How we know that there are UNIQUE form of a complex number

Question

As the title described, i need to proof this theorem :

let $Z$ be a complex number

there are only unique couple of reals $a$ and $b$ where $Z=a+ib$.

my question is how did you know if there are a complex number that can't be written like $Z=a+ib$

i searched over the web but all of sources let it without proving

take a look here

Edit

i defined a complex number as any function that including the unit $i=\sqrt{-1}$

then $Z= f(i) $

simple example: $Z= f(i)=\arctan(i)^{i \ln(i)} $

so again how we can proof that this expression or any other can be written as $f(i)= a+i b $ where $a$ and $a$ are reals

Answer 1

"i defined a complex number as any function that including the unit $i=\sqrt{-1}$

then $Z=f(i)$

simple example: $Z=f(i)=\arctan(i)^{iln(i)}$

so again how we can proof that this expression or any other can be written as $f(i)=a+ib$ where $a$ and $b$ are reals"

We can't using your "definition" because it is incorrect.

According to your definition, $i\cdot \begin{bmatrix}1&1\\0&1\\0&0\end{bmatrix}$, $i\oplus \heartsuit$, $5+i-j+2k$, $\{i,\{2\},\emptyset\}$, and many other things should count as "complex numbers" but many of those most certainly can not be written in the form $a+bi$ with $a$ and $b$ both real numbers. Under more standard definitions, none of those I just listed are considered complex numbers but are instead considered something else entirely.

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One of the correct and common ways to define the complex numbers would be as

$$\Bbb C=\{(a,b)~:~a,b\in\Bbb R\}$$ with the operations of addition and multiplication defined as:

$$(a,b)+(c,d)=(a+c,b+d)$$

$$(a,b)\times (c,d)=(a\cdot c-b\cdot d,b\cdot c+a\cdot d)$$

where the addition and multiplication of real numbers is otherwise the usual addition and multiplication we are used to.

Further, equality of complex numbers is defined as $(a,b)=(c,d)$ if and only if $a=c$ and $b=d$, where equality of real numbers is defined as normal.

For convenience sake, we may choose to represent a complex number $(a,b)$ in the form $a+bi$, potentially omitting either the $a$ or the $bi$ if exactly one of them is zero.

From this definition, letting $z\in\Bbb C$, the existence of a real $a,b$ such that $z=a+bi$ is immediate since the only elements in $\Bbb C$ are by definition those which can be written in that form. Further, that such a representation is unique is again immediate from the definition, because that is how equality was defined in the first place.