Why do many math books have so much detail and so little enlightenment? (2010)
I own and love the book, and did a lot of pure math in my math undergrad degree. But SFM is closer to the tradition of Martin Gardner's long-running "Mathematical Games" column in Scientific American than real higher pure math education.
I think that's because it hinges on giving you tools to "hack" problems in physics, geometry and calculus that leverage intuition about quantities and space. There are branches of pure math where this matters a lot, but frankly it's not going to help you learn about Lie groups or category theory. It's not clear to me that the style and approach of SFM is helpful in this kind of very abstract context where a detailed understanding of unfamiliar sorts of concepts and mechanics is crucial.
Meanwhile, with the "teens" trick, the local minimum is 29 (68.9%) and then at 299, 2999, etc, and the mean underestimation is boosted to... huh, only 91%?
Then again, I'm assuming a completely even distribution, which probably won't be true in some contexts.
This has been the case for me too. I never learned this in a class in school, but one of the most important things I've always done when learning some mathematic topic is try to #1 associate it with something in the real world and #2 find a way to "guestimate" without a calculator what the answers will be. For instance, take square roots. The sqrt of 137 will be between 11 and 12 more precisely about 11.7 and I can visualize it in my head as one side of a 2D square shape. Also squares and square roots allow me to easily think about the area covered (size/shape) by N-dimensions in relation to the size of just one dimension. A real world application which is interesting to note is that the time it takes a falling object to hit the ground is equal to the square root of the distance traveled (really 2 times the square root divided by gravity but the general relationship is the same).
Can't wait to read the entire work.
This is within 5% of the real answer, but does not not involve music!
Here's another one, less accessible to attack: Find e^7. Or 25000^(1/8).
The trick to using the musical scale is that it's logarithmic in frequency but linear in key (ABCDEFG). The frequency step between adjacent notes is 2^(1/12); the frequency jump over an octave is 2; the frequency jump over 40 notes is 10. There are other handy ratios to remember, like that going up a whole step is a frequency increase of 9/8, and the more you know the more powerful this method is.
To solve 25000^(1/8) = x, you're looking for an interval x on the keyboard such that if you were to go up 8 of those intervals, the frequency would jump by 25000. Well, 25000 is just 10^5 / 2^2, or in terms of keyboard intervals, "Go up 40 notes (repeat four more times), then down two octaves". This means to represent a frequency increase by a factor of 25000, I need to go up 105-212 = 176 half steps on a keyboard.
Divide by 8 to find x = 22 steps, which is two octaves minus a whole step on the keyboard. The frequency change over this 22-step interval is thus (2*2) [up two octaves] / (9/8) [down a whole step] ~ 3.5. So that's the final answer: 3.5
The actual value of 25000^(1/8) is 3.546, so this approximation is good to within a couple of percent.
Your trick is cool too, by the way.
For your other challenge, I would use the enormously useful 2^10 = 1024 ~ 1000 (which the musical trick implicitly uses too, in the 40 notes is 10 times the frequency mnemonic). So 25000 is close to 5^2 * 2^10, and it's square root is 5 * 2^5 = 160 (we can always keep in mind that 2^10 ~ 1000 is an overestimate by 2.4%, so our 160 ~ sqrt(25000) is an overestimate by 1.2%, so an improved approximation of sqrt(25000) is 158). 158 is very close to the midpoint of 144 and 169, so its square root is not far from the midpoint of 12 and 13. So my approximation of 25000^(1/4) is 12.5. Finally I need to take a sqrt of that. I know the square of 3.2 is 10.24 (our old friend 2^10 = 1024), and that's about 20% too low. Ok, we need to bump our 3.2 up by 10%, that makes is 3.52, and that's our answer. It's only 0.7% off, but it's only by luck. For a more general problem, keeping a tab of your first estimate as an absolute number and your current correction as a percentage gives an error of less than 2%.
log10 2 ~= 0.3
log10 2.5 ~= 0.4
log10 3 ~= 0.5
log10 4 ~= 0.6
log10 of 25000 is 4.4
4.4 / 8 is 0.55
antilog10 of 0.55 is approximately 3.5
Obviously, you could do it some other time, but the important part is to make it a habit, so that it feels almost automatic to do, or that you feel like something's missing/feel guilty if you skip it.
You're right about making a habit. I am trying to read an hr before bedtime or an hr right after office (when I reach home) nowadays.