Johnny wrote a long number sequence: $1,  2,  3,  4,  5,$ $$\cdots $$ $111,  112,  113$ on a piece of paper. He tries to find the number of digits he wrote by doing the following:  $$9\times 1=9$$  $$99-10=89$$  $$89\times 2=178$$  $$113-100=13$$  $$13\times 3=39$$  $9+178+39=226$  Is he correct? If not, write down the correct calculations. 
Overseas Competition->Knowledge Point->Counting Modules