# 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.