Could somebody please explain the following notation to me (if possible, please describe them both mathematically - so I can use them correctly - and in laymen's terms - so I understand them!):

O,o,~,> ,< ,= (the last three should be made up of curved, rather than straight lines).

I've just started a book on number theory, and I felt that it would be a good idea to have a good understanding of the symbols being used before I started trying to prove hypotheses involving them!
Thanks in advance for any help.

Philip,

I've used TeX to illustrate what these symbols mean because I can't get them to work in HTML!

$f(x)=O(g(x))\iff |f(x)|<Ag(x)$ $\forall x$

$f(x)=o(g(x))\iff f(x)/g(x)\to 0$, as $x$ tends to some limiting value

$f(x)~g(x)\iff f(x)/g(x)\to 1$, as $x$ tends to some limiting value

$f(x)\prec g(x)\iff f(x)=o(g(x))$

$f(x)\succ g(x)\iff f(x)/g(x)\to \infty$, as $x$ tends to some limiting value

$f(x)\approx g(x)\iff Ag(x)<f(x)<Bg(x)$ $\forall x$.
I hope this is useful for you.

Paul

Thanks for the help... however, I'm still confused! What, for instance, does
o(1)=O(1) mean?

It means that if we have f(x)=o(1) - ie f(x) approaches zero in a limit, then also f(x)=O(1).

f(x)=O(1) means we may construct a constant A such that f(x) < A for all x. So basically, f(x)=o(g(x)) implies f(x)=O(g(x)).

An obvious example is x=O(x² ) for x tending to infinity since we may set our A at 1.1 and clearly 1.1x² > x for any x greater or equal to 1.


Julian

Ah, thanks... that helps a lot. I was unsure of what they meant in conjunction with each other.
Thanks again to both of you for sorting this out for me.