.999... Equals exactly 1.
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And why would we want to divide by 1/9 ? Wouldn't this work the same with we substituted 9 with 8?
No-one is "dividing" anything by 1/9. Look again at the OP; you are multiplying 9 by 1/9.
What are 2 halves? What are three thirds, what are four quarters......? Nine ninths = 1, one ninth is 0.11111111...... as your calculator tells you. Surely even you can see that the rest follows.
Enmos
Valued Senior Member
Oh I see, I wasn't really paying attention the math.
Well hmm..
LOL this must be the fifth or so thread about this (hence the aggh
)
Wikipedia says this equation proves it..
Where does multiplying 0.111 come from?
I don't get it!
That equation proves very little. if you want to get technical, I guess we can (even though I am lazy).
BY DEFINITION, $$ 0.111... = \lim_{n \rightarrow \infty} \sum_{i=1}^{n} \frac{1}{10^i}$$.
Ok? Also, BY DEFINITION, $$\lim_{n \rightarrow \infty} \sum_{i=1}^{n} \frac{1}{10^i}$$ is equal to the limit of the sequence of partial sums $$(s_1, s_2, ... )$$ where $$s_n = \sum_{i=1}^{n} \frac{1}{10^i}$$. Does EVERYONE see this? We are NOT summing infinite terms by this definition... people seem to think that because summing an infinite number of terms is physically impossible we cannot describe what such a sum may look like.
OK, we notice two things people. The first... $$s_n = \sum_{i=1}^{n} \frac{1}{10^i}$$ is an non-decreasing. What does this mean? $$s_{n+1} > s_{n} \ for \ all \ n \geq 1$$. So, clearly... $$s_1 < s_2 < ... < s_k < ... $$.
What's the second thing we notice?!?!? For all subsequences $$s_n$$, we have that $$s_n < 1$$.
These two points are very significant. Why? Think about the real world for a second... if you fill a balloon with helium it floats, right? Letting go of the balloon and we see that its height is NON-DECREASING (in other words, its height is strictly increasing). If you let this balloon float up and up and up in a building with a roof, it's going to stop floating when it hits the roof.
This sequence $$s_n$$ is just like this balloon. It increases... but it's bounded by a "roof." We know that this roof is at MOST 1. However, we can show that 1 is the shortest roof for this sequence (least upper bound, people). How?
I'll explain later. My class starts in a few minutes.
Enmos
Valued Senior Member
lol I'm sure he gets it now..
This isn't really for him. I want to get my point across to all the idiots that will eventual invade this thread with their stupid claims about 1 != 0.999...
Enmos
Valued Senior Member
The mathematicians like everything absolute, like it is definite as a rock. The so called limit formula is the definition that implies that a limit to the .9999 to the infinite equals 1. This definition is self contradicting because INFINITY DOES NOT HAVE A LIMIT but they dismiss it because just as they say that photon does not have mass (which it does 10^-50g) they dismiss this as well because as they say it is NEGLIGIBLE. I really hate this but if you were asked what is the .9999 really equal to, understand that by human mathematics definition it is equal to 1, when in reality it does not.
Absane learned the definition, every book has it allright, every mathematician says it allright, well its all an assumption, its all based on an assumption that infinity does have a limit.
AND NO I DON'T PULL THIS STUFF OUT OF MY ASS: http://www.aip.org/pnu/2003/split/625-2.html
Absane learned the definition, every book has it allright, every mathematician says it allright, well its all an assumption, its all based on an assumption that infinity does have a limit.
AND NO I DON'T PULL THIS STUFF OUT OF MY ASS: http://www.aip.org/pnu/2003/split/625-2.html
Oh, so now I won't get it?
from someone who has been so many eons in mathematics, we are like bugs to squash
Well if a photon has mass, how does light illuminate behind glass.. or how does light shine thru glass?
transfer of energy from atom to atom. the light you see shining is not made of the photons from the original source but photons that were created/knocked out of the atoms of glass.
nothing changed, what happened is that the photons were absorbed into atoms of the glass and the energy waves transfered from atom to atom to knock out a new photon particle from a new glass atom on the other side of the glass cup.
and come on...you asked a question...I am answering, I am not hijacking anything.
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