# Is there any rule of solving problems, having some values very small

1. Dec 16, 2011

### Sumedh

Is there any rule of solving problems, having some values very small
for example
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if current in the circuit by a battery (of EMF= and internal resistance=r)

=ε/r+R

if r is very small as compared to R , can we write it as
=ε/R
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Are there any rules (or something like that) or any information on the net?

I will be very thankful even if some hint or web address is given.

2. Dec 17, 2011

### 256bits

That would depend upon how much error you are willing to accept in a calculation.
AND how much error your instruments can measure ie its accuracy and precision.

In some cases, with your example, if r is 1/10 of R, then an error of 10% might be acceptable. Then again, there could be a situation where you would want 0.1% error, or 0.01% error, or less.

I think you will have to figure out yourself as to how much error you can tolerate in your design, and then decide whether or not the equation can be "simplified".

3. Dec 19, 2011

### Thundagere

I think one thing to keep in mind is that derivatives have similar applications here. They can be used to estimate such results. I've estimated squares using derivatives, and as one would expect, the margin of error was about equal to the amount I estimated squared. About. With that in mind, it depends on the degree of your equation as to how accurate your answer will be.
Now, if you have something like 50, and you're adding 0.0000000000001, then you can ignore the small numbers, since significant figures demand that it wouldn't change.
The way I always do it is I look at my margin of error, guess how much the result will be, and decide if it's within significant figures. If it is, good. If not, I don't ignore small numbers.

4. Dec 20, 2011

### Nabeshin

The rigorous way is always to taylor expand. Rewrite as:

$$I=\frac{\epsilon}{r+R}=\frac{\epsilon}{R}\left(1+ \frac{r}{R} \right)^{-1}=\frac{\epsilon}{R}\left(1-\frac{r}{R}+\left(\frac{r}{R}\right)^2- \cdots\right)$$

In doing this, I identified the small parameter as $\frac{r}{R}$. Now, as the above posters have said, it's a matter of identifying how precise you need your expansion to be. In almost all circumstances, you will be keeping either the first or second terms in the series, and dropping all the rest. Here, for example, we obviously just keep the first term. In practical applications, this is determined by your sensitivities and whatnot (i.e. if r/R ~ 10^-3, and you need things accurate to 10^-3, it suffices to keep the first two terms in the series).

This same procedure applies in any problem with an identifiable 'small parameter'. Just rewrite so that you have something you can taylor expand in the small parameter, and truncate the series accordingly.