0.9999.... = 1.0

[Jaydi]

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.