Do it on paper using the method you learned in third grade. How many times above do you combine a single pair of digits? On the left it's 4 (0x0, 0x1, 1x0, and 1x1). Which is O[n^2] and n=2 above.
On the right it's 9 pairwise operations for the ones column plus 9 for the tens column (but it would usually be 10 if we had any carries) for a total of 18. A lot more than 4.
Oh but you say "the computer doesn't have to do those adds digit by digit. It can do all the digits in parallel." That's only true when the numbers fit into a register. If your numbers are not 64 bits wide but 64 thousand bits wide, the computer has to effectively do addition digit by digit. Which is why the left side above is quicker than the right, and the O[n logn] method in the article is better still.
On the right it's 9 pairwise operations for the ones column plus 9 for the tens column (but it would usually be 10 if we had any carries) for a total of 18. A lot more than 4.
Oh but you say "the computer doesn't have to do those adds digit by digit. It can do all the digits in parallel." That's only true when the numbers fit into a register. If your numbers are not 64 bits wide but 64 thousand bits wide, the computer has to effectively do addition digit by digit. Which is why the left side above is quicker than the right, and the O[n logn] method in the article is better still.