I got a bit carried away in a comment over on Dan Burfiend's blog: Quadrant Dan. As an opener for his geometry class, he asks about some large numbers. I suggested a couple of follow-up questions, for those who wanted to pursue the opener further:
Are there any numbers (feel free to restrict to integers) where a < b but ab < ba? What are they?
I will leave you to play with this one.
Approximating big powers
A rough approximation that can be really helpful is 210 is close to 1000, aka 103. For 4234, you could approximate:
42 is approximately 40 = 4 * 10
So 4234 could be close to 268 * 10 34
Using our approximation of 1000 for 1024, replace 268 by 28 * 10006, so we get
256 * 1052 or 2.56 * 1054
Of course, that's still only about 1/6 the precise value calculated by worlfram alpha, but seems pretty good for such simple calculation.
Approximating compound interest
Let's say you want to do better than the previous approximation (we do, we do!) Can we make a useful adjustment to correct for replacing 42 by 40? Well, 40 = 40 * 1.05, so 4234 = 40^34 * 1.0534.
That second term looks like a calculation for compound interest, right? One rule of thumb (the rule of 70) is that a compounding process will double in approximately (70/rate) periods. In other words, the time it takes your money to double at interest rate r% is about 70/r years. At 5%, about how many doubling periods do we get when we compound 34 times? About 34/70 * 5 which is about 2.5. So, we can approximate 1.0534 by 22.5
Depending on your love for √2 , you ignore that bit and end up with a final estimate of 1055 (approximately 2.56*1054 * 4). However, for those playing along who want to say √2 is close to 1.5, then we get a final approximation of 1.5*1055.
Exercises for the reader
Try to approximate 3442. Is your approximate result larger or smaller than the approximation we got above? How confident are you that this allows you to determine which is larger, 3442 or 4234?
What did we learn?Well, in cleaning up this post, I learned how to do exponents and square roots in html, so that's cool. More seriously I feel this example shows something important about the roles of manual calculations and computer based math.
First, this wasn't blind calculation following an algorithm. At each step, we were thinking about relationships, albeit approximate ones, and ways to short-cut the direct calculation.
On the other hand, the sequences of approximations could easily have taken Dan's whole class. Would it have been fun for the students? The use of the calculation engine brings this into scope as a 5 minute class opener for a class that will eventually be about something else entirely (I guess).
Even if you wanted to talk about the approximations in class, I think seeing the answer from Wolfram Alpha actually makes the hand calculations a lot more fun. The kids would be thinking something analogous to this: "sure he can fly over that building in an airplane, but can he really jump over it?!"