Journal of Formalized Mathematics
Volume 15, 2003
University of Bialystok
Copyright (c) 2003 Association of Mizar Users

## Inner Products and Angles of Complex Numbers

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

### Summary.

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.

#### MML Identifier: COMPLEX2

The terminology and notation used in this paper have been introduced in the following articles [10] [15] [12] [14] [16] [5] [9] [17] [7] [8] [1] [11] [3] [13] [4] [2] [6]

#### Contents (PDF format)

1. Preliminaries
2. More on the Argument of a Complex Number
3. Inner Product
4. Rotation
5. Angles

#### Bibliography

[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.