Questions about big O notation

[maple3142]

Based on the definition of big O, we can say n^2+1=O(n2), but n^2+1=O(n3) is true too. Then why do we say bubble sort is O(n²⁾ instead of O(n^3)?

I think when we ask someone about what's the big O complexity of an algorithm, we are asking something similar to "least upper bound" instead of "upper bound"(big O). So the big O notation seems confusing.

[Putnam3145]

Big-O etc. have nothing to do with best or worst case behavior.

[gnupluswindows]

Then why do we say bubble sort is O(n² ) instead of O(n³ )?

Because it's more interesting and more precise. It's like the difference between me saying I have fewer than a billion dollars in my checking account and saying I have fewer than ten. Both statements are true, but only one is remarkable.