# Approximating Division for X1, X2 & ΔX Expressions

• d.arbitman
In summary, the approximation (X-(ΔX/X)) / (X+ (ΔX/X)) ≈ 1- 2ΔX/X² is derived by using long division or the Padé approximation technique. It is a numerical approximation for e^-x and is valid for all positive values of x. The higher order terms in (ΔX)2 and above can be discarded due to the smallness of ΔX compared to X. This approximation is also known as a Padé approximant.
d.arbitman
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

Thanks fellas!

EDIT: Revised the expression.

Last edited:
(X-(ΔX/X)) / (X+ (ΔX/X)) ≈ 1- 2ΔX/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.

You're correct. I revised the expression to make sense.

Solved.
There's a series expansion and the two terms I listed are just the first two terms of the series.

d.arbitman said:
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.

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.

Thanks.

Mark44 said:

How do I handle the fractional parts? or do I split the numerator into two?

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.

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.

## What is division approximation for X1, X2 & ΔX expressions?

Division approximation for X1, X2 & ΔX expressions is a method of estimating the result of a division calculation using values for X1, X2, and ΔX. This is often used in scientific and mathematical calculations where precise values are not necessary, but a general idea of the result is needed.

## Why is division approximation for X1, X2 & ΔX expressions useful?

Division approximation for X1, X2 & ΔX expressions is useful because it allows for a quick estimation of a division calculation without having to perform the full calculation. This can save time and effort, especially when dealing with complex equations.

## How is division approximation for X1, X2 & ΔX expressions calculated?

Division approximation for X1, X2 & ΔX expressions is calculated by first finding the approximate values of X1, X2, and ΔX. Then, these values are used in a simplified version of the division formula, where X1 is divided by X2 and multiplied by ΔX. The result of this calculation is an approximation of the actual division result.

## What are the limitations of division approximation for X1, X2 & ΔX expressions?

Division approximation for X1, X2 & ΔX expressions is not always accurate and should not be used for precise calculations. It is only an estimation and may not give the exact result. Additionally, the accuracy of the approximation depends on the values chosen for X1, X2, and ΔX.

## When should division approximation for X1, X2 & ΔX expressions be used?

Division approximation for X1, X2 & ΔX expressions should be used when a general idea of the result is needed and precise values are not necessary. It is commonly used in scientific and mathematical applications but may also be useful in everyday situations where quick estimations are needed.

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