algebra-precalculuscomplex numbers
I don't really understand why the absolute value of a complex number is defined as its magnitude.
The absolute value for a real number, I think has two sensible interpretations: the magnitude of this number from the origin or the positive value of this number.
When extrapolating this to complex numbers, why is the first definition chosen, and not simply that the absolute value of a complex number is a complex number in the first quadrant (of the complex plane).
To Henry's comment here is the drawing:
The standard inner product of two complex numbers $z_1,z_2 \in \mathbb{C}$ is defined to be $z_1\bar{z_2}$, so the norm it induces will be $||z_1||:=\sqrt{z_1z_1}=\sqrt{z_1\bar{z_1}}=\sqrt{(a+bi)(a-bi)}=\sqrt{a^2+b^2}$. As a vector space over $\mathbb{R}$ (which it is almost never considered to be in practice), $\mathbb{C}$ has dimension two and thus is isomorphic to $\mathbb{R}^2$, the standard basis might be chosen to be $\{(1,0), (0,i)\}$ where we represent a number $z=a+bi$. The canonical isomorphic then sends $(1,0) \to (1,0)$ and $(0,i) \to (0,1)$, thus if you take a vector in $\mathbb{C}$, say $z=a+bi$, and "transport" it to $\mathbb{R^2}$ to investigate its norm in that inner product space (where the inner product is defined to be $(x_1,y_1)(x_2,y_2)=x_1x_2+y_1y_2$), we have $z=a+bi=a(1,0)+b(0,i)$, apply the canonical isomorphism and we get $a+bi$ is transported to $(a,b)$, (and not (a,bi)!), so the inner product here is different, there being no conjugation in the second coordinate, but that's fine because we're dropped the $i$ anyways by moving to $\mathbb{R^2}$. Pythagoras is fine.
One further comment: viewing $\mathbb{C}$ as a two-dimensional vector space over $\mathbb{R}$ is convenient merely for visualizing the complex plane, not for investigating its properties as a vector space, consider the following excerpt from Rudin's Functional Analysis:
Best Answer
In general, we invent definitions and notation because we intend to make use of them somewhere. For example, the term "absolute value" and the corresponding notation $|-|$ exist because we regularly have occasion to refer to it in e.g. the definitions of variance of a random variable, limit of a sequence, and other constructions. For every one of these applications, the corresponding concept in the complex numbers is captured by magnitude. In contrast, I cannot think of a single case where I have needed to refer to "the number in the first quadrant differing from this one by a factor of a power of $i$."
Moreover, the useful algebraic properties of the absolute value function on the reals are not true of the function you've described. For example, if we denote your function by $[-]$, it is not the case in general that $[xy] = [x][y]$ -- how could it be, since the first quadrant isn't even closed under multiplication?
The point is, you can define whatever function you want, but if it's just a curiosity and not something that comes up naturally then it's probably not worth endowing with its own special notation and terminology.