We define a complex number as a number of the form $a+ib$, and then we can equip the set of complex numbers with addition ($+$) and multiplication ($*$), these two operations have good properties, thus $\mathbb{C}$ is a field. But it seems we did not define what is the meaning of $ib$, does it mean $i*b$ even though we define complex numbers first and then the multiplication?
In fact, we seem to always treat it as $i*b$. For example, by polar transformation $x=rcos\theta, y=rsin\theta$, we have $z=rcos\theta+irsin\theta$, but we also denote it as $r(cos\theta+isin\theta)$. Then by Euler's identity, we can get $z=re^{i\theta}$
Best Answer
This is why formal definitions are so important. As you surely can find on this forum, a more rigorous definition is to define $\mathbb{C}$ as a set to be simply $\mathbb{R^2}$, and define two operations on it:
$(a,b)+(c,d)=(a+c,b+d)$
$(a,b)\cdot (c,d)=(ac-bd, ad+bc)$
These operations turn $\mathbb{C}$ into a field, which can be easily checked. Finally, we denote the pair $(a,b)$ by $a+ib$. This is just a notation, nothing else. For example, when we write $ib$, what we really mean is the pair $(0,b)$. For example, $i=(0,1)$ by definition.
Thing is, this is a very nice notation, as we get that $ib$ is indeed equal to the product of $i=(0,1)$ and $b=(b,0)$. (as you can easily prove using the definition of multiplication)