summaryrefslogtreecommitdiffstats
path: root/math/numbers
diff options
context:
space:
mode:
Diffstat (limited to 'math/numbers')
-rw-r--r--math/numbers54
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.