nonsense 1 is greater than anything less than one even by a bit
Round 2
Pro
1 is greater then anything less then 1, even by a bit. But 0.999... is clearly = to 1. And anything equal is not less then 1, even by a bit.

0.333 is not 1/3rd. 0.333333333 is not 1/3rd.
0.3 and any finite number of zeros is still less then 1/3rd. Even if there are billions of trailing 3's, still less then 1/3rd (even if by only a bit).
But 0.3 with *infinite* 3s is 1/3rd.
Just like 0.9 with *infinite* 9s is 1.

Your error lies in the fact that infinity is not a number. Infinity is not simply alot of 9s. Its never ending. To infinity. If 0.999... =/= 1, then 0.333... =/= 1/3, which it most certainly does.

And finally, appeals to logic are always weaker then objective fact, like a mathematical proof. Heres another proof:

0.333... = 1/3
0.333... (x3) = 1/3 (x3)
0.999... = 3/3 = 1
Con
you are splitting hairs , literally!
Round 3
Pro
Disagree. This is not a question of semantics or anything like that. This is fact, as certain as 1 = 1.

I thought that 10x in my original formula was conveniently easy. If it is an iron fact rule, any multiplication of both sides of the equation should work.

First i tried 5, assuming wrongly that 5x would be half of 9.9999999.... as 4.5454545.... forgetting to carry remainders. If you punch in 0.99999 ×5 into a calculator you will get 4.99995. Add more 9s, the answer adds more 9s. No other changes. So assuming infinite 9s, 5x would be 4.99999.... and the final 5 would never come. Thus:

X = 0.999...
5x = 4.999...
5x - x = 4.999... - 0.999...
4x = 4
x= 4/4=1

Again with 3.

3x=2.999...
3x-x=2.999... - 0.999
2x=2
X=1

Again with 23
23x= 22.999...
22x=22
Con
Forfeited