Negative energy levels in the diagram for a hydrogen atom 
The higher the number of the shell (n), the higher is the energy level of the electron. However, why was it necessary to have negative values. So for example, when $n=1$, the energy could be $5 eV$ and for $n=2$, $6 eV$... having positive values could also have supported the idea that as $n$ increases, energy of electron increases. What is the point of having negative numbers, does it somehow aid calculations?
 A: 
What is the point of having negative numbers, does it somehow aid calculations?

Setting the energy of a free electron to zero does indeed aid calculations because it establishes a convenient reference point. For example suppose you are calculating the energy change in the reaction:
$$ Na + Cl \rightarrow Na^+ + Cl^- $$
Since the energy of a free electron is zero for both atoms the energy change is just the ionisation energy of sodium plus the electron affinity of chlorine.
A: We say that a free, unbound electron has zero energy (that's convention, you could just as well put another number there). This means that the level $n = \infty$ is fixed at $E_\infty = 0 \text{eV}$. Since the other levels lie lower, i.e. possess less energy, this forces all other bound states to have negative energies - which then represent that we need to add energy to make the bound state free, which corresponds to raising its energy to zero.
A: The negative sign of energy means that the energy of the electron in the atom is lower than the energy of a free electron at rest. A free electron at rest is an electron that is at sufficiently far away from the nucleus and its energy is assumed to be zero. The negative sign also indicates that the electron is bound to the nucleus.$_1$  

Credits: $_1$ Moderns abc of Chemistry, Dr.S.P. Jauhar.
