Yesterday’s was all about Perfect number. It’s not how pretty the number is, or how well you like its form. It’s not even about if the number is your favorite. Perfect numbers have a distinct definition. A perfect number is a number whose factors (not including the number itself) add to give the number itself. Six is the first perfect number. The factors of 6 are 1, 2, 3, and 6, but we aren’t including 6. 1 + 2 + 3 = 6 and that makes it perfect.
I hadn’t even heard of perfect numbers until Waldorf math caught my eye in my son’s sixth year of formal homeschooling. This is where Waldorf geometry introduces a compass and a straightedge to the student just like in the days of Euclid. In case you aren’t familiar, Waldorf teaches geometry like none I’ve ever seen, and you guessed it, I LOVE it!!
So apparently perfect numbers were studied even in Pythagorean’s school, which is pretty interesting and worth looking into. Those Greek guys were pretty fascinating. Well, the ones that were into geometry anyway. But the credit of course goes to Euclid. I say, “of course,” because I’m pretty sure he gets credit for almost everything mathematical. Okay, okay, a lot. He gets credit for a lot.
Actually, Euclid’s ingenious act was writing down all the postulates and theorems (stuff they had figured to be true in math to that point) of all geometry (and some math,) and then expanded on the work. It’s still the material that we teach children in geometry today. It’s really quite fascinating that it has lasted and endured so long. Kind of shows how universal it is.
Anyway, he wrote the formula for finding perfect numbers (not like this equation I’m giving you; They wrote them all as word problems. Can you imagine?!? The Greeks didn’t have the simple formula type expressions. That was a gift from the Arabs. Also fascinating, and I don’t know too much, but can’t wait to find out.)
This is how it goes. If 2p − 1 is prime, then 2p−1(2p − 1) is a perfect number. So, if we raise 2 to a given power (you get to choose what power) and subtract one and get a prime number, then we can raise 2 to a power less than we did the first time and multiply it by that prime number we got a second ago, and we get…you guessed it! A Perfect number.
Let’s do 6! So, if I take 2 and raise it to the second power, then I get 4. When I subtract 1 from 4, I get 3. Three is prime. I can only divide 3 by 1 and 3; it has no other factors. So then if I take 2 to the first power (one less than I raised 2 to the first time) which is 2 and then multiply it by the 3 that we got earlier, I get 6.
Unfortunately, we don’t see another perfect number until 28, and it goes up quickly. So, what are all the other numbers called? Well, if the sum of their factors is less than the number itself it’s consider deficient (poor little numbers and doesn’t that say a lot about the Greeks?) If the sum is greater, it’s considered abundant.
So, the lovely thing about doing this lovely sum exercise is that we are practicing much of the math that we have done previously this year and past years. We are definitely factoring and definitely doing some mental addition. Indirectly though we are also dividing and learning the rules of division. Great practice, and no worksheets.
So, here’s my question to you: given all that we know right here, which numbers do you think are mostly to be abundant, and which are more likely to be deficient? Why? There are no right answers. Consider it a “Notice and Wonder.”