There is no basic method for adding them or multiplying. For one thing countinued fractions sometimes a very elegant way to express numbers. For example the square roots irrationals have no repeating decimals but they have a repeating countinued fraction! Another place where countinued fractions are useful is when you want to approximate an irrational number by rationals. The beauty of this is that using countinued fractions you get the best possible fraction with that denominator as little as it can be. Further because,Originally Posted by SkyWatcher
the rate of convergence is extreme. Thus, as few convergents after the expansion and you have a super accurate approximation to 10 decimal places!