What is XLI * XXIII?

In my Ancient Rome class this week we reviewed the use of the Roman Numeral System. This system as almost everyone knows was fairly advanced for its era but lacked the digit and concept of Zero. Because it doesn't have placemarkers many times we think that certain arithmetic operations would have been so complex as to be impossible. Our instructor pointed out that, on the contrary, they had worked out systems that were workable and in some ways better or at least as good as ones we work with.

For example take the problem 41 * 23. Before we all got so used to calculators we'd set up and work the problem like this
41
23
___________
123
82
___________
943

Note that we have an "implied" 0 after 82 in the fourth row so it is actually 820.

Now according to our Professor the Romans did the following starting with the same two numbers in Roman Numerals:


This method required the ancient Roman have only a knowledge of doubling and halving. With each step the first number is cut in half and any remainder thrown away. Similarly, at each step the second number is doubled. Taking the number from the second row if and only if the corresponding number in the first row is an odd number, we obtain three numbers to add together. Try it with another pair of numbers. Now tell us how and why it works!