0.9999... never happens anyway. There's no fraction that gives you that decimal unless you fuck up the long division. So don't even bother with this shit.
>>7
Actually, it happens every time you eg divide a number by itself, since it is *gasp* the exact same thing as 1. The '1' notation is just more convenient to write.
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Anonymous2006-05-25 21:35
If you rewrite .9999999... as .9 + .09 + .009 + .0009 + ..., and then rewrite those numbers as fractions, you get 9/10 + 9/100 + 9/1000 + 9/10000 + ... So therefore .9999... is really just an infinite geometric series with the first term = 9/10 and the rate =1/10. Using the formula for the sum of an infinite series, which is a/(1-r), you'll get
(9/10)/(1-1/10)
= (9/10)/(9/10)
= 1
Q.E.D.
I thought this up myself at two in the morning one time. Other people have probably done it this way too, since it's the only really solid proof I've ever seen of .999999... = 1.
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Anonymous2006-05-25 23:16
>>9
No need to go through all that rubbish. You just have to show that there is no real number between .99999999... and 1.
Why would you ever want to write 0.99999... instead of 1 in the first place? It's not like you will ever encounter it in arithmetic. It's just a curiousity, a strange construct.
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Anonymous2006-05-26 3:00
>>8
Nonsense, it never happens. I'm not talking value, I'm talking the representation of the value. You never see 0.99999... when you divide an integer by itself, tough guy.
It can be even simpler.
1/3 = .33333...
3 * 1/3 = 1
3 * .333333... = .99999...
∴ .9999... = 1
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Anonymous2006-05-28 0:07
>>1
>What the fuck. Why is that true. They got different numbers in them.
Lets list a few different ways we can express the value 1 without using that number, shall we?
3-2=1
2/2=1
cos(0)=1
8935^0=1
|√(0.5)+√(0.5)i|=1
i^4=1
ln(e) = 1
So why in the world would the fact that .999... doesn't include the number 1 prevent them from being equal?
(my intention was not to prove .999...=1, but rather to dissuade the assumption that since they include different numbers, they cannot be equal)
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Anonymous2006-05-28 2:00
|√(0.5)+√(0.5)i| = 1
Could you prove this? I'm not so good with imaginary numbers and would like to see if this is true... Unless you meant to write
||√(0.5)-√(0.5i)||
Which I can see how it would work
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Anonymous2006-05-28 20:13
No, it's correct, |√(0.5)+(0.5i)|=|(√2+√-2)/2| right.
Take √2 out of the bracket, you get |(√2(1+i))/2|
Cancel the √2 you get |(1+i)/√2|
Square the fraction you get |(1+2i-1)/2|
Collect like terms |2i/2|
Simplify, end up with |i| which we know = 1.
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Anonymous2006-05-28 20:20
it depends on what norm you are using lol.
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Anonymous2006-05-28 21:27
>>18
Or you could use a calculator.
Or you could defer to the complex number plane, plot the point, and then realize that the point is exactly π/4 on the unit circle in such a plane, and then realize that absolute value means distance from zero, which in this case is 1 because its the fucking unit circle.
In other words, you shouldn't have had to do all that algebra.
Yes, but he did say prove and while plotting the point on the complex plane is simpler, I can't easily do that on a text board. I could've described it like you did but I wasn't sure if that would be satisfactory. Besides, it's not that much algebra.
It's a flaw in the Number System since you can't apply these laws into reality and are impossible to co-erce with realistic physics. Since, in realistic physics, there are no fractions and all matter/energy exists in integers.
So my point is, such numbers are non-applicable to reality, which is where the confusion exists.
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Anonymous2006-05-30 21:47 (sage)
wen eye git a nerection, my peenis is so long an hard yu cud poke it wit a needle and it would burst and the exploshun wud be enuff to destroy teh world!!!!!
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Anonymous2006-05-31 3:48
gad
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Anonymous2006-05-31 3:48
adg
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Anonymous2006-05-31 3:48
adf
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Anonymous2006-05-31 3:49
.9999999... to my knowledge, does not equal to one. It is always less then one, always.
but, it is infinitly close to one, just never exactly one