Decimal Approximations of the Real Number by Excess and Defect

Let′s take the irrational number `sqrt(2)` and we have:

`1^2<2<2^2` ; `1<sqrt(2)<2` ;

`1.4^2<2<1.5^2` ; `1.4<sqrt(2)<1.5` ;

`1.41^2<2<1.42^2` ; `1.41<sqrt(2)<1.42` ;

`1.414^2<2<1.415^2` ; `1.414<sqrt(2)<1.415` ;

`1.4142^2<2<1.4143^2` ; `1.4142<sqrt(2)<1.4143` .

The numbers 1; 1.4; 1.41; 1.414; 1.4142 are called the decimal approximations of number `sqrt(2)` by defect accurate within according to 1; to 0.1; to 0.01; to 0.001; to 0.0001. The numbers 2; 1.5; 1.42; 1.415; 1.4143 are called the decimal approximations the number `sqrt(2)` by excess accordingly with the same accuracy.

For number `sqrt(2)` we can use the form of the infinite decimal fraction: `sqrt(2)=1.4142` ... .

In general,we can consider any real number in the form of infinite dicimal fraction and besides periodic, if the number is rational and terminate, if the number is irrational.

For example, `14/55=0.2(54)=0.2545454` ... . (see note converting infinite periodic decimal into proper fraction)

The decimal approximation of number `14/55` accurate within `0.001` by defect equals 0.254, and by excess equals 0.255.