so now we can rewrite our multiplication problem xy as followswhen we expand and gather like terms we get the followingmore convenientlywe can write it like thiswhereit should be pointed out that this suggests recursive approach to multiplying two numbers since this procedure does itself involve multiplication specificallythe products acadbcand bd all involve numbers smaller than the input number and so it is conceivable that we could apply the same operation as partial solution to the overall problem this algorithmso farconsists of four recursive multiplication steps and it is not immediately clear if it will be faster than the classic long multiplication approach what we have discussed so far in regards to the recursive approach to multiplicationhas been well known to mathematicians since the late th century the karatsuba algorithm improves on this is by making the following observation we really only need to know three quantitiesz ac =ad +bcand bd to solve equation we need to know the values of abcd only in so far as they contribute to the overall sum and products involved in calculating the quantities and this suggests the possibility that perhaps we can reduce the number of recursive steps it turns out that this is indeed the situation since the products ac and bd are already in their simplest formit seems unlikely that we can eliminate these calculations we can however make the following observation