## Real Exponents

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.

^{3.1}An

*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 as

^{3.2}

*rational*number. That is, it can be rewritten as an integer divided by another integer. For example,

*limit*of as goes to infinity is . Since is defined for all , we naturally define as the following mathematical limit:

*real*exponents.

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A First Look at Taylor Series

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Rational Exponents