Can you explain how to multiply two numbers when their range exceeds the normal integer range in an algorithm?

Algorithm: Multiplying Two Numbers with Range Exceeding Integer Limit

When dealing with multiplying two numbers whose range exceeds the normal integer limit, a common approach is to use the Karatsuba algorithm. This algorithm is an efficient multiplication algorithm that can handle large numbers by breaking them down into smaller sub-problems.

The basic idea behind the Karatsuba algorithm is to split the two numbers into smaller halves, perform multiplications on these halves recursively, and then combine the results using addition. This reduces the complexity of multiplication from O(n^2) to approximately O(n^log2(3)), making it more suitable for handling large numbers.

Here is a basic outline of the Karatsuba algorithm for multiplying two numbers with a range exceeding the normal integer limit:

1. Split the two numbers into halves, x = (x₁ * 10^n/2 + x₀) and y = (y₁ * 10^n/2 + y₀).
2. Recursively calculate the products of x₁ * y₁, x₀ * y₀, and (x₁ + x₀)(y₁ + y₀), combining them to get the final result.
3. Combine the results using the formula: x * y = x₁ * y₁ * 10ⁿ + (x₁ * y₀ + x₀ * y₁) * 10^n/2 + x₀ * y₀.

By using the Karatsuba algorithm or other similar multiplication algorithms, it is possible to efficiently multiply two numbers with a range that exceeds the normal integer limit in an algorithmic manner.