# Does time or space complexity of arithmetic operations get affected by the number of digits?

Suppose I have two 5-digit numbers (A and B) and two 50-digit numbers(C and D). Do the operations A+B and C+D have equal complexity in terms of time and space? or C+D is more complex due to the size of digits?

## 2 Answers

The complexity of arithmetic operations is typically measured in bit operations, though not everybody agrees this is the correct measure. There is a Wikipedia page listing the complexity of the best algorithms currently known for various operations, including addition and multiplication.

It depends on the model you use. Yuval Filmus's answer assumes you use a multitape Turing machine. This is also mentioned in the Wikipedia page:

Here, complexity refers to the time complexity of performing computations on a multitape Turing machine.

However, for example, if you use a random access machine, the time and space complexity of an arithmetic operation of any-digit numbers is considered to be $$O(1)$$.