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I have some questions about the construction of the complex numbers in this Wikipedia article, especially of the exponents of complex numbers.

$1$. Is it enough to define it as $a+bi$, where a,b are real numbers and $i^2=-1$. I mean, is this the only definition we need to do? In my book, if z is a complex number they define $e^z=e^{a+bi}=e^a(\cos b+i\sin b)$, so it seems they make two definitions when defining the complex numbers.

However, at Wikipedia, they only state the first definition, and they prove eulers formula: $e^{ib}=\cos(b)+i\sin(b)$. But here they seem to use the exponent rules with complex numbers.

$2$. Is it correct that there are two ways of doing this? One is by defining $e^z$ and then showing that the exponent rules work. Or the opposite assuming that the exponent rule works and then proving eulers formula? This part is confusing.

$3$. In the formal construction they use ordered pairs: Is it possible to deduce the exponent rules from this? I mean, here they do not seem to assume either Eulers rule, or that expoent rules work, or have they done so implicitly?

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  • $\begingroup$ You can type \sin and \cos to have the functions typeset properly. $\endgroup$ – user147263 Aug 9 '14 at 2:19
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There are many ways to define the complex numbers. Each such definition should consist of the following:

  1. A "list" of all complex numbers. Two examples are the formal expressions $x+yi$ for real $x, y$ (or, which is the same, pairs $(x, y) $), and polar representation, i.e. $(r,\theta) $ for real $r> 0$ and $\theta$ plus the number $0$. Each complex number should have a unique representation. Alternatively, we can provide a rule for deciding when two complex numbers are equal, but that complicates the further steps of the construction.
  2. An embedding of the real numbers into the complex numbers. In other words, for each real number there should correspond some unique complex number.
  3. A definition of the field operations of addition and multiplication. These are easier in the non-polar representation.
  4. An identification of one of the roots of $-1$ as $i$ (the field operations don't distinguish $i$ from $-i$).

Given these, one should be able to derive the usual representations of the complex numbers, and from these define the function $\exp z$; this is a definition. One way of defining this function is through the formula you mention, and another is using an infinite series (this requires more work since you have to define limits).

More abstract definitions are possible. For example, we can define the complex numbers as the algebraic closure of the reals (the smallest field containing the reals in which every polynomial has a root).

Using the uniqueness of the algebraic closure, one can show that all definitions of the complex numbers are the same, that is they result in the same notion of complex numbers. In fact, suppose we have a field extending the complex numbers, possessing a root $i$ of $-1$, such that every number can be written in the form $x+yi$; then this field is isomorphic to the complex numbers.

Summarizing, there are many ways of defining the complex numbers, and they all result in the same notion of complex numbers. The exponential function is a definition and doesn't form a part of the construction. Other derived concepts are the complex conjugate, the norm, and convergence.

The difference between essential and derived notions is more philosophical than practical. You might as well say that the derived notions form part of the construction. This is just another point of view, which perhaps makes sense from the perspective of formal logic.

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To define the complex numbers as a field, one only needs the definition of them as $a+bi$ where $i^2 = -1$. The definition of a complex exponential is not necessary to construct the complex numbers.

As far as the definition of the complex exponential, this is usually done by extending the definition of the exponential function as the power series $$ e^z = \sum_{n = 0}^{\infty}\frac{z^n}{n!} $$ into the complex numbers, possible exactly because this power series has an infinite radius of convergence. By examining the power series definitions of $cos$ and $sin$, Euler's formula can be proven, and from this definition of $e^z$ follow the exponent rules for complex numbers.

In other words, you do not need to assume Euler's formula--it follows naturally from the extension of power series into the complex plane.

As far as the formal definition, it is equivalent to the $i^2 = -1$ definition (they give rise to isomorphic fields), so, though it may be notationally difficult, you would be able to prove Euler's formula, once you come up with a way to define what a power series on $(x,y)$ means, exactly. (Alternatively, you could just define $e^{(a,b)}$ with Euler's formula.) Wikipedia doesn't seem to assume either because neither is terribly important to the formal definition; the formal definition exists for purposes related to the complex numbers as an algebraic structure, not an analytical one.

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