"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.
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?
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.
i is defined to be the square root of -1 isn't it? well, then -1=1 (mod 2) and the polynomial x^2-1 = (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.
I like the title of a monograph by nineteenth-century 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.
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.
Matt grime, It almost seems that you're not listening to me. But first, I will assume that I am very confused about mod2 (being not a math person). First of all, I thought that mod2 could only deal with two distinct discrete objects, for instance 0 and 1. Then, the combination rule is defined such that 1 + 1 = 0, so that you always stay in the group. I didn't know that multiplication, exponentiation, imaginary, or irational numbers were allowed in this scheme. I read through your bit about what i is in mod2 over and over again, and I still don't get it. If i = 1, then why write i? Does it only equal 1 when it is considered by itself, but it equals √-1 when it is in a product or exponent? It seems like there is 0 and there is 1, and then there is an infinitude of other "values" (not mod2) that you either have to identify with 0 or with 1. Why not say that i = 0? I never asked about n. I think you may have seen the html "pi" in my post and interpretted it as an "n." Anyway, I'm curious what you would call pi in mod2. And now I realize that I forgot to ask about e. Both of these are irrational. That seems like it would be a problem for any discrete number system. Tell me where I have gone way off. You said something about a ring. What is that? edit: Well, I don't know why I screwed this whole thing up so bad. For some reason, I thought we were talking about Euler's identity. But you were just talking about -x = x this whole time (I just read through the thread again). No wonder you think I'm whacko. I am TERRIBLY sorry about that. I would still like to hear what you have to say about the questions/concerns that I posed above, though.
for turin indeed i was only talking about x=-x, nothing to do with e or pi. Firstly, F_2 is the set {0,1} as you know, with the addition and mult. as you stated. Now, this is all I was using. Now, i is defined as the square root of -1 in C, here there is a analogous element in the field because -1=1 has a square root. Notice however that there is a polynomial x^2+x+1 that has no root in F_2. Just like in the real/complex case, we simply define A to be a root of this polynomial (the other root is 1+A), then there is an extension F_2[A] which we call F_4, it is the field with 4 elements, it is still not posible to find roots of every polynomial in F_4, so we can extend again and again. Each extension has 2^r elements for some r. The 'limit' of this construction, we'll call F, and it is the algebraic closure of F_2. It is infinite. In all these fields 1=-1. In the same way as there is a surjection from Z to F_2, there is a way of relating algebraic integers to "an" algebraic closure of F_2. Let (2) be the ideal generated by 2 inside the ring of algebraic integers (that is the set of solutions of all monic polynomials with integer coeffs), the ideal is just the set of all things 2x, where x is an algebraic integer. Just like mod 2 arithmetc, we can declare two elments in the alg, integers to be the same if their difference lies in (2) (actually we require a maximal ideal containing (2) but that's a technicality). Thus it is possible to define the image of any algebraic integer in a field of characteristic 2. However, neither e nor pi are alg. integers so i can't define their images. To explain the high faluting maximal thing - 2 must get sent to zero, and so sqrt(2) must also get sent to zero if this were to make any sense, but sqrt(2) is not in the ideal (2). The maximal thing there corrects that problem and makes sure that the quotient is a field. the vast majority of algebraic integers are irrational, in fact the only rational ones are the integers. it's a useful exercise to prove that interestingly, the golden ratio, a root of x^2+x+1 must get sent to A, or 1+A in this scheme. note that A has no value as a real number! just as i doesn't, it's just a symbol we manipulate according to the rule A^2=A+1 Matt