TITLE: Redundancy of numbers
QUESTION [3 upvotes]: I searched for this on the internet and could not find any answers.
Assuming we are trying to be as less redundant as possible in mathematics (even if it's not the case, assume so for the moment), aren't real numbers redundant since we have complex numbers? Because every real number is a complex number with 0 imaginary component. If there is a larger set (I don't know the correct terminology here) that includes complex numbers, fine then I'll say complex numbers are redundant.
In this manner, can't we have the same mathematics without even defining real numbers? Of course, it would be harder. But I'm just asking

REPLY [3 votes]: It is technically correct to say that a real number is just a complex number whose imaginary part is 0.  (Similarly, it is technically correct to say that an integer is just a fraction whose denominator is 1.)
But there are operations that you can perform with real numbers that you can't do with general complex numbers.  In particular, $\mathbb{R}$ has a natural total ordering that $\mathbb{C}$ does not, allowing real numbers to use the relational operators $<$, $\le$, $\ge$, and $>$, as well as the floor $\lfloor x \rfloor$ and ceiling $\lceil x \rceil$ operators.
Real numbers also have a much simpler concept of “sign” or “direction” on a number line/plane.  If you're given that $x \in \mathbb{R}$ and $|x| = 1$, then $x = \pm 1$ — only two possibilities.  But if you're given that $z \in \mathbb{C}$ and $|z| = 1$, then there are an infinite number of solutions, in the set $\{\cos\theta + i\sin\theta : \theta \in [0, 2\pi) \}$.
In my opinion, this kind of thing makes the reals “special” enough to deserve their own name and definition, rather than just $\{z \in \mathbb{C} : \Im(z) = 0\}$.  I'm not sure how you'd even define the concept of “real and imaginary components” of a complex number without either having a circular definition or defining “real number” first.
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