Implications of quantum hypothesis and uncertainty principle

Quantum hypothesis implies light behaves as if it were composed of particles: it can be emitted or absorbed only in packets, or quanta. 

Uncertainty principle implies that particles behave in some respects like waves: they do not have a definite position but are “smeared out” with a certain probability distribution.

The theory of quantum mechanics is based on an entirely new type of mathematics that no longer describes the real world in terms of particles and waves; it is only the observations of the world that may be described in those terms.

There is thus a duality between waves and particles: for some purposes it is helpful to think of particles as waves and for other purposes it is better to think of waves as particles.