What would happen if someone invented a way to measure both position and momentum precisely? If it is impossible why?
I sense that the mathematical derivation of the uncertainty principle may not be a sufficient explanation... :) But if you're interested, look at the answer here: Heisenberg Uncertainty Principle scientific proof
Think about how we look at something in macro space - we shine a light on it, such as with a microscope, and see what reflects back. In the case of a very small particle, this light could be as small as one photon - but one photon has enough momentum to push the object we're examining when they collide. So if we want to know where a particle is in space, we bounce a photon off of it - but this accelerates the particle, so now its momentum is "uncertain" (or changed). Likewise if we want to know what its momentum is - now its position has changed.
Whatever technique is used to measure a quantity (photons are just an example), it necessarily has an impact on the subject of that measurement. Thus it is impossible to measure these quantities simultaneously to a degree of precision which exceeds the uncertainty principle's limitations.
You've probably heard the observational reason for the uncertainty principle which basically states that if you measure either position or momentum very accurately your measurement changes the other. That's a really nice hand-wavy explanation but the uncertainty principle is fundamental to waves and is not about measurement.
Check out the Wikipedia article on this. The second paragraph states:
[...] the uncertainty principle actually states a fundamental property of quantum systems, and is not a statement about the observational success of current technology
The article goes on to spell out in detail why this is so.