I'm trying to understand what Landau and Lifshitz mean in their $\S31$ of "Quantum mechanics. Non-relativistic theory" about vector model of addition of angular momenta:
... This result can be illustrated by means of what is called the vector model. If we take two vectors $\mathbf L_1,\mathbf L_2$ of lengths $L_1$ and $L_2$, then the values of $L$ are represented by the integral lengths of the vectors $\mathbf L$ which are obtained by vector addition of $\mathbf L_1$ and $\mathbf L_2$; the greatest value of $L$ is $L_1+L_2$, which is obtained when $\mathbf L_1$ and $\mathbf L_2$ are parallel, and the least value is $|L_1-L_2|$, when $\mathbf L_1$ and $\mathbf L_2$ are antiparallel.
As I understand, I should take vectors $\mathbf L_1=(\sqrt{L_1^2-M_1^2}\;\;\;M_1)^T$ and $\mathbf L_2=(\sqrt{L_2^2-M_2^2}\;\;\;M_2)^T$, and get an integral length $L=|\mathbf L_1+\mathbf L_2|$. But when I take e.g. $L_1=3$, $M_1=1$, $L_2=5$, $M_2=-3$, I get
$$L=2\sqrt{7+4\sqrt2}\approx7.12,$$
which is by no means integral. What am I missing? Should I round the result to integer? Or does the vector model actually work only qualitatively?