Big O, what is the complexity of summing a series of n numbers?

Question

I always thought the complexity of:

1 + 2 + 3 + ... + n is O(n), and summing two n by n matrices would be O(n^2).

But today I read from a textbo

Answer 1

answer of sum of series of n natural can be found using two ways. first way is by adding all the numbers in loop. in this case algorithm is linear and code will be like this

 int sum = 0;
     for (int i = 1; i <= n; i++) {
     sum += n;
   }
 return sum;

it is analogous to 1+2+3+4+......+n. in this case complexity of algorithm is calculated as number of times addition operation is performed which is O(n).

second way of finding answer of sum of series of n natural number is direst formula n*(n+1)/2. this formula use multiplication instead of repetitive addition. multiplication operation has not linear time complexity. there are various algorithm available for multiplication which has time complexity ranging from O(N^1.45) to O (N^2). therefore in case of multiplication time complexity depends on the processor's architecture. but for the analysis purpose time complexity of multiplication is considered as O(N^2). therefore when one use second way to find the sum then time complexity will be O(N^2).

here multiplication operation is not same as the addition operation. if anybody has knowledge of computer organisation subject then he can easily understand the internal working of multiplication and addition operation. multiplication circuit is more complex than the adder circuit and require much higher time than the adder circuit to compute the result. so time complexity of sum of series can't be constant.