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yes because the number line never stops
|dw:1349359330358:dw| Is infinity a number?
not a natural number and not real number either.
infinity is something which cant be defined...
no .. irrational number is a real number.
Is it a number?
IS infinity a number?
it's a number is http://en.wikipedia.org/wiki/Hyperreal_number and http://en.wikipedia.org/wiki/Surreal_number ... I don't know much about this ... but this is not a real number and natural number.
If 100000000000000... is not a natural number then the set of natural number is finite. right?
@experimentX how do u say that it is not a real number??
no ... the set of natural number is infinite but does not contain infinity.
Infinity is a concept, not a number (I know I'm basically just restating, but it is worth emphasizing)
you always write |dw:1349359774760:dw|
you should have noted.
then why 1/3 is considered a number?
0.3333.... as the 3's continue the number is not getting any bigger, as opposed to making an infinitely number before the decimal which would create an infinitely large number.
hm... I phrased that poorly :/
natural number is defined as inductive set that begins with 1 and |dw:1349359861034:dw|
indeed what would 10000.....+1 be? could you write it? where would you put the 1 ?
Why 0.333333333333333333....... is a number?
that is different from 0.3333....+1=1.3333... not problem there
1/3 is repetitive ... if you take 1m string ... fold it thrice, you can always pinpoint this is 1/3 . Therefore it lies inside real number line.
you can add any number to 0.333.... and create a new number on the real line where would 1000...+1 be on the real line if you can point it out then I will let you call it a number ;)
I like this video to help with understanding repeating decimals, though it does not directly answer your question http://www.youtube.com/watch?v=TINfzxSnnIE
@abdul_shabeer u can find the number 0.333333333 or atleast u can locate the point 0.33333 number between 3 and 4 , but u cant locate the infinity .......
We can't locate 0.333333.......
yes we can
atleast u know that 0.33333 is present between 0.3 and 0.4 .... in case of infinity u cant ...... can u ????
now you show me what two numbers 1000..... lies between :)
What is (0.333333......)* 10?
0.333..... * 10 = 0.333...+0.3333....+0.3333....( 10 times) Can we add this?
I think it is a lot easier to see it as 1/3+1/3+1/3+...(ten times) =10/3
I want to add it without converting it into 1/3
As 0.3333.... = 1/3, in whichever way i add I must get the same result
100000000000......... lies between 99999999999...... and 11000000000000...... where all three of the numbers stretch out infinitly.
0.333...+0.333...=0.666... 0.666...+0.333...=0.999... 0.999...+0.333...=1.222... do that ten times and you will get 3.333...
and @Razzputin that is wrong, how can you show me that 9999... is not bigger than 1000... ?
are there more digits in one number or the other? no
what if you where to state that 999.... was 1 less than infinite?
ad a 3 to each 9
oh my bad, .999.+.333...=1.333....
the 1's carry over
and the last digit would be 2
yeah but there are an infinite number of 3's and 9's so every digit carries a one from the previous decimal place
there is no last digit, that's why it's an infinitely repeating decimal
There is no last digit, I just took a small value
which makes it not the same as 1/3+1 which is what you are trying to do
this is why we don't add numbers with infinite decimals very often; It's ugly, confusing, and entirely unnecessary
what do you get when you add 0.9999...+0.3333... without taking 0.99999...=1 and 0.333... = 1/3
math is about making things as simple as possible, not over-complicating things and I told you 0.999....+0.333...=1.333... ^ why is this not a 2? because there is another 3 after it, though we did not write it and another after that, and another after that, ad infinitum
at no point does the decimal terminate, hence EVERY digit carries a 1 from the digit after it
But still you just can't neglect a 2
yes you can because there is absolutely no 2 how many decimal points down would you expect to find it?
Which means it is not equal to 1+ 0.3333....
it would have to be in the same decimal place as the remaining 1 you might expect by subtracting 1-0.999... you might say, "well where is the remaining 1?" well, it is literally *infinitely* far down the decimal, which means it ain't there
how do you figure that 0.999...+0.333... is not the same as 1+0.333... ?
When we add, we add the numbers from Right hand side
you want to start at the far right side I presume? but what is the farthest right digit, and what decimal place is it in?
you are trying to use math skill you learned in grade school to deal with simple, finitely long numbers on an infinitely long decimal representation
How do you add two numbers?
either by grasping the concept that there is no last digit and recognizing that the 1 will ALWAYS carry over from the previous digit, or by representing it as a fraction which is perfectly valid an makes life way easier
I suggest you stop trying to hurt your brain dealing with infinitely long decimals and and work on deepening your understanding as to why these infinite decimals are exactly equivalent to fractions
0.9999.... = 1, I want to disprove this
you can't disprove something that is true
Can you prove it?
again, what is 1-0.999... ?
prove it? yes would you like me to?
0.999...*10 = 0.999...+0.999...+0.999...(10 times) How do you add this?
again, the way I showed you adding 0.333... every time we get the following succession 0.333... 0.666... 0.999... 1.333... 1.666... 1.999... 2.333... 2.666... 2.999... 3.333...
But what about the last digit. Though we don't get till last digit we observe that the last digit gets a different value
oh I did it with 0.333... but the idea is the same
where is this mythical last digit? what decimal place is it in? the tens, the thousands, the ten-millions?
what is the last digit of pi?
It is an irrational number
but the point is the same, where is the last digit of 0.999... ? what decimal place is it in?
How would you add 8467539+10384?
normally, from right to left but this is not a comparable problem because each number has a last digit, which is where you start adding from can you do that if there is *no last digit*???
how can you start at the far right when there is no far right? you can't
But you say that 0.3333..... is a rational number
yes and I can prove it what is your point?
When you can't start at the far right, how do you take 0.999...*10 = 9.999...
by having a deeper understanding of what it means to add two infinitely long decimal representations, or by understanding that this is the same as 1*10 your grade school adding methods are powerless here, you must accept that
Why it is same as 1*10?
What is the standard way of adding two numbers?
Not the grade school adding methods
brb @experimentX feel free to take over if you like I will be back in a sec
i lost track of it ... looks like conversation reached almost infinity ... where do you have problem?
What is the standard way of adding two numbers?
think of addition as addition of distance.
How would you add 8912394+1398124?
draw a real line .. |dw:1349367528137:dw|
How would you add 0.999.... and 0.333...?
I would agree that is a good standard way to think of addition^ but you want to do 0.333...+0.999... in which case you have to observe a pattern, which you seem to be reluctant to accept
you can always locate these points on the real line. and if you continue 0.999.... .... to up infinity this is 1 and for 0.33333.... you might think that you can never locate this point on real line. you can actually locate it. and there is ONE and only ONE point on the line i drew. add these distances.
This problem started in the proof of 0.999... = 1
yeah ... it seems that this is not equal to 1 ... but if you continue this up to infinity ... still this is non intuitive. here a short reason to believe. 0.9 ~ 1.1 <--- let's find a pair of points 0.99 ~ 1.01 0.999 ~ 1.001 <--- 0.9999 ~ 1.0001 <--- in similar fashion you put infinite zeros between 1 and the last 1 what would you get 1.00000000000.. infinite zeros.....................1
to do 0.999...*10 you can either 1) accept that multiplication by 10 moves the decimal place over by one space (it is perfectly okay to utilize that, if we did not utilize powerful concepts as givens then math would be a monstrosity to do!) or 2) add each corresponding digit ten times and recognize that there is not last digit, hence every digit will carry a one from the digit to its right, of which there are infinitely many I strongly suggest option 1
Multiplication by 10 is nothing but adding it 10 times
if you say this is 1 1.00000000000.. infinite zeros.....................1 then this must be equal to 1 too 0.99999999999999.... infinite nines
or as I put it, 1-0.999...=? if we say 1-0.999...=0.000...1 in what decimal place would the 1 be? infinitely far out, i.e. there isn't one
I think you are misunderstanding infinity with undefined, infinity is greater than you imagine.
1.00000000000.. infinite zeros.....................1 this means 1.00000000000.. here are more zeros than you imagine ....................1
@abdul_shabeer if you want to utilize option 2 that is fine, but be careful about how the digits carry over from infinitely far to the right
You mean something which is at infinity is not there.
don't play fast and loose with ideas like "infinity means it's not there" this is a subtle and tricky issue, and cannot be captured in a phrase like that
yep. kinda something like that http://math.stackexchange.com/questions/11/does-99999-1 also that video by ViHart is very nice ... but few the argument she used had been downvoted quite badly on MSE. Let the experts know.
Do you remember the question on whether a point is dimensionless? If we compare the size of earth with universe, how big do you think the earth would be?
the same size it would be if the universe was only as big as the solar system comparison does not make a thing bigger or smaller
if you are out of Milky way and you look at earth, would you be able to find it?
If it isn't 1 * 10^n, what is it?
with the right equipment, theoretically yes
With no equipments
with the human eye (which is a piece of equipment itself in many respects)? no of course not, why does that matter?
If we see the space out of earth, would it appear like a 2D image?
the ancients thought it was, so I guess you could argue yes, but that is just a matter of the limitations of human perception
there is no obvious perspective point in the night sky, so our brains don't do so well gauging distance why are we talking about this all of a sudden?
Okay Thank You Max and ExperimentX
no probs at all ...