I'm taking 4 classes this semester: Affordable Design and Entrepreneurship, Renewable Energy, Intro to Moral Philosophy (at Wellesley), and the Art of Approximation, or since that's surely far too many syllables for a name, "Approx".
Approx is real-world, back-of-the-envelope type math (our professor sometimes calls it "Street-Fighting Math". We cover questions like "What's the bandwidth of a 747 packed with CDs flying from Boston to London?", thinking on a logarithmic number line, and converting 7/16's to decimal without a caluculator. The class is way more than just numbers-we also talk a lot about creating mental models and how to handle data.
Our class made this numberline to collect and compare information we find ourselves using frequently.
As we spend so much time in the logarithmic world, one of my favorite tricks we've learned so far is this method for converting to base 10. It's based on pianos, and the assumption that 2^10 (1024), is basically 10^3 (1000). Here's how it works:
Some preliminary math:
Each octave on a piano is a doubling of requency: to the C above middle is at twice as high a frequency as Middle C. The intermediary notes between C's are equally spaced (by frequency), and, counting both white and black keys, there are are 12 of them: 12 halfsteps or semitones = 2X frequency. Also, there's a 3:2 frequency ratio between G and C which are separated by 7 semitones. With those two starting points, we can figure out how many semitones it takes to make any desired frequency ratio.
Getting this chart takes a bit of creativity. For example, we know that a 3-fold increase in frequency is 19 semitones,
because 2(3/2)=3 -- so we add the 12 semitones to the 7 semitones to get 19. The trick is in remembering that the semitones are equally distributed but are already on a multiplicative scale. 2*2=4; or 12+12=24 semitones, and so forth. 2*3 = 6, so 6X frequency is 12+19=31 semitones and so forth. 7 requires a bit of fudging, so we saved it for last and simply placed it between 6 and 8. We're going to use this chart to write an exponential number in terms of semitones, and then convert it to factors of 10, or any other base we want.
We know that each factor of 4 is 24 semitones, so we have 13x24=312 semitones. 40 semitones is a factor of 10, so we can say that 312 semitones is ~7.8 factors of 10.
I know 10^7.8 is something a bit larger than 6 X 10^7, 60+million*. For reference 4^13 is actually 67,108,864. Pretty good!
*Because I know that 10^0.8 is 6.something. It's useful to have a couple known conversion points out of the logarithmic scale: the easiest and most useful to remember are probably that 10^0.5 ~3 and 10^0.7 ~ 5.