<p>I was wondering where the CC community at large stands on this issue.</p>
<p>Does .999..... equal 1, or is it just a lie told by brash mathematicians? If you disagree, I'd like to see why!</p>
<p>it's a lie. get over it.
math is a precise science, so everything is just how it is. rounding is herecy.</p>
<p>Here are the proofs as to why .999...=1.</p>
<p>.99... is the same as 9/10+9/100+9/1000.... and the sum of any such series is a/(1-r), in this case a=9/10, and r=1/10, so the sum is (9/10)/(1-1/10)=9/9=1.</p>
<p>There are many more. Mathematically, it seems irrefutable that the two values are equal. And yet people love to say they aren't. Why?</p>
<p>x=.9~
10x=9.9~
10x-x=9.9~-9~
9x=9
x=1</p>
<p>.999~ = sigma(.9*[.1]^[n-1]). </p>
<p>.999~ = .9 + .09 + .009 + .0009 + .00009 + .000009 .... (ad infinitum)
Infinite geometric progression: a + ar + ar^2... (ad infinitum)
Let a, 1st term = .9
Let r, common ratio = 10^-1
Sum to infinity = a / (1 - r) = .9 / (1 - .1) = .9/.9 = 1
.999~ = 1</p>
<p>sigma(n = 1, n -> inf.) 9/(10^n) = Definition of geometric series.
9 * sigma(n = 1, n -> inf.)(1/10)^n = Property of a series.
9 * 1/9 = (r/[1 - r], r = .1)
3/9 = 1/3
.999~ = 1</p>
<p>.999~ = .9 + .09 + .009 + .0009 + .00009 + .000009 .... (ad infinitum)
sigma<a href=".9*.1%5Ei">i:0 -> inf.</a>
omega -> sigma = .9/(1 - .1) = .9/.9 = 1
.999~ = 1</p>
<p>The sequences (.9, .99, .999... ad infinitum), and (1, 1, 1... ad infinitum) are equivalent, so they have the same limit, .999~.
S = .999~
S = .9 + .09 + .009 + .0009 + .00009 + .000009 .... (ad infinitum)
S = 0.9 + (1/10)(0.9 + 0.09 + 0.009 + .0009... [ad infinitum])
S = 0.9 + (1/10)S
(9/10)S = .9
S = 1
.999~ = 1</p>
<p>.333~ = sigma(n = 1, n -> inf.) 3/(10^n) = Definition of geometric series.
3 * sigma(n = 1, n -> inf.)(1/10)^n = Property of a series.
3 * 1/9 = (common ratio, r/[1 - r], r = .1)
3/9 = 1/3
1/3 = .333~
1/3 * 2 = 2/3
.333~ * 2 = .666~
2/3 = .666~
.333~ + .666~ = .999~
1/3 + 2/3 = 3/3
3/3 = .999~
3/3 = 1
.999~ = 1</p>
<p>.(46)~ = 46/99
.000~ = 0/9
.111~ = 1/9
.222~ = 2/9
.333~ = 3/9
.444~ = 4/9
.555~ = 5/9
.666~ = 6/9
.777~ = 7/9
.888~ = 8/9
.999~ = 9/9
9/9 = 1
.999~ = 1</p>
<p>An infinite geometric series:
Sn = a + ar + ar² = ar³ +...+ ar^(n-1)</p>
<p>Sn = sum ; a = 1st term ; r = ratio ; n = term number</p>
<p>The series for .9~:
Sn = .9 + .9(.1) + .9(.1)² + .9(.1)³ + ... + .9(.1)^(n-1)</p>
<p>Sum of an infinite geometric series:
Sn = a/(1-r)</p>
<p>Sum for .9~:
Sn = .9/(1-.1)
=.9/.9
=1</p>
<p>I wasn't looking for proof. I know exactly why .99...=1. I was wondering why people disagree with said proof.</p>
<p>JCoveney
it's a paradox. just like this one <a href="http://hlavolamy.szm.sk/images/zmiznutie.gif%5B/url%5D">http://hlavolamy.szm.sk/images/zmiznutie.gif</a></p>
<p>a much more concise proof can be found in abstract algebra. the rational numbers are dense in the reals, so since there is no number "between" .9999... and 1, they are identically the same.</p>
<p>That's a more sophisticated proof, iostream, and once again, proof is not the point.</p>
<p>And dma, the problemw ith that is that on the second figure the lines are not the same slope. It's not an actual triangle.</p>
<p>I don't think that it's true cuz where's the extra .000000...0000001 go? ;-)</p>
<p>Are you being serious Mr. Hairy?</p>
<p>Lol no, but that's the argument that you usually see against this kind of proof.</p>
<p>Arguments for disbelief by people who believe are seldom effective :)</p>
<p>i think a big reason why people like to argue against it is because they are trolls.</p>
<p>I remember when I first saw this I thought it was soooo cool. So, I started showing my friends and family. I went through and proved it, but in reality, the only thing I proved was the fact that I had no life, haha.</p>
<p>The geometric series implies that that series converges to 1. Doesn't mean that it equals one. As for everything else, sure, essentialy, they're the same. But .999repeating is not a rational number. Thus, it's hard/stupid to compare it to 1, which is actually a rational number.</p>
<p>Im more interested in that triangle. Whats the deal with that?</p>
<p>i second that</p>
<p>I found it. </p>
<p>Pretty cool; I didn't notice when I looked at it.</p>
<p>ah...678910characters</p>
<p>Dammit. .999999.... IS the limit of the partial sums of said series. It converges yes, but that limit is exactly .999999999....</p>
<p>And .9999... IS a rational number as I just showed you. Rational meaning expressible as a ratio of two integers. 8/9 = .8888.... 9/9 = 1.</p>