The closest we can actually get to most real numbers is to compute a rational number that is as close as we need. It can be shown that rational numbers are dense in the real numbers; that is, between every two real numbers there is a rational number, and between every two rational numbers is a real number.3.1An irrational number can be defined as any real number having a non-repeating decimal expansion. For example, is an irrational real number whose decimal expansion starts out as3.2
Other examples of irrational numbers include
Their decimal expansions do not repeat.
Let denote the -digit decimal expansion of an arbitrary real number . Then is a rational number (some integer over ). We can say
Since is defined for all , we naturally define as the following mathematical limit:
A First Look at Taylor Series