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I decided to learn analysis over the summer for fun, but I'm really confused by the field properties. Why is the 2 element set of 0 and 1 a field? Addition wouldn't be satisfied, because 1 + 1 = 2, which isn't in the set.

Also, if there are a set of properties that completely determine the real number system, why does that imply that there is only one? My book says something about a one to one correspondence between the reals and some other real system that preserves the functions of + and *, but I don't understand this at all.

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    @Zack Please ask separate question on separate pages. The above should have been two questions.2011-07-20

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The field $\mathbb{F}_2$ is indeed a field. The important thing is to remember that addition is done modulo $2$. Thus,

$ 1 + 1 = 2 \equiv 0 \pmod{2}, $

and $0$ is indeed in $\mathbb{F}_2$.

Furthermore, the real numbers are the unique complete totally ordered field, and they can be constructed by say, completing the rational numbers with the metric $d(x,y) = |x - y|$, where $|\cdot|$ is absolute value. If you share with us what is written in the book you are using, we may be able to help a bit more.

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    One important property o$f$ the reals is that every number is either zero, a non-zero square, or the negative of a non-zero square. The non-zero square numbers are therefore precisely the positive numbers. The set of positive numbers in an ordered field determines the order (to compare two elements, take the difference and see whether it is positive, zero or negative).2011-07-20