Why is real part written first in complex numbers? While expressing complex numbers as $a + \iota b$, is there a specific reason for writing the real part before the imaginary part?
Who introduced this notation first? Is it a case where it just hung up with us due to the guy you introduced us to this?
 A: It is a historical convention, and Euler himself chose it:

But there are some pretty good mathematical reasons to prefer it, which might explain why Euler chose it.
$
\def\lfrac#1#2{{\large\frac{#1}{#2}}}
\def\rr{\mathbb{R}}
$
At first, it might seem more natural to write $ai+b$, like "$ax+b$" where $x$ is the variable, but that is mostly useful only when de-emphasizing the constant terms. Here it is not the case, because $i$ is the constant. $\{1,i\}$ form the standard basis of the complex numbers over the reals, so in this setting one can think of a complex number as $a·1+b·i$, and we drop the "$1$" because it is redundant. If this was all, then there would be no reason to prefer one ordering of the basis elements over the other. However, $1$ is in fact very special, as it is the multiplicative identity.
This specialness of $1$ results in various facts that favour putting it first:


*

*The reals embed into the complex numbers as numbers of the form $a+bi$ where $b = 0$. Thus it makes sense to put the real part first. The reals form a field, while the purely imaginary numbers do not, so we can see the complex numbers as a field extension of the reals, whereby it is natural to likewise extend the representation by appending a "${}+bi$".

*$\exp(iz) = \cos(z) + i·\sin(z)$ for any complex $z$. Obtained from the Taylor series, $\cos(z)$ is the asymptotically significant term approximating $\exp(iz)$ as $z \to 0$. So it makes sense to put it first. Note also that we usually write "$i·\sin(z)$" to emphasize the viewpoint that this equation relates $\exp,\cos,\sin$, in which $i$ is merely a coefficient. Typographical issues established this form in history as $\operatorname{cis}$, but I think it is very likely that anyone who rediscovers the identity will choose the same form.

*Related to the above, $r·\exp(it)$ for real $r,t$ is not only the polar form of a complex number but also can be seen as a spiral transform (scaling plus rotation) about the origin. The identity spiral transform is just $1 = \cos(0)+i·\sin(0)$, so again it makes sense to have the $\cos(0)$ first.

*When defining the complex logarithm as the inverse of the complex exponential function, namely via $\ln(r·\exp(it)) := \ln(r)+it$ for real $r,t$ where $r > 0$ and $t \in (-π,π]$, it makes sense to put the scaling $r$ first before the rotation $\exp(it)$, because scalings are in some sense simpler. For example they preserve the reals, and have diagonal transformation matrices.
