Units

Photo courtesy of Dr. Fredrick Olness, a professor of physics at SMU: My Favorite One-Liners: Part 92

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

This is one of my favorite quote from Alice in Wonderland that I’ll use whenever discussing the difference between the ring axioms (integers are closed under addition, subtraction, and multiplication, but not division) and the field axioms (closed under division except for division by zero):

‘I only took the regular course [in school,’ said the Mock Turtle.]

‘What was that?’ inquired Alice.

‘Reeling and Writhing, of course, to begin with,’ the Mock Turtle replied; ‘and then the different branches of Arithmetic — Ambition, Distraction, Uglification, and Derision.’ Arithmetic with big numbers: Index

I’m doing something that I should have done a long time ago: collect past series of posts into a single, easy-to-reference post. The following posts formed my series on doing basic arithmetic with very large numbers that exceed the character displays of most calculators.

Part 2: Multiplication

Part 3: Division

Arithmetic with big numbers (Part 1)

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). 