This is the post I was going to write this morning, when I got sidetracked looking at Doodle 4 Google. I finally did complete the search I meant to start, finding out about “abundant” numbers.
Occasionally when I help Al with his math homework (usually for his ELP class, though occasionally for his regular math class also), he refers to a term I’m not familiar with. Most of the time I know the concept, it’s just the terminology that has changed over the decades since I was in school.
This week, though, there was a concept that was new to me. Abundant numbers? Well, of course numbers are abundant – they’re infinite. But the paper explained what the term means: an abundant number is one that is less than the sum of its “proper divisors” (factors of the number other than the number itself).
The homework assignment didn’t mention “deficient numbers” but they are the opposite – where the sum of those proper divisors is less than the number itself. And then there are “perfect numbers” where the sum of the proper divisors is equal to the number itself.
The concept may have been new to me (or maybe it was mentioned in a long-forgotten math lesson but not required to learn for a test), but it’s been around since the time of the ancient Greeks. The terms abundant, deficient, and perfect in reference to numbers were coined by Nicomachus.
What I really wanted to know, though, was what significance there was to identifying numbers as abundant (or deficient or perfect). I went through several pages of hits on Google before I found my answer in this quote by Martin Gardner:
One would be hard put to find a set of whole numbers with a more fascinating history and more elegant properties surrounded by greater depths of mystery–and more totally useless–than the perfect numbers.
On the other hand, on the same page (which contains a wonderful collection of quotes about education, books, mathematics, and more), I found this statement by Nicolai Lobachevsky:
There is no branch of mathematics, however abstract, which may not someday be applied to the phenomena of the real world.
Who knows? Maybe by the time Al someday has kids and they come to him for help with their math homework, someone will have found a practical use for the concept of abundant numbers.
But in the meantime, I found another concept: weird numbers. These are abundant numbers that are not also semiperfect, meaning that they can not be expressed as the sum of some of their proper divisors. They probably aren’t very useful either. But I like the name.