What does "compare" mean, when we are comparing an experimental number with a theoretical number? It means two things: absolute error and relative error.

Absolute error means simply the difference between the two numbers. Relative error means the absolute error divided by the theoretical value, if available. Often times people like to multiply the relative error by 100 to get "percent" error.

The reason relative error is better than absolute error is this: Suppose I measure something and tell you I am "off by 10." Is this good or bad? Well it depends. If I was measuring how tall a certain person was and I was off by 10 feet, this would be really bad. On the other hand, if I were measuring a concentration and was off by only 10 parts per million, this would be good. So absolute error becomes meaningful only when you put it in perspective, which is what relative error does.

For example, suppose we estimate the number 1/3 with 0.333.

What is the error?
The absolute error is

>

The relative error is

>

The percentage error in the approximation is 0.1%.

See the discussion in the textbook on page 309, where the author writes about error analysis in the context of differential approximations.