Approximations and errors

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

For example, if we know that someone has earned 303.215 € in the lottery, we say around 300.000 €; when we measure a table with a tape measure, we can approximate to the centimetres or millimetres, but we can’t measure it with more precision.

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  a lower order. It can be:
- Lower approximation, if all the digits are exact (the resulting number is lower than the number)
- Higher approximation, if all the digits are exact except the one that indicates the order of units, which is a unit greater (the resulting number is greater than the number)

For example: the approximations of π = 3.1415.. to the nearest hundredth are: 3,14 (lower) and 3,15 (higher).

The rounding of a number is the closest approximation to that number. Remember that 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.

Example: 3,14 is the rounding of π to the nearest hundredth.

When we give an approximate number we are making an error, which is the subtraction between the exact number and the approximate one. This is called absolute error.

For example, if we use 3.14 to approximate π, then:

AE = |π – 3.14| = 0.001592…<0.01

In this example, we don’t know the exact error but we can control it. We can say the error is lower than a hundredth.

To compare errors, we use the relative error: Exercises:

1.- Find out the approximations and the rounding of 1/3, √2 and e to the nearest hundredth.

2.- Calculate the errors when we approximate 1/3 by 0.3 or e by 2.7

Solutions:

1.- lower approximations: 0.33;1.41; 2.71; higher approximations: 0.34; 1.42; 2.72; rounding: 0.33; 1.41; 2.72

2.- absolute errors: 1/30; 0.018 < 0.1; relative errors: 1/10 = 10%; 0.00662...< 0.01 = 1%