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Forums - General - 0.9999.... = 1.0

 

Are you convinced?

Yes 34 58.62%
 
No 20 34.48%
 
not sure 1 1.72%
 
Total:55
dsgrue3 on 02 May 2013
Jay520 said:


I've shown it was correct mathematically as you agreed was valid. Your squaring example is not applicable counter-proof. All you've proven is that sometimes things don't work out, but you've done nothing to show that what I did wouldn't work out.

I'm saying it works because it is true, not that it's true because it works. Important distinction - and it isn't a proof.



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ebw on 02 May 2013
Jaydi said:
ebw said:

[One exception to this claim: in my own area of math, we often (ab)use the = sign in a way that is not symmetric when describing error estimates.  We would write "x = O(n)" to mean that x is bounded by a multiple of n, when formally it would be more appropriate to use element inclusion.  Consequently, even if one has "x = O(n)" and "y = O(n)" one cannot deduce that "x=y".]

 

That's why the notation x=O(n) is not so efficient and shouldn't be used. You should say that x= n h(n) with h(n) tending to a constant when n is getting large.

Technically h(n) would not be tending to a constant — in typical usage it would be bounded by a constant but not convergent.

To be fair, the Landau notation was invented precisely for its efficiency.  If I'm adding together 50 error terms of size O(n) it is a major inconvenience to have to give each of them a separate name, especially as this draws attention away from the main term.  Despite its flaws I definitely think it should be used, it's just not for beginners.



Zkuq on 02 May 2013
dsgrue3 said:
Jay520 said:


We're not solving that x = 0.999. We are showing 1 = 0.999..

...and that is not a valid way of showing it.

What if we forget x and simply write:

    9.999... = 10*0.999... (I believe we can agree on this.)

    9 = 9*0.999... (subtract 0.999...)

    1 = 0.999... (divide by 9)

All right here, right? Well then, let's make writing the thing easier and say x = 0.999... The exact same process written in terms of x:

    9.999... = 10*0.999... = 10x

    9 = 9x (subtract 0.999...)

    1 = x (divide by 9)

Now do you claim that when I decide I want to write the equations in a more convenient form using symbols, the whole thing loses validity?



Jaydi on 03 May 2013
ebw said:
Jaydi said:
ebw said:

[One exception to this claim: in my own area of math, we often (ab)use the = sign in a way that is not symmetric when describing error estimates.  We would write "x = O(n)" to mean that x is bounded by a multiple of n, when formally it would be more appropriate to use element inclusion.  Consequently, even if one has "x = O(n)" and "y = O(n)" one cannot deduce that "x=y".]

 

That's why the notation x=O(n) is not so efficient and shouldn't be used. You should say that x= n h(n) with h(n) tending to a constant when n is getting large.

Technically h(n) would not be tending to a constant — in typical usage it would be bounded by a constant but not convergent.

To be fair, the Landau notation was invented precisely for its efficiency.  If I'm adding together 50 error terms of size O(n) it is a major inconvenience to have to give each of them a separate name, especially as this draws attention away from the main term.  Despite its flaws I definitely think it should be used, it's just not for beginners.

Yes, you're totally right. I was writing an equivalence, not a O. 

When I said it lacks efficiency, I meant that students don't even really understand the meaning of a o, which is far more easier to use than O. So I try to avoid these notations as more as possible.