Number Theory and its application Need a detailed explanatio

Number Theory and its application
Need a detailed explanation. thx.

3. Give an accurate estimate of the bitlength of each of the following numbers: (a) the sum of a million different numbers, each roughly equal to one million; (b) the product of a million different numbers, each roughly equal to one million.

Solution

3

Part a)

Sum of a million different number, each roughly equal to one million:

Since each number is roughly equal to one million, therefore for estimation purpose, we can consider that each of the number is exactly equal to one million. One million being added to itself one million times can be found by multipying one million to itself.

Therefore, number of bits = 2*(number of bits in one million)-1

We know that there are 7 bits in one million, therefore, the required bit length is = 2*7-1 = 14-1 = 13

Therefore, bitlength would be 13 for part (a).

Part b)

Product a million different numbers, each roughly equal to one million:

This quantity can be written as one million raised to one million. That is, 1000000^1000000

We know that number if bits in n^n is equal to (n+1). Therefore, number of bits in 1000000^1000000 are (1000000+1) = 1000001

Therefore, bitlengh for part b is 1000001.