Quote:
Originally Posted by Kirby
I understand what Infinity means. Never-ending and all of that jazz.
The difference between "1.999..." and "2" is "0.000...001"
And I also understand the whole "not logically being able to be anything but 1" also, but there's still a difference in 1.999... and 2.
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Look, I know it seems counter-intuitive to believe that 0.9 recurring = 1, but you have to trust me on this one
Your reasoning follows on from this:
2 - 1.9 = 0.1 = 10^-1 (which is more than 0)
2 - 1.99 = 0.01 = 10^-2 (ditto)
2 - 1.999 = 0.001 = 10^-3 (ditto)
General Form:
2 - 1.9{n} = 10^-n where 9 is repeated n times (which is more than 0)
That reasoning is fine for a finite n. But as n tends towards infinity, limit(10^-n) = 0. In other words, that 0.000...001 number you quoted is basically 1/∞ which is considered to be zero, meaning that 2 - 1.999... = 0 (which makes sense since 2
is 1.999...)
Here's another simple proof that 0.9... = 1. This is a method commonly used to convert recurring decimals into fractions and it demonstrates that 0.9... is 1/1 as a fraction.
x = 0.999...
10x = 9.999...
10x - x = 9.999... - 0.999...
9x = 9
x = 9/9 = 1, therefore 0.999... = 1
The important thing to understand is that 1.000... and 0.999... are both decimal representations of them same number (1). This applies to other numbers too (i.e. 0.8324 = 0.8323999... ) but not to all numbers (i.e. 1/3 = 0.3... and there is no other decimal representation).