TITLE: What is the norm of a complex number?
QUESTION [26 upvotes]: Many articles on the web state that the norm of $z = a +bi$ is 
\begin{align}
\sqrt{z*z^*} 
&= \sqrt{(a + bi)(a - bi)}\\
&= \sqrt{a^2 - abi + abi - b^2i^2}\\
&= \sqrt{a^2 - b^2\sqrt{-1}^2}\\
&= \sqrt{a^2 + b^2},\\
\end{align}
$$$$ 
where $z^* = a - bi$ is the conjugate of $z$. However, my textbook states that the norm of a complex number $z = a +bi$ is $a^2 + b^2$, i.e. without the square root. 
What is the norm of a complex number? Which of the two versions is correct?

REPLY [10 votes]: Let me give you a more geometric perspective of the norm.
If we treat multiplication by $a + bi$ as a linear transformation on the space $x + yi$, then, under the usual basis of $\{1, i \}$, multiplication by $a+bi$ is written as the matrix multiplication:
$$\left(\begin{matrix}a &-b\\b&a\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}ax-by\\ay+bx\end{matrix}\right)$$
In number theory, the "norm" is the determinant of this matrix. In that sense, unlike in analysis, the norm can be thought of as an area rather than a length, because the determinant can be interpreted as an area (or volume in higher dimensions.) However, the area/volume interpretation only gets you so far. The reality is that the determinant of a matrix is an "algebraic" quantity which has the nice property that it is independent of the basis chosen, so it is well defined without picking a basis.
In particular, if $A$ and $B$ are two $n \times n$ matrices, then $\det AB = \det A \det B$, which means that the norm defined this way has the same property: $N(z)N(w)=N(zw)$. That nice property follows through to other cases of "number fields" and their rings of integers where the "area" interpretation is less clear-cut.
In particular, this "algebraic norm" is not measuring distance, but rather measuring something about the multiplicative behavior of $a+bi$. That it turns out to be the square of the geometric norm in this case is a deep geometric fact about the geometry of complex numbers.