Diophantine approximation

Diophantine approximation studies how well real numbers can be approximated by rational numbers. It explores questions like: Given a real number, how close can a fraction with whole-number numerator and denominator get to it? The field investigates the quality of these approximations, quantifying how small the difference can be relative to the size of the denominator. It has deep connections to number theory, analyzing the properties of real numbers and their relationships with rational numbers, and plays a key role in understanding how "irrational" or "rational" a number is.