what is zero
a simple question
what is zero "0" does it count as a number? any answers 
"0" is the cardinality of the empty set (that's one definition).
"0" is the additive identity (that's another). Yes, 0 is a number just like 1, π, √(2) and i. 
0 is the mark i will get on my linear exam if i don't study ;)

Zero is the only solution of the equation [tex] x = x[/tex].
Also, zero is both the limit of the largest negative numbers and the limit of the smallest positive numbers... Naturally, even more important (due to Euler): [tex] 0 \; = \; e^{i\pi}1[/tex] Had enough? [:D] 
0 was transferred from India to Arabians and to world
I dont know much of its importance given by Our Ancient Seers But mathematically it is an essence in every field Metaphysically it represents DEATH,GLOOMY,INAUSPICIOUS 
Zero is the identity element in addition (of vectors, integers, etc.)
edited to add: looks like HallsofIvy has beaten me to the punch on that defintion. 
Only by 19 days! [:D]

zero makes mathematics full of identities and definitions....

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[tex] 0 \; = \; e^{i\pi}+1[/tex] 
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Not in mod 2 arithmetic. 
What is i or π in mod2?

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i is defined to be the square root of 1 isn't it? well, then 1=1 (mod 2) and the polynomial x^21 = (x+1)(x+1) mod 2 so the answer is i=1 and n is either 0 or 1 depending on n odd or even resp. 
Some people consider 0 to be an asymtote. Not saying i do. but some do.

I like the title of a monograph by nineteenthcentury German mathematician Richard Dedekind.
"Was sind und was sollen die Zahlen?" This can be rendered roughly in English by the following. What are numbers, and what should they be?" I think that is the fundamental question behind this topic. 
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I'm still trying to understand that bit about i. 
do you know what the integers are mod 2? where did pi come from?
if that's all too much, then you probably don't want to know about maximal ideals in the ring of integers by n I assumed you mean 1+1+1...+1, n times. The key thing to understand is that when i introduced mod 2 arithmetic, i was pointing out that the question, and many of the answers were assuming that it was posed in the the real numbers. that is notthe only place where zero occurs. 
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