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Just out of curiosity - when we define a field, why bother mention multiplication, when its nothing more then repeating the same addition operation?

Here's the definition we were taught in calculus for physicists class:

A field F is a non-empty set on which two binary operations are defined: an operation which we call addition, and denote by +, and an operation which we call multiplication and denote by $\cdot$ (or by nothing, as in a b = ab). The operations on elements of a field satisfy nine defining properties, which we list now...

And then of course you have the axioms of field.

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    So... multiplication is nothing but repeated addition... So, in the real numbers, how do you add $\sqrt{3}$ to itself $\sqrt{2}$ times in order to compute $\sqrt{2}\times\sqrt{3}$? In the complex numbers, how do you add $i$ to itself $i$ times to get $-1$? In the field of rational functions with coefficients in $\mathbb{Q}$, how do we add $1+x$ to itself $x^3-2x+\frac{1}{2}$ times?2012-03-08
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    Sounds like **the** answer if it will be fleshed out. : )2012-03-08
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    A previous question: [If multiplication is not repeated addition](http://math.stackexchange.com/q/64488/742)...2012-03-08
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    In other words, multiplication is *not* "nothing more than repeating the same addition operation". Repeated addition is merely computational technique that works in a very specialized case. Except it's not a very good technique -- the reverse is far more useful: in a situation where you happen to be interested in repeated addition, multiplication is a useful computational technique.2012-03-08

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