Diophantine approximation is a branch of number theory that focuses on how closely real numbers can be approximated by rational numbers. It studies the relationships between integers and their ratios, aiming to find rational numbers that are close to a given irrational number. This field is named after the ancient Greek mathematician Diophantus, who worked on equations that have integer solutions. One key concept in Diophantine approximation is the idea of continued fractions, which provide a systematic way to express real numbers as fractions. These approximations can reveal important properties of numbers, such as their irrationality and how well they can be approximated by simpler fractions.