strange? (1 Viewer)

[onebytwo]

i saw this paradox on some website:
let x=0.99999........
so, 10x = 9.99999.....
then, 9x = 9
so, x=1
therfore 1 = 0.99999......
how can one number equal another, if the above working is not flawed, which i dont think is.
can anyone explain

 

[A l]

There has been a post about this before.
In fact it is widely accepted to be equal. There is no number between 0.9999... and 1 so it can assumed to be equal.
Interpret the recurring decimal as a sum to infinity or limiting sum.
i.e. S = 0.9 + 0.09 +0.009 + 0.00009 +..............
a = 0.9
r = 0.1
.: S = a/(1 - r)
S = 0.9/0.9
S = 1

As the number of terms in the series approaches infinity (i.e. as we go through each fraction the recurring decimal), the value of S or the recurring decimal gets closer and closer to 1. Therefore, it is equal to 1.

 

[SoulSearcher]

Not so strange, it just takes a bit of thinking.

 

[onebytwo]

ok, but what if you consider the number before 0.9999999.........
oh yeah, there is no number.
but i still cant comes to terms with having one number equal to another
thanks anyway

 

[GaDaMIt]

0.111... = 1/9

0.555... = 5/9

0.999... = 9/9 = 1

 

[darkliight]

ok, but what if you consider the number before 0.9999999.........
oh yeah, there is no number.
but i still cant comes to terms with having one number equal to another
thanks anyway

But it's not a different number equal to another Just another way of writing the same number.

1 can also be written as 2/2, (-2)/(-2), cos(0), the multiplicative identity, 2*int (0 to 1) x dx etc

 

[Templar]

They're both the decimal expansion of unity, that's all.

 

[GaDaMIt]

i saw this paradox on some website:
let x=0.99999........
so, 10x = 9.99999.....
then, 9x = 9
so, x=1
therfore 1 = 0.99999......
how can one number equal another, if the above working is not flawed, which i dont think is.
can anyone explain

Also.. after more thought..

1 doesn't equal 0.9999...

think about it...

the working has a minor flaw..

if x = 0.999...

then 10x = 9.999...

well.. at the very far right hand corner of the 0.999... when its multiplied by 10 the digits need to shift to the left .. the rounding obtained here is not considered because it is so slight.. i think.. not really sure how to word this better.. sorry

not sure.. just a thought

 

[Raginsheep]

If I understand your question properly....

Theres no number between 1 and 0.999....
Therefore 1=0.999....

Similarly,
There is no number between 10 and 9.999.....
Therefore 10=9.999.....

 

[darkliight]

Also.. after more thought..

1 doesn't equal 0.9999...

think about it...

the working has a minor flaw..

if x = 0.999...

then 10x = 9.999...

well.. at the very far right hand corner of the 0.999... when its multiplied by 10 the digits need to shift to the left .. the rounding obtained here is not considered because it is so slight.. i think.. not really sure how to word this better.. sorry

not sure.. just a thought

But there is no far right hand corner

Maybe this will make sense, 1/3 = 0.3333333...... by primary school maths.

Now, multiplying both sides by 10 gives

10/3 = 3.33333333...........

There is no end to those 3's either, more importantly, no right hand corner for there to be a zero or rounding error But, for the sake of argument, let's say that there was a right hand end and a rounding error, then ...

10/3 must be 3.33333....... minus some rounding error right? That is, we've added an extra 3 on to the 'end'. I'll call that rounding error RE.

So we have,

10/3 - 1/3 = 3.33333333.......... - RE - 0.333333333.........
9/3 = 3.000000....... - RE
3 = 3.0000000.......... - RE

By kindergarten maths, that rounding error must be zero. So there really wasn't a rounding error after all

I choose a third because people are used to that number by now, but the same argument applies to 0.99999.... ofcourse.

Hopefully that makes sense and helps.

 

[hyparzero]

Think about it....

0.99999....... = 0.88888...... + 0.11111...

But

0.88888...... = 8/9 and

0.11111...... = 1/9

Therefore 0.99999..... = 8/9 + 1/9

Hence 0.9999..... = 1

 

[airie]

lol I remember my maths teacher went through the whole 1/3 thing with the whole class trying to get ppl's heads around it Weirdest things happen at infinity...O.O

 

[Templar]

Once you go into infinite sets with different cardinality it gets pretty messy.