# Approximations and errors

Frequently, we use approximate numbers because it isn’t necessary or convenient to give an exact quantity that we know, or maybe we cannot measure it exactly.

An **estimate** of a real number is another real number that is still close enough to be useful. An **approximation of n-order of units** is an approximation of the number in which we remove the digits of units of greater order. We can use the lower approximation or the higher one.

For example: the approximations of *e* = 2.718.. to the nearest hundredth are: 2,71 (lower) and 2,72 (higher)

**rounding**of a number is the closest approximation to that number. Remember that in order to round a number to a particular order of units, we remove all the digits on the right of this order and, if the first substituted digit is greater than or equal to 5, we round up the previous digit too.

**absolute error**.

**relative error**:

**Exercise**: Calculate the errors when we approximate 1/3 by 0.3 or *e* by 2.7

Solutions: a) AE = 1/30 = 0.0333...< 0.1; RE = 1/10 = 0.1 = 10%; b) AE = 0.018...< 0.1; RE = 0.00672...< 0.01 = 1%

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