The discussion centers on the derivation and validity of the approximation (X-(ΔX/2)) / (X+(ΔX/2)) ≈ 1 - ΔX/X, with ΔX being small compared to X. It is noted that the original approximation was incorrect, and a revised expression was provided. Participants discuss the use of series expansion and long division techniques to derive approximations, emphasizing that higher-order terms can be discarded for practical analysis. Additionally, there is a mention of capitalization conventions in mathematics and the context of Padé approximations for numerical solutions. The conversation highlights the importance of dimensional consistency in expressions involving physical measurements.
#1
d.arbitman
100
4
What is the name and where can I find the derivation for the following approximation?
(X-(ΔX/2)) / (X+ (ΔX/2)) ≈ 1- ΔX/X
Assuming ΔX << X, and X = (X1 + X2) / 2 and ΔX = X1 - X2
in addition, if the X1 and X2 are physical measurements, like distances,
then (X-(ΔX/X)) and (X+(ΔX/X)) are invalid expressions because X and ΔX/X do not have the same dimensions.
#3
d.arbitman
100
4
You're correct. I revised the expression to make sense.
#4
d.arbitman
100
4
Solved.
There's a series expansion and the two terms I listed are just the first two terms of the series.
Solved.
There's a series expansion and the two terms I listed are just the first two terms of the series.
You can do this by using long division (which is another technique to produce an infinite series). Since Δx is small in comparison to x, (Δx)2 will be miniscule, so you can discard all terms in (Δx)2 or higher powers.
#6
d.arbitman
100
4
Mark44 said:
You can do this by using long division (which is another technique to produce an infinite series). Since Δx is small in comparison to x, (Δx)2 will be miniscule, so you can discard all terms in (Δx)2 or higher powers.
Yes, the higher order terms are minuscule and it's okay to discard them for hand analysis. How would I go about performing long division to achieve an infinite series? I've never done it.
When I worked on this yesterday, I noticed that the stated approximation wasn't working out to what you had, so I abandoned my efforts.
It's hard to lay out long division on a computer, but I'll do the best I can
Code:
____________
x + 1/2 Δx ) x - 1/2 Δx
1. Divide x in the dividend (the numerator) by x in the divisor (the denominator). The partial quotient is 1.
Code:
1
____________
x + 1/2 Δx ) x - 1/2 Δx
2. Multiply the partial quotient (1) times the divisor, and put the answer beneath the dividend
Code:
1
____________
x + 1/2 Δx ) x - 1/2 Δx
x + 1/2 Δx
________________
3. Subtract. You should get -Δx
4. Now divide -Δx by x to get -Δx/x.
5. Continue this process until you get tired of doing it.
$$ 1 + \frac{-Δx}{x} + \text{other terms}$$
The "other terms" are those in (Δx)2 and higher powers, which as I mentioned, can be discarded.
BTW, except in probability, variables are almost always written in lower case. IOW, x rather than X. In probability, so-called random variables are usually written in upper case.
#10
d.arbitman
100
4
Mark44 said:
When I worked on this yesterday, I noticed that the stated approximation wasn't working out to what you had, so I abandoned my efforts.
It's hard to lay out long division on a computer, but I'll do the best I can
Code:
____________
x + 1/2 Δx ) x - 1/2 Δx
1. Divide x in the dividend (the numerator) by x in the divisor (the denominator). The partial quotient is 1.
Code:
1
____________
x + 1/2 Δx ) x - 1/2 Δx
2. Multiply the partial quotient (1) times the divisor, and put the answer beneath the dividend
Code:
1
____________
x + 1/2 Δx ) x - 1/2 Δx
x + 1/2 Δx
________________
3. Subtract. You should get -Δx
4. Now divide -Δx by x to get -Δx/x.
5. Continue this process until you get tired of doing it.
$$ 1 + \frac{-Δx}{x} + \text{other terms}$$
The "other terms" are those in (Δx)2 and higher powers, which as I mentioned, can be discarded.
BTW, except in probability, variables are almost always written in lower case. IOW, x rather than X. In probability, so-called random variables are usually written in upper case.
Thank you, I mean it. I just copied the variables from one of my engineering textbooks which omits the derivation.
I didn't even know that in mathematics that there's a capitalization convention. I'm used to lower case for small signals and upper case for DC (+AC).
Alternatively, remember that ##1/(1 - a)## is the sum of the geometric series ##1 + a + a^2 + a^3 + \dots## and multiply the series by ##(1+a)##. (And then let ##a = -x/2##).
You didn't say what context this came from, but it is an example of a Padé approximation, which is a technique used to create numerical solutions of differential equations etc. Specifically, ##(1-\frac x 2)/(1+ \frac x 2)## can be a nice numerical approximation to ##e^{-x}##, for all positive values of ##x##, not just for "small" values of ##x## like the first few terms of a Taylor series. http://en.wikipedia.org/wiki/Padé_approximant http://en.wikipedia.org/wiki/Padé_table