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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

merovingian
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    Complex numbers are usually defined from the real numbers, like $z=a+bi,$ where $a,b\in\mathbb R,$ so it's pretty hard to circumvent the real numbers even existing. We can do math in the setting of the complex numbers pretty easily, but a lot of mathematics is actually about making our lives easier by simplifying proofs, much as it may seem otherwise :D. (and sometimes we want statements like "all nonconstant linear functions have real roots") – Holden Rohrer Nov 11 '22 at 22:34
  • Yes but for this question, we could define it as z = a + bi where both a and b are complex numbers with 0 complex coefficients. And yes, I also understand we really need real numbers for making things easier, but whenever I read any advanced mathematics, you know, they are as fancy, as general as possible. So I thought maybe one could argue that these real numbers may be redundant. @HoldenRohrer – merovingian Nov 11 '22 at 22:38
  • You're absolutely right about higher math trying to generalize as much as it can, but you might be interested in some things that happen on the real line but not on the entire complex plane. For example, a "bump" function (0 except for a small region in the domain) can have an infinite number of continuous 1st, 2nd, etc derivatives ("smooth") on the real line, but such a function can't exist on the complex plane (if it is smooth and nonconstant, its range actually would have to include every complex number if I'm remembering correctly) – Holden Rohrer Nov 11 '22 at 22:51
  • Also if you want a field even bigger than the complex numbers, we can look at the quaternions and the octonions and the sectonions BUT these lose many properties in general compared to the real line and complex plane. – Holden Rohrer Nov 11 '22 at 22:55
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    Modern math is now built on set theory, and in set theory we can demand the "least redundant" set called the Grothendieck Universe that contains every single set/object we can define in ZFC, but this set is HUGE and very hard to work with, which is why most of our work is in settings with incomparably more restrictions on their definitions. – Holden Rohrer Nov 11 '22 at 22:57
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    When you wrote in your comment "where both $a$ and $b$ are complex numbers with $0$ complex coefficients" you implicitly conceded that you need to be able to identify the reals as a subset of the complex numbers, but you can't do that just using complex number algebra. The first-order theory of the real field is richer than the first-order theory of the field of complex numbers. – Rob Arthan Nov 11 '22 at 23:26

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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.

Dan
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  • Very good explanation. But as far as I understand, the main argument is "it is useful to define real numbers" and that is correct. But my question is mostly "Is it theoretically possible to not have the real numbers and end up with the same math?" or "defining real numbers from complex numbers, not vice versa". I also understand the importance and necessity of real numbers. But if you want to be as general as possible, wouldn't you start with complex numbers? Then every theorem regarding real numbers would be a special case of the general theorem. @Dan – merovingian Nov 12 '22 at 12:54
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    For example, one can say " Complex numbers have natural ordering iff their imaginary component is 0 ". And I would define 0 as a complex number. I feel like this definition is as valid as your definition. That is actually what I mean by redundancy. – merovingian Nov 12 '22 at 12:58
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    @merovingian when we talk about generalizing something in math, we usually mean making our assumptions apply to way more situations, and writing a theorem in the complexes instead of the reals can (sometimes) be a way to make it apply to more situations, but a better type of generalization is "all sets that have a multiplication and addition operator," and this is called the theory of fields – Holden Rohrer Nov 28 '22 at 21:44