I understand that 0.999.. equals one
ecks equals 0.9999...
(ecks times ten) - ecks equals ecks times nine
nine times ecks equals nine
divide both sides by nine
ecks equals one
I understand the formula but for some reason, it just doesn't feel right.
can someone explain why it works on a fundamental level?
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>ecks
That isn't a rigorous proof desu, this is better.
0.9999.... = sum x=0->inf of 0.9*0.1^-x
a=0.9
r = 0.1
0.9999... = a/(1-r) = 0.9/0.9 = 1
get a fucking cat, you puss. meow (=ↀωↀ=)
That's because that argument is question begging desu
>it just doesn't feel right.
It doesn't feel right because you are down by one number, but don't worry there an infinite number of numbers.
1/3 = 0.33333333333333333333333...
1/3 times 3 = 0.99999999999999999...
3 thirds = 1
>Sees bulge
ↀωↀ
Because there are no infinitessimals in the real numbers.
The difference [math]1 - 0.\overline{9}[/math] is, depending on the number system, either zero or infinitessimal. Only one of these are real.
if there's an infinite number of numbers, it's not a real number, like 1.
Could it be said that they are not equal because 1 is a real number that totals 1 while 0.999... is not a real number that totals 1? That while their titals agree, they are not the same because 1 is real and the other is not?
Not an argument. That is only true in base 10.
In base 3 1/3 is 0.1
Why do we need a base other than 10?
That's not rigorous either. The proper definitive proof is to construct the particular real numbers to see that they indeed are equal.
why is base 10 the relevant one?
Shut up
Proofs of some theorems are easier in base other than 10 for example.
Not him, but nice argument.
It's rigorous as long as the proof that the sum of a geometric series is a/(1-r) as long as the absolute value of r is less than 1 is given, and most people learn this proof in high school math/calculus
>why is base 10 the relevant one?
its not. but it does shine a light on our biases toward rational numbers. ignoring fraction notation, if a rational number has to be written with repeating digits, it's only because your choice of base doesn't have the right prime factors.
>In base 3 1/3 is 0.1
Only if you're a moron who doesn't convert the fraction to base 3.
>I understand the formula but for some reason, it just doesn't feel right.
It doesn't feel right because you don't know what a notation like 0.999... actually means. Do you have a consistent model of what infinite decimals represent? If not, then you should not even be having an opinion on the value of a particular one of them.
Intuitively, 1-0.999... = 0.000...1 but 0.000...1 is impossible, it's equal to 0
Mathematically, it's a basic calculus limits problem
0.999... base 10
0.FFF... base 16
Both equal to 1?
Equal to each other?
Is one closer to 1 than the other?
In base 3, 0.222... is about equal to 2/3
In base 2, 0.111... is about equal to 1/2
And yet the argument is these are also equal to 1.
0.999... is not equal to 1.
>these are also equal to 1.
yup
What about the rest?
In base 2, 1/2 is 0.1
0.111 is exactly equal to 0.555 in base 10.
So 0.111... is equal to 0.555...?
And somehow also 0.999... and 1?
0.111_2 = 1/2 + 1/4 + 1/8
2 does not exist in base 2.
There is only 1 and 10.
1
10
11
100
101
111
1000
>0.111_2 = 1/2 + 1/4 + 1/8
looks like the beginning of an infinite series we should all know.
It's an argument for there being an infinitely small nonzero fraction between 0.111... and 1.0
Which is an argument AGAINST 0.111... and 1.0 being equal.
Which is an argument AGAINST 0.999... and 1.0 being equal.
Samefag
>most people learn this proof in high school math/calculus
not condescending at all
Suppose a function has a range of (∞,1)
What is the greatest possible y value it can output?
>0.111 is exactly equal to 0.555 in base 10
Wrong
>arguing with bait
Shut up piggot
>0.9999.... = sum (x=0->inf) of 0.9*0.1^-x
>The proper definitive proof is to construct the particular real numbers to see that they indeed are equal.
Thats what I did you absolute retard.
you didn't construct diddly dick
>0.999... base 10
>0.FFF... base 16
>Both equal to 1?
Yes.
>Equal to each other?
Yes.
>Is one closer to 1 than the other?
No.
>In base 3, 0.222... is about equal to 2/3
No. 0.222... in base 3 is 1. 0.2 in base 3 is 2/3.
>In base 2, 0.111... is about equal to 1/2
No. 0.111... in base 2 is 1. 0.1 in base 2 is 1/2.
>And yet the argument is these are also equal to 1.
Indeed.