How would you approximate these without using a calculator?

Question

23.42/42.94 and 5.2/4.256 How?

In most classes like Physics/Calculus you'll just use a calculator if you need to compute these, so I forgot how to do them by hand. 23.42 is smaller than 42.94, and they're both decimals, this makes it tricky for me.

My last question is how would I approximate something like cos(2) or ln(3) without using a calculator?

Answer 1

For the decimal ones you just multiply the top and bottom by whatever makes them not decimals (if that is what's tripping you up) and then do long division. For example, multiply 23.42/42.94 by 100/100, to get 2342/4294. You're just multiplying by 1 so it doesn't change anything.

For Cos(x) and Ln(x), you really are just better off using a calculator. In approximating those guys, generally, one uses a series (if you haven't had what is generally the second or third college level Calculus course, and not just Pre-Calc, you won't know what those are, if you have had that Calc course then look them up) and that is just NO FUN by hand.

If you have had calc. classes (and refreshed your memory as to how to approximate functions) then it should be pretty obvious how to go about the approximation process. If you haven't had that level of Calc. then I can't explain it to you in a simple (and general) fashion.

If you are thinking of going into the science field and you aren't yet comfortable with series and approximations, just give it a little time. You will be once you've had more classes.

P.S. If you have had Calculus Courses that included series, then just email me and I'll clear things up for you a little better regarding the series stuff.

Answer 2

For the 23.42/42.94, I would round to 23.5/43 or 47/86. Note that half of 86 is 43, so this is 1/2 + 4/86, and 4/86 should be just over 4%, so your answer will be somewhere between 0.54 and 0.55. A quick check on the actual division gives .5454122... , so our approximation is quite accurate.

For the second, I'd probably just estimate this as a little under 5/4 and be done with it. If I need a more accurate answer than that, I probably have time to compute it exactly. The actual answer, btw, is 1.221804511..., so I'm not too far off.

2 is just under 2/3 of pi, so I'd place the cosine of 2 as a little over -1/2. In fact, it's closer to -.4 than -.5, so approximations of trig functions won't get you very far. Better to keep a table. For the ln, note that e is just under three, so you probably want something just over 1, say, 1.1 for your approximation. In fact this is fortuitously accurate: the actual value of ln 3 is 1.0986123.... Usually top-of-the-head estimates for logarithmic functions won't be that good.

Answer 3

Round things so as to reduce the number of decimal places. E.g., 23.42/42.94 -> 23.5/43 -> 0.45. 5.2/4.256 -> 5/4 -> 1.25.

cos(2) depends on whether 2 is radians or degrees. If degrees, the angle in radians is approximately 2/60 or 1/30, whose sine is also approximatety 1/30, so use cos^2 (x)x + sin^2 (x) = 1 to get the cosine. If 2 is radians, the angle is approximately 120 degrees, so use 30-60-90 triangle. As for ln(3), it is a bit bigger than ln(e), which is 1, so call it 1.1 and go on.

Answer 4

23.42/42.94 = about 23 / 43 = about 0.5

5.2/4.256 = about 5 / 4 = about 1.3

Any cos is between -1 and +1. So I guess 0
Tayler: cos x = about 1 - x2/2!
x2 2 leads to about 1 - 4/2 = -1

Tayler: ln(1 + x) = about x
x=2 gives ln(3) is about 2

Th