A continued fraction is a way to represent a number through a sequence of fractions. It starts with an integer, followed by a series of fractions where the denominator is also a fraction. This method can express both rational and irrational numbers, making it a useful tool in number theory and approximation. Continued fractions can provide better approximations of real numbers than simple decimal expansions. For example, the continued fraction for the irrational number π reveals its structure and can help in calculating its value more accurately. They are also used in various mathematical fields, including algebra and dynamical systems.