TITLE: Big-O Interpretation
QUESTION [13 upvotes]: I have trouble understanding what the "Big O" notation, or asymptotic notation means. For instance, if you have $\sin(x)=x+O(x^3)$, what does this mean? Can anyone describe it in a simple way? I tried looking it up but it the explanations didn't help much. 
Thanks.

REPLY [3 votes]: First, the Big O (and its cousins: little o, Big and little theta, etc) notation is a mathematical device used for asymptotic quantification of the growth rate of a function near a point, often 0 or infinity.  If you don't specify which point, infinity is assumed.  That is why in your $sin x$ example, you should have added that x approaches 0.  
As for why $sin(x)=x+O(x^3)$ as x approaches 0, it comes from the Taylor expansion of the $sin$, which is of course an infinite series.  The O notation simply gives you the "bottom line". Specifically, it says that $sin x$ near 0 is equal to the argument $x$ itself plus a quantity whose vanishing (not growth) rate (not value, realize) as $x$ approaches 0 is no slower than the vanishing rate of $x^3$.
Finally, it is pathetic that the manipulation of the Big O notation is plagued with huge abuse.  To imagine the size of the abuse, recall that the O denotes a set, not a quantity nor even a function. As such it should not be equaled to a function, as in $7n^2+5=O(n^3)$. Nor should it be added to a variable as in your example $sin(x)=x+O(x^3)$, or even to itself as in what is sometimes called "Big O algebra".  But this abuse has accumulated over the years anyway and it is hard to change it now.