The type of continued fractions that emerge from this process are intrinsically important. When we approximate an irrational number y by rationals we naturally turn to the decimal representation of y. This is excellent for general calculations but,being tied to a particular base, is not mathematically natural.Essential to the nature of y is how well our number y can be approximated by fractions with relatively small denominators. Is there anyway to find a series of fractions that best deals with the conflicting demands of approximating yto a high degree of accuracy while keeping the denominators relatively small? The answer lies in the continued fraction representation of a number that does this through its truncations at ever lower floors.