100000000000................ continues infinitely. Is it a number?

- anonymous

100000000000................ continues infinitely. Is it a number?

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- anonymous

yes because the number line never stops

- anonymous

|dw:1349359330358:dw|
Is infinity a number?

- experimentX

not a natural number and not real number either.

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## More answers

- anonymous

infinity is something which cant be defined...

- experimentX

no .. irrational number is a real number.

- anonymous

Is it a number?

- anonymous

IS infinity a number?

- experimentX

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.

- experimentX

*in

- anonymous

If 100000000000000... is not a natural number then the set of natural number is finite. right?

- anonymous

@experimentX how do u say that it is not a real number??

- experimentX

no ... the set of natural number is infinite but does not contain infinity.

- TuringTest

Infinity is a concept, not a number (I know I'm basically just restating, but it is worth emphasizing)

- experimentX

you always write |dw:1349359774760:dw|

- experimentX

you should have noted.

- anonymous

then why 1/3 is considered a number?

- TuringTest

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.

- TuringTest

hm... I phrased that poorly :/

- experimentX

natural number is defined as inductive set that begins with 1 and
|dw:1349359861034:dw|

- TuringTest

indeed
what would 10000.....+1 be?
could you write it? where would you put the 1 ?

- anonymous

Why 0.333333333333333333....... is a number?

- TuringTest

that is different from 0.3333....+1=1.3333...
not problem there

- experimentX

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.

- TuringTest

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 ;)

- TuringTest

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

- anonymous

@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 .......

- anonymous

We can't locate 0.333333.......

- TuringTest

yes we can

- anonymous

atleast u know that 0.33333 is present between 0.3 and 0.4 ....
in case of infinity u cant ......
can u ????

- TuringTest

|dw:1349360426152:dw|

- TuringTest

now you show me what two numbers 1000..... lies between :)

- anonymous

What is (0.333333......)* 10?

- TuringTest

3.333333...

- anonymous

0.333..... * 10 = 0.333...+0.3333....+0.3333....( 10 times) Can we add this?

- TuringTest

I think it is a lot easier to see it as
1/3+1/3+1/3+...(ten times)
=10/3

- anonymous

I want to add it without converting it into 1/3

- TuringTest

why?

- anonymous

As 0.3333.... = 1/3, in whichever way i add I must get the same result

- anonymous

100000000000......... lies between 99999999999...... and 11000000000000...... where all three of the numbers stretch out infinitly.

- TuringTest

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...

- TuringTest

and @Razzputin that is wrong, how can you show me that 9999... is not bigger than 1000... ?

- anonymous

How 0.999...+0.333...=1.222...?

- TuringTest

are there more digits in one number or the other?
no

- anonymous

what if you where to state that 999.... was 1 less than infinite?

- TuringTest

ad a 3 to each 9

- TuringTest

add*

- TuringTest

oh my bad, .999.+.333...=1.333....

- TuringTest

the 1's carry over

- anonymous

|dw:1349364346874:dw|

- anonymous

and the last digit would be 2

- TuringTest

yeah but there are an infinite number of 3's and 9's so every digit carries a one from the previous decimal place

- TuringTest

there is no last digit, that's why it's an infinitely repeating decimal

- anonymous

There is no last digit, I just took a small value

- TuringTest

which makes it not the same as 1/3+1 which is what you are trying to do

- TuringTest

\[0.333\neq0.333...=1/3\]

- TuringTest

this is why we don't add numbers with infinite decimals very often; It's ugly, confusing, and entirely unnecessary

- anonymous

what do you get when you add 0.9999...+0.3333... without taking 0.99999...=1 and 0.333... = 1/3

- TuringTest

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

- TuringTest

at no point does the decimal terminate, hence EVERY digit carries a 1 from the digit after it

- anonymous

But still you just can't neglect a 2

- TuringTest

yes you can because there is absolutely no 2
how many decimal points down would you expect to find it?

- anonymous

Which means it is not equal to 1+ 0.3333....

- TuringTest

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

- TuringTest

how do you figure that 0.999...+0.333... is not the same as 1+0.333... ?

- anonymous

When we add, we add the numbers from Right hand side

- TuringTest

so?

- TuringTest

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?

- TuringTest

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

- anonymous

How do you add two numbers?

- TuringTest

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

- TuringTest

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

- anonymous

0.9999.... = 1, I want to disprove this

- TuringTest

you can't disprove something that is true

- anonymous

Can you prove it?

- TuringTest

again, what is
1-0.999...
?

- TuringTest

prove it?
yes would you like me to?

- anonymous

yes

- TuringTest

\[x=0.\overline9\]\[10x=9.\overline9\]\[10x-x=9x=9\]\[x=1\]

- anonymous

0.999...*10 = 0.999...+0.999...+0.999...(10 times) How do you add this?

- TuringTest

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...

- anonymous

But what about the last digit. Though we don't get till last digit we observe that the last digit gets a different value

- TuringTest

oh I did it with 0.333... but the idea is the same

- TuringTest

where is this mythical last digit?
what decimal place is it in?
the tens, the thousands, the ten-millions?

- TuringTest

what is the last digit of pi?

- anonymous

It is an irrational number

- TuringTest

but the point is the same, where is the last digit of 0.999... ?
what decimal place is it in?

- anonymous

How would you add 8467539+10384?

- TuringTest

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*???

- TuringTest

how can you start at the far right when there is no far right?
you can't

- anonymous

But you say that 0.3333..... is a rational number

- TuringTest

yes and I can prove it
what is your point?

- anonymous

When you can't start at the far right, how do you take 0.999...*10 = 9.999...

- TuringTest

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

- anonymous

Why it is same as 1*10?

- TuringTest

because 0.999...=1

- anonymous

What is the standard way of adding two numbers?

- anonymous

Not the grade school adding methods

- TuringTest

brb
@experimentX feel free to take over if you like I will be back in a sec

- experimentX

i lost track of it ... looks like conversation reached almost infinity ... where do you have problem?

- anonymous

What is the standard way of adding two numbers?

- experimentX

think of addition as addition of distance.

- anonymous

How would you add 8912394+1398124?

- experimentX

draw a real line .. |dw:1349367528137:dw|

- experimentX

|dw:1349367566686:dw|

- anonymous

How would you add 0.999.... and 0.333...?

- TuringTest

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

- experimentX

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.

- anonymous

This problem started in the proof of 0.999... = 1

- experimentX

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

- TuringTest

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

- anonymous

Multiplication by 10 is nothing but adding it 10 times

- experimentX

if you say this is 1
1.00000000000.. infinite zeros.....................1
then this must be equal to 1 too
0.99999999999999.... infinite nines

- TuringTest

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

- experimentX

I think you are misunderstanding infinity with undefined, infinity is greater than you imagine.

- experimentX

1.00000000000.. infinite zeros.....................1
this means
1.00000000000.. here are more zeros than you imagine ....................1

- TuringTest

@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

- anonymous

You mean something which is at infinity is not there.

- TuringTest

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

- experimentX

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.

- anonymous

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?

- TuringTest

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

- anonymous

if you are out of Milky way and you look at earth, would you be able to find it?

- anonymous

If it isn't 1 * 10^n, what is it?

- TuringTest

with the right equipment, theoretically yes

- anonymous

With no equipments

- TuringTest

with the human eye (which is a piece of equipment itself in many respects)?
no of course not, why does that matter?

- anonymous

If we see the space out of earth, would it appear like a 2D image?

- TuringTest

the ancients thought it was, so I guess you could argue yes, but that is just a matter of the limitations of human perception

- TuringTest

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?

- anonymous

Okay Thank You Max and ExperimentX

- TuringTest

very welcome!

- experimentX

no probs at all ...

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