Why is the notion of magnitude of a complex number ,$a = x+iy$ , defined as $|a|^2 = \bar{a}a$ ?
Should I think of the conjugate as the dual space and it is used to define an inner product where we can define a notion of length? I am unsure if this is the correct interpretation.
Please feel free to refer me to a text that covers this with some rigor.
An algebraic/geometric view: Consider the representation of a complex number $z = a + ib$ on the complex plane. By Pythagoras' theorem, its length is given by $|z|^2 = a^2 + b^2$. By some calculations, $z \overline{z} = (a + ib)(a - ib) = a^2 + b^2$, so $|z|^2 = z \overline{z}$.
Another point of view is that $\langle z, w \rangle = \overline{z} w$ defines an inner product on $\mathbb{C}$, and so the norm given by $\langle z, z \rangle = \overline{z} z$ is a reasonable notion of length. Note that this agrees with the more elementary view given above, and that it also generalises well to norms on $\mathbb{C}^n$ given by $\langle x, y \rangle = x^*y$ for $x, y \in \mathbb{C}^n$.