encountering the worst case decreases geometrically thusthe skip list illustrates tension between the theoretical worst case (in this caseth(nfor skip list operation)and rapidly increasing probability of average-case performance of th(log )that characterizes probabilistic data structures numerical algorithms this section presents variety of algorithms related to mathematical computations on numbers examples are activities like multiplying two numbers or raising number to given power in particularwe are concerned with situations where built-in integer or floating-point operations cannot be used because the values being operated on are too large oftensimilar operations are applied to polynomials or matrices since we cannot rely on the hardware to process the inputs in single constanttime operationwe are concerned with how to most effectively process the operation to minimize the time cost this begs question as to how we can apply our normal measures of asymptotic cost in terms of growth rates on input size firstwhat is an instance of addition or multiplicationeach value of the operands yields different problem instance and what is the input size when multiplying two numbersif we view the input size as twothen any non-constant-time algorithm has growth rate that is infinitely high compared to the growth of the input this makes no senseespecially in light of the fact that we know from grade school arithmetic that adding or multiplying numbers does seem to get more difficult as the value of the numbers involved increases in factwe know from standard grade school algorithms that the cost of standard addition is linear on the number of digits being addedand multiplication has cost when multiplying an -digit number by an -digit number the number of digits for the operands does appear to be key consideration when we are performing an algorithm that is sensitive to input size the number of digits is simply the log of the valuefor suitable base of the log thusfor the purpose of calculating asymptotic growth rates of algorithmswe will consider the "sizeof an input value to be the log of that value given this viewthere are number of features that seem to relate such operations arithmetic operations on large values are not cheap there is only one instance of value there are instances of length or less the size (lengthof value is log