Ready for an elementary arithmetic problem? Here it is:
Nothing to it… just add the two numbers. Of course, we’d rather not add them by hand, so let’s use a calculator instead:
Uh oh… the calculator doesn’t give the complete answer. It does return the first nine significant digits, but it doesn’t return all 16 digits. Indeed, we can’t be sure that the final 7 in the answer is correct because of rounding.
So now what we do (other than buy a more expensive calculator)?
When I pose this question to students, the knee-jerk reaction is to just start adding one digit at a time. Though that’s not the worst possible response, it is possible to use modern technology to make ordinary grade-school addition move a lot quicker. One way to do this is to take three digits at a time while using a calculator:
Notice that the 1 in 1369 gets carried over to the next block of three digits in much the same way that a sum greater than 10 has the tens digit carried over to the next digit. Continuing:
This is logically equivalent to using base 1000 to add these two numbers (as opposed to base 10) and is certainly a lot faster than using only one digit at a time. Of course, it’d go even faster if we use up to nine digits a time (which is equivalent to using base one billion).