For systems that need to multiply numbers in the range of several thousand digits, such as computer algebra systems and bignum libraries, long multiplication is too slow. These systems may employ Karatsuba multiplication, which was discovered in 1960 (published in 1962). The heart of Karatsuba's method lies in the observation that two-digit multiplication can be done with only three rather than the four multiplications classically required. This is an example of what is now called a "divide-and-conquer algorithm". Suppose we want to multiply two 2-digit base-"m" numbers: "x"1" m + x"2 and "y"1" m + y"2: