0.99~=1

[bananasiong]

How about an alternate proof?

    0.9999....
   -----------
9 ) 9.0000....
    8 1
    ---
      90
      81
      --
       90
       81
       --
        90
        81
        --
         9....

So what's the answer for this?
0.99...99 PLUS 0.00...09 numerator and 9 denomenator, so the answer is still 1.

c = 0.999~
10c = 9.999~
10c - c = 9.9999~ - 0.999~
9c = 9
c = 1

How about this:

c = 0.99...99
10c = 9.99...90
10c - c = 9.99...90 - 0.99...99
c = 0.99..99

 

[Roff]

So what's the answer for this?
0.99...99 PLUS 0.00...09 numerator and 9 denomenator, so the answer is still 1.


How about this:

c = 0.99...99
10c = 9.99...90
10c - c = 9.99...90 - 0.99...99
c = 0.99..99

I think you miss the point. The string of 9's is infinitely long. You can't multiply by 10 and have a 0 show up at the end. If you think this will happen, then you don't understand the concept of infinity (which, admittedly, is difficult to comprehend).

 

[bananasiong]

I think you miss the point. The string of 9's is infinitely long. You can't multiply by 10 and have a 0 show up at the end. If you think this will happen, then you don't understand the concept of infinity (which, admittedly, is difficult to comprehend).

You're right too. But sometimes this can be done by imaging it, but not writing it out.
For example, does anyone know about Heaviside's rule of partial fraction? There is a step which is something like this:

2(infinity)/infinity = 2

But this cannot be written on the paper, and my lecturer call this as 'illegal job'

 

[3iMaJ]

infinity/infinity is an indeterimant form, IE you can't say 2*inf/inf = 2. The rules dont work like that.

 

[bananasiong]

infinity/infinity is an indeterimant form, IE you can't say 2*inf/inf = 2. The rules dont work like that.

Yes, you can't say 2*inf/inf = 2, but this is done when the Heaviside's rule of partial fraction is applied. Something like this:

A = lim(x-->inf)  [2x/x  +  (x+B)/x]
A = 3 [b]edited[/b]

Straight away as shown above, but the infinity over infinity cannot be written.

 

[3iMaJ]

Thats completely different. You're thinking of l'Hospital's rule where you have an indeterimant form involving limits it goes as:

if lim g(x)/f(x) is either 0/0 or inf/inf then the following applies

lim g(x)/f(x) = lim g'(x)/f'(x) where g'(x) and f'(x) are the derivatives which is what you would apply in your problem for both sections.

Applying to your problem:

lim (x->inf) 2*x/x + (x+B)/x = inf/inf + inf/inf as above l'Hospital's rule applies so:
d/dx of above = 2 + 1/1 = 3

lim (x -> inf ) [2*x/x + (x+B)/x] = 3

 

[bananasiong]

Yes, you're right, that is 3. I didn't notice that. Why you call that as hospital's rule??

So, do you think 0.99...99 =1?

 

[3iMaJ]

So, do you think 0.99...99 =1?

Yep. I posted the proof above. I'll post it again here.

 

[HudzonHawk]

Wait, people are still arguing over the whole 0.99.. = 1 equation despite the fact that a 5 minute search on the Internet will show thousands of credible sources which prove that the equation is true?

No offense to anyone, but do you guys also disprove that the Earth is round?

 

[Roff]

Wait, people are still arguing over the whole 0.99.. = 1 equation despite the fact that a 5 minute search on the Internet will show thousands of credible sources which prove that the equation is true?

No offense to anyone, but do you guys also disprove that the Earth is round?

I've been trying to get the naysayers to rewrite the Wikipedia entry. No takers yet.

 

[bananasiong]

I've been trying to get the naysayers to rewrite the Wikipedia entry. No takers yet.

So you can proof that 0.99...999 is not 1?

 

[Roff]

So you can proof that 0.99...999 is not 1?

No, my statement was meant as sarcasm. What I meant was, if the naysayers believe so strongly that 0.999... is not equal to one, they should correct the Wikipedia entry. In English, we have a saying: "Put your money where your mouth is". You probably know that saying, or have a similar one in your culture.

 

[Nigel Goodwin]

The most amazing thing about this entire thread is that I've not deleted or locked it yet!

 

[Sceadwian]

Please Nigel! PLEASE!

 

[marplots]

1 - .999... is zero. The reason for this is that between any two different rational or irrational numbers, there is an infinite number of other numbers. The only way that there can be no difference between two numbers is if they are, in fact, the same number.

Hence, there is no number between 3 and 3, but an infinite number of them between 3.141414... and 3.1399999...

Since, for the expression .999... and 1 there is no number 'between', they are, in fact, the same number. Any difference you come up with is an artifact of your calculator or truncation.

 

[neoandrewson]

there is no number 1 in this world... it is almost 1, but not 1... x->1 (i don't know the correct english term for it, sorry).it's 0.999..9
same for 2, 3 and all other numbers.

its tends to... level of significance always matters in practical stuff... but in math, no idea where it leads to...

 

[sheng_james]

huh..
all i know always like the current gain of a common-base configuration of BJT transistor 0.9999...aproximately equal to 1

 

[mcs51mc]

Here's another one
Imagine an archer 16 meters away from a target.
When the arrow leaves the bow it has 16 meters to go.
When half the way is done, still 8 meters to go
After another half is done, still 4 meters to go
After another half is done, still 2 meters to go
After another half is done, still 1 meters to go
After another half is done, still 0.5 meters to go
After another half is done, still 0.25 meters to go
After another half is done, still 0.125 meters to go
After another half is done, still 0.0625 meters to go
After another half is done, still 0.03125 meters to go
After another half is done, still 0.015625 meters to go
After another half is done, still 0.0078125 meters to go

When will it reach the target?
Never ... ...

 

[Papabravo]

This is a variation of Zeno's paradox. It involved a runner in a race against a turtle with a modest head start. In the paradox the poor runner was doomed to lose the race.

 

[Robot builder 101]

Lol, feels like im awlays half way done with my projects, aka they never get finished.

In the human mind, we can percieve the number 1 with objects. For example, we see a orange as 1 orange. Therefore, we can have the whole number one, with only the number one as long as the human mind can percieve objects as only the number one.

Other words, we are usually use to seeing only 1 object, meaning we thinnk the number one means 1.00000000000, not when it is actually 0.9999999999999999999999999999. We only see 1 object, so we get use to 1. Imagine seeing 0.9999999999 of a peice of paper... It would appear as 1, so we get familarized with one, when we dont actually see that it could be 0.9999999999.