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I am learning complex analysis now, but I still can't tell the difference between $\mathbb{C}$ and $\mathbb{R}^2$. If we define a map:$F:\mathbb{C}\rightarrow\mathbb{R^2}$, $$x+iy\rightarrow (x,y),x,y\in R.$$ Further, we define a product in $\mathbb{R}^2$:$(a,b)\cdot(c,d)=(ac-bd,ad+bc)$.So why we still need complex number？

Will Jagy
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    Indeed, $\mathbb{R}^2$ with that product is a way to define $\mathbb{C}$. That said, I don't understand your question. I mean, the real numbers can be modelled as equvialence classes of Cauchy sequences of rational numbers. So why do we need real numbers, then? – Harald Hanche-Olsen Aug 28 '13 at 19:23
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    Maybe look here the question was already asked – Dominic Michaelis Aug 28 '13 at 19:27
  • (I'm not posting an additional answer because I think this is a duplicate.) When you write $\mathbb{R}^2$ you could also mean a product of rings - so $(a,b) \cdot (c,d) = (ac,bd)$ - and thus using $\mathbb{R}^2$ for complex numbers would generate some confusion. You'd have to have two different signs for product. Using $\mathbb{C}$ and defining $i$ the way we do is just a convention, but it's a good convention. – Caleb Stanford Aug 28 '13 at 19:36
  • http://en.wikipedia.org/wiki/Complex_number#Matrix_representation_of_complex_numbers I prefer to put the minus sign in the lower left, myself. But that does not really matter. – Will Jagy Aug 28 '13 at 19:38
  • Well, you can consider $\mathbb{C}$ to b equivalent to $\mathbb{R}^2$ with the relevant product operation. However, it is a bit like asking why don't we just deal with $\mathbb{Q}$ and a limiting operation, so we don't need $\mathbb{R}$. – copper.hat Aug 28 '13 at 19:38

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It would not suffice to define only the product. You would have to define lots of other functions, such as what it means to take a logarithm of the (complex) number, how to define $e^{(a,b)}$, etc. The structure of $\mathbb{C}$ implies much more than the product definition you provide.

exk
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  • But I thought that those other functions are defined from the definition of the product. For example, exponential can be charaterized from its power series, and using the new product in this power series will give the extended definition of exponential function on $\mathbb C$. – Xoff Aug 28 '13 at 19:30
  • Not all functions can be defined thus. For instance, any function that is not analytic cannot. – exk Aug 28 '13 at 19:32