Volume 15, 2003

University of Bialystok

Copyright (c) 2003 Association of Mizar Users

**Wenpai Chang**- Shinshu University, Nagano
**Yatsuka Nakamura**- Shinshu University, Nagano
**Piotr Rudnicki**- University of Alberta, Edmonton

- An inner product of complex numbers is defined and used to characterize the (counter-clockwise) angle between ($a$,0) and (0,$b$) in the complex plane. For complex $a$, $b$ and $c$ we then define the (counter-clockwise) angle between ($a$,$c$) and ($c$, $b$) and prove theorems about the sum of internal and external angles of a triangle.

- Preliminaries
- More on the Argument of a Complex Number
- Inner Product
- Rotation
- Angles

- [1]
Grzegorz Bancerek.
The ordinal numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [2]
Czeslaw Bylinski.
The complex numbers.
*Journal of Formalized Mathematics*, 2, 1990. - [3]
Library Committee.
Introduction to arithmetic.
*Journal of Formalized Mathematics*, Addenda, 2003. - [4]
Krzysztof Hryniewiecki.
Basic properties of real numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [5]
Jaroslaw Kotowicz.
Real sequences and basic operations on them.
*Journal of Formalized Mathematics*, 1, 1989. - [6]
Anna Justyna Milewska.
The field of complex numbers.
*Journal of Formalized Mathematics*, 12, 2000. - [7]
Anna Justyna Milewska.
The Hahn Banach theorem in the vector space over the field of complex numbers.
*Journal of Formalized Mathematics*, 12, 2000. - [8]
Robert Milewski.
Trigonometric form of complex numbers.
*Journal of Formalized Mathematics*, 12, 2000. - [9]
Konrad Raczkowski and Pawel Sadowski.
Topological properties of subsets in real numbers.
*Journal of Formalized Mathematics*, 2, 1990. - [10]
Andrzej Trybulec.
Tarski Grothendieck set theory.
*Journal of Formalized Mathematics*, Axiomatics, 1989. - [11]
Andrzej Trybulec.
Subsets of real numbers.
*Journal of Formalized Mathematics*, Addenda, 2003. - [12]
Andrzej Trybulec and Czeslaw Bylinski.
Some properties of real numbers operations: min, max, square, and square root.
*Journal of Formalized Mathematics*, 1, 1989. - [13]
Michal J. Trybulec.
Integers.
*Journal of Formalized Mathematics*, 2, 1990. - [14]
Wojciech A. Trybulec.
Vectors in real linear space.
*Journal of Formalized Mathematics*, 1, 1989. - [15]
Zinaida Trybulec.
Properties of subsets.
*Journal of Formalized Mathematics*, 1, 1989. - [16]
Edmund Woronowicz.
Relations and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989. - [17]
Yuguang Yang and Yasunari Shidama.
Trigonometric functions and existence of circle ratio.
*Journal of Formalized Mathematics*, 10, 1998.

[ Download a postscript version, MML identifier index, Mizar home page]