I managed to solve the math question the grinding way (get out a piece of paper, expand everything, add and subtract all the terms from each other, get a result, pick the leading order term), then remembered that you can just use the approximation (1 + a) / (1 + b) = a/b for a and b much less than 1. There's some other terms as well of course, but we'll throw them out in the end, so approximating to the leading term during the calculation isn't a truncation error. With that approximation, the question simplifies to "-10^[-4] + 3*10^[-4]", which is just 2*10^(-4), which is D. So there's the answer: just get an intuition that only the leading term matters when dealing with an expression that uses the likes of 10^(-4) vs. 10^(-8).
Theory of Instruction: Principles and Applications by Siegfried Engelmann goes more into this and see Shepard Barbash's Clear Teaching for an accessible introduction.
I got it wrong in the math question but my idea is that 1.0001 and 1.0003 are very small numbers, practically 1, that division will produce the difference only in the four number after the decimal point. And that difference in that number will be approximately 3 times more in the second case as in the first. Then subtracting 0.999 will cancel out and we will get something like 0.000X or closer to 10^-4 and not 10^-8.
your posts would be easier to read if you made your footnotes into links, so people could jump down to them and then jump back, instead of having to scroll-- or have them pop up when the mouse hovers over the number.
I managed to solve the math question the grinding way (get out a piece of paper, expand everything, add and subtract all the terms from each other, get a result, pick the leading order term), then remembered that you can just use the approximation (1 + a) / (1 + b) = a/b for a and b much less than 1. There's some other terms as well of course, but we'll throw them out in the end, so approximating to the leading term during the calculation isn't a truncation error. With that approximation, the question simplifies to "-10^[-4] + 3*10^[-4]", which is just 2*10^(-4), which is D. So there's the answer: just get an intuition that only the leading term matters when dealing with an expression that uses the likes of 10^(-4) vs. 10^(-8).
This is suggested under the second principle, Devise a Plan, in Polya's classical heuristic, How to Solve It.
Theory of Instruction: Principles and Applications by Siegfried Engelmann goes more into this and see Shepard Barbash's Clear Teaching for an accessible introduction.
1/(1+a)-1/(1+b) is roughly equal to a-b if a and b are very small. The other terms in the exact answer are very tiny, so it must be D.
I got it wrong in the math question but my idea is that 1.0001 and 1.0003 are very small numbers, practically 1, that division will produce the difference only in the four number after the decimal point. And that difference in that number will be approximately 3 times more in the second case as in the first. Then subtracting 0.999 will cancel out and we will get something like 0.000X or closer to 10^-4 and not 10^-8.
your posts would be easier to read if you made your footnotes into links, so people could jump down to them and then jump back, instead of having to scroll-- or have them pop up when the mouse hovers over the number.
Fair. I'll work on that.