Euclidean plane vs the complex plane

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February 16th 2010, 08:45 AM
guildmage

Euclidean plane vs the complex plane

I know that a complex number may be viewed as an ordered pair of real numbers. I'm just wondering what is the main reason for studying complex analysis when complex numbers can be viewed as an ordered pair of real numbers? I mean, what can we do in the complex plane that we can't do in the 2 dimensional real plane?
February 16th 2010, 10:28 AM
TheEmptySet

Quote:

Originally Posted by guildmage

I know that a complex number may be viewed as an ordered pair of real numbers. I'm just wondering what is the main reason for studying complex analysis when complex numbers can be viewed as an ordered pair of real numbers? I mean, what can we do in the complex plane that we can't do in the 2 dimensional real plane?

In my opinion the main difference is the algebraic structure. You can't multiply two vector in $\mathbb{R}^2$ but you can in $\mathbb{C}$ . This gives the complex plane more structure. There are many theorems that are true in the complex plane that do not hoold in $\mathbb{R}^2$
February 16th 2010, 12:55 PM
Drexel28

Quote:

Originally Posted by guildmage

I know that a complex number may be viewed as an ordered pair of real numbers. I'm just wondering what is the main reason for studying complex analysis when complex numbers can be viewed as an ordered pair of real numbers? I mean, what can we do in the complex plane that we can't do in the 2 dimensional real plane?

One can show that $\mathbb{R}^2\approx\mathbb{C}$ ( $\approx$ means homeomorphic with the usual topologies on both) so they are equivalent topologically but that says nothing, as TheEmptySet they are different in their algebraic structure.