Can 1/(1 + x) be simplified to 1-x for a very small value of x?

[Matuku]

If you have a fraction, for example,
$$\frac{1}{{1.0091532\times10^{-12}} + 1}$$

Is there a simple way to convert it to a more easily calculated form, specifically, 1-x (where x is a very small number)

 

[CRGreathouse]

1/(1+a) = 1 - a/(1 + a), which for tiny a is about 1 - a. More precisely,
1/(1+a) = 1 - a/(1 + a) = 1 - a + a^2/(1 + a)
which is about 1 - a + a^2 for tiny a.

 

[symbolipoint]

If you have a fraction, for example,
$$\frac{1}{{1.0091532\times10^{-12}} + 1}$$

Is there a simple way to convert it to a more easily calculated form, specifically, 1-x (where x is a very small number)

Just arrange an equation based on that.
1 - x = $$\frac{1}{{1.0091532\times10^{-12}} + 1}$$

Determine an expression for the value of x, and then rewrite the left-hand expression.

 

[adriank]

Expanding on what CRGreathouse says, the Taylor series (around 0) for 1/(1 + x) is 1 - x + x² - x³ + ... (and it converges if |x| < 1). You can approximate it by truncating the Taylor series, and since you have x ~ 10^-12, for practical purposes 1 - x should be enough (any higher-order terms will be smaller than the precision you give anyway).