I was studying microeconomics and I confess I am not the brightest person for maths and sorry if this is very dumb but I get that we CAN normalize the income and I get where it comes from and how it is done, but I don't understand WHY do we have to do that. Can't we still do the maximization problem? I read further and understood that it is to "fix" a line in the box with slope Px/Py (with Py being the normalized price), I still don't understand why. What is the problem of finding the Pareto optimum in a freely moving point? Is it only to represent the budget constraint in the box? I am probably missing a fundamental calculus/microeconomic fact.
If you by normalizing income mean normalizing one of the prices then yes you do not have to normalize it, but it makes everything easier since only the relative price matters. All price vectors (2 dimensions) that solves the problem will have the same ratio. There are therefore usually infinitely many price vectors that solves the problem.