# The Real Number System

The real number system, $\mathbb{R}$, is a ring, $R,$ definined on the **quotient set** of all rational Cauchy sequences such that the operations of addition and multiplication are defined as follows:

$$[a_n] + [b_n] = [a_n + b_n] \\ [a_n][b_n] = [a_n b_n] $$

The Cauchy sequences are in brackets to denote the **equivalence classes**. For example, 1/2 and 3/6 would be in the same equivalence class. The quotient set of $a_n$ would be the set of all equivalence classes whose limit is $a$.

From the above definition it follows that the real number system is a field because it can easily be shown $\mathbb{R}$ is a commutative ring with unit. For example, the property of commutativity follows from the way the real numbers were defined above. The same goes for commutative multiplication and associativity.

The element 0 is an element of $\mathbb{R}$ since $a_n$ can be defined as the sequence consisting of all zeroes. It can also be shown if $a_n \neq 0,$ then $a_n$ has a multiplicative inverse. We can make $\mathbb{R}$ an ordered field by defining a set of positive numbers which are positive Cauchy sequences. The positive Cauchy sequences have the same properties as defined in the positive cone earlier.

## Book traversal links for The Real Number System

Title | Authored on | Content type |
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R Dataset / Package Stat2Data / Nursing | March 9, 2018 - 1:06 PM | Dataset |

R Dataset / Package datasets / attitude | March 9, 2018 - 1:06 PM | Dataset |

R Dataset / Package DAAG / elastic1 | March 9, 2018 - 1:06 PM | Dataset |

R Dataset / Package KMsurv / kidney | March 9, 2018 - 1:06 PM | Dataset |

R Dataset / Package Ecdat / Participation | March 9, 2018 - 1:06 PM | Dataset |