Here is an example . Note that if we start with a decimal expression, then there is always a real number that is represented by that decimal expression. The addition, subtraction and multiplication of integers results in forming another integer. With division, however, we sometimes generate non-integers. We note that N is a subset of Z which is a subset of Q which is a subset of R , denoted .
Every rational number has a decimal representation that has a repeating component.
Also, any decimal representation of a number with a repeating component can be changed to a rational number. The real numbers which are not rational are called irrational. For the irrationals, we can not determine their full decimal representation. We can get a close representation of the irrationals by doing the same kind of procedure that we did in the example on the cube root on 10 in the section Decimal Representations of Real Numbers. Exercises - Introductory .