**diff options**

author | bryan newbold <bnewbold@snark.mit.edu> | 2008-06-10 10:48:24 -0400 |
---|---|---|

committer | bryan newbold <bnewbold@snark.mit.edu> | 2008-06-10 10:48:24 -0400 |

commit | 397cefef7929e6cd49959db1d891f5b0654ebd05 (patch) | |

tree | ec197475d62c9149288863f89c97e26131959815 /math/numbers | |

parent | 80b4628e5a94dfedd2c6ee91bdb608c531a78598 (diff) | |

download | knowledge-397cefef7929e6cd49959db1d891f5b0654ebd05.tar.gz knowledge-397cefef7929e6cd49959db1d891f5b0654ebd05.zip |

added a bunch of math content based on alaska notes. added more ethernet content

Diffstat (limited to 'math/numbers')

-rw-r--r-- | math/numbers | 54 |

1 files changed, 54 insertions, 0 deletions

diff --git a/math/numbers b/math/numbers new file mode 100644 index 0000000..541d174 --- /dev/null +++ b/math/numbers @@ -0,0 +1,54 @@ +======================== +Numbers +======================== + +.. note:: + incomplete + +.. note:: + Most of the definitions and notation in the section are based on [rudin]_ or [meserve]_ + +.. contents:: + +*incommensurable* + objects are incommensurable when their ratio isn't rational + +Real Numbers +================== +The *real numbers* are defined via Dedakind cuts in [rudin]_, or [meserve]_ +(1-12). + +Complex Numbers +================== +The *complex numbers* are constructed as an ordered pair of real numbers. + +Algebraic and Transendental Numbers +=============================================== +*Algebraic numbers* are solutions of polynomials, such as x in +:latex:`$a_0 x^n + a_1 x^{n-1} + a_2 x^{n-2} + ... a_n = 0$`, where all a are +real numbers. *Transcendental numbers* are not solutions to any such +polynomials. + +All real numbers are either algebraic or transcendental. + +Some algebraic numbers aren't real (such as :latex:`$i = \sqrt{-1}$`). They +can be rational or irrational. All transcendental numbers are irrational; +some are not real. + +Exersize: is the square root of 5 algebraic or transcendental? + +e +======== +:latex:`$e = \lim_{x \rightarrow 0} (1+x)^{\frac{1}{x}}$` + +Infinities +================== +*aleph-zero* (:latex:`$\aleph_0$`) is the countably infinite set. + +Positive integers, integers, and rational numbers are all countably infinite. + +It is unproven that the real numbers are *aleph-one* (:latex:`$\aleph_1$`). + +.. [rudin] `Principles of Mathematical Analysis (3rd ed)`:title:, by Walter Rudin. McGraw-Hill, 1976 + +.. [meserve] `Fundamental Concepts of Algebra`:title:, by Bruce Meserve. |