There is no difference. When you subtract the two, you get 0.000... = 0. There is no 1 at the end, because there is no end. That's what "an infinite number of nines/zeroes" means. They are the same number.
This is not correct. Consult a text on 'real analysis' and/or 'complex analysis'. If your claim were true, vast amounts of solved mathematics would vanish.
.9 recurring = 1, an infinite series. 1 plus half plus quarter etc =2, an infinite series
Which part of the quote is not correct? And go ahead; I'll just give you a link to Wikipedia and you can start browsing from its references.
0.9999...
by definition is:
sum_[i=1]^\infty 9/10^i := lim_(n -> infty) sum_[i=1]^n 9/10^i = 1.
Similarly,
0.000...
by definition is:
sum_[i=1]^\infty 0 := lim_(n -> infty) sum_[i=1]^n 0 = 0.
The sum of an infinite series can be a well-defined number. The sum of reciprocals of powers of 2 (1+ 1/2 + 1/4 + 1/8 +...) to infinity is 2. Yes, the whole number 2, nothing more, nothing less. Not one infinitesimal less. Likewise, 0.9999... is the sum of 9/10 + 9/100 + 9/10^3 +... to infinity, which gives the well-defined exact answer 1. Not an iota off.
@steve_bute: I'm using nice simple language to explain a nice simple result in the nice simple real numbers. Real numbers. That's R, not C or R* or what have you. I'm aware of real analysis as well as other constructed hyperreal/infinitesimal systems. What is being argued here is in neither of those.
.9 recurring = 1, an infinite series. 1 plus half plus quarter etc =2, an infinite series
Well, I still can't see why I should differentiate 2 from 1 + 1/2 + 1/4 + 1/8 + ..., besides that the latter needs a (relatively easy) proof to be shown to be equal to 2.
0.000...
by definition is:
sum_[i=1]^\infty 0 := lim_(n -> infty) sum_[i=1]^n 0 = 1.
This is genius. :P
However, besides for that glaringly obvious typo, what he said is correct.
@steve_bute: I'm using nice simple language to explain a nice simple result in the nice simple real numbers. Real numbers. That's R, not C or R* or what have you. I'm aware of real analysis as well as other constructed hyperreal/infinitesimal systems. What is being argued here is in neither of those.
Even in the seemingly simple domain of good ol' R^1, the difference between an open set and its boundary is important.
There is no difference. When you subtract the two, you get 0.000... = 0. There is no 1 at the end, because there is no end. That's what "an infinite number of nines/zeroes" means. They are the same number.
This is not correct. Consult a text on 'real analysis' and/or 'complex analysis'. If your claim were true, vast amounts of solved mathematics would vanish.
What in real analysis contradics anything with .999...=1? Please enlighten me. I'm grabbing my real analysis book as we speak.
@steve_bute: I'm using nice simple language to explain a nice simple result in the nice simple real numbers. Real numbers. That's R, not C or R* or what have you. I'm aware of real analysis as well as other constructed hyperreal/infinitesimal systems. What is being argued here is in neither of those.
Even in the seemingly simple domain of good ol' R^1, the difference between an open set and its boundary is important.
The problem is that real numbers don't have infinitesimals, so sequences tending to a point (or, taking your way, points in a set tending to a boundary) has the limit of that point, exactly. You won't say 0.999... = 1 minus some infinitesimal number, because that infinitesimal number simply doesn't exist in R.
x = 0.999...
10x = 9.999...
10x - x = 9.999... - 0.999...
9x = 9
x = 1
(Taken from the wikipedia article for 0.999...)
x = 0.333...
10x = 3.333...
10x - x = 3.333... - 0.333...
9x = 3
x = 1/3 (or 3/9)
Works for any repeating number:
x = 0.777...
10x = 7.777...
10x - x = 7.777... - 0.777...
9x = 7
x = 7/9
Generally speaking, 0.nnn.... (where n is an integer between zero and nine) = n/9. Consequently, 0.999... is 9/9, or 1.
Works with two digit repeaters and on up from there as well, it's just that the denominator changes:
x = 0.7575...
100x = 75.7575...
100x - x = 75.7575... - 0.7575...
99x = 75
x = 75/99
.9 repeating is one if yo dissagree watch this http://www.youtube.com/watch?v=TINfzxSnnIE
211.000013/99 is not rational.
Um, yes it is:
211.000013/99
= 211.000013 x 1000000
99 x 1000000
= 211000013
99000000
What's the sum of all natural numbers?
Spoiler: the answer is crazier than the question
-1/12. Euler was the man.
good post
Be wary of a non-peer-reviewed publication like wikipedia. There are many ongoing disputes about what's been written on a variety of mathematical topics. It can be a great source at times, but occasionally misleading.
What's the sum of all natural numbers?
Spoiler: the answer is crazier than the question
-1/12. Euler was the man.
Um, no:
http://scientopia.org/blogs/goodmath/2014/01/17/bad-math-from-the-bad-astronomer/
Thanks for that.
Be wary of a non-peer-reviewed publication like wikipedia. There are many ongoing disputes about what's been written on a variety of mathematical topics. It can be a great source at times, but occasionally misleading.
I told you that you can start browsing from the references. Yes, I do know Wikipedia is not perfect, but I know Wikipedia has references.
Regarding 1+2+3+4+...: In the usual sense, the series is divergent, but some mathematician made an interpretation to them.
The sum of an infinite series is, unfortunately, a number too (or undefined because the series diverge). I'm not sure I get what you mean by not confusing a number and a sum.