## The complex numbers (1990)

Venue: | Journal of Formalized Mathematics |

Citations: | 104 - 1 self |

### BibTeX

@ARTICLE{Byliński90thecomplex,

author = {Czesław Byliński},

title = {The complex numbers},

journal = {Journal of Formalized Mathematics},

year = {1990},

volume = {2}

}

### Years of Citing Articles

### OpenURL

### Abstract

Summary. We define the set C of complex numbers as the set of all ordered pairs z = 〈a,b 〉 where a and b are real numbers and where addition and multiplication are defined. We define the real and imaginary parts of z and denote this by a = ℜ(z), b = ℑ(z). These definitions satisfy all the axioms for a field. 0C = 0 + 0i and 1C = 1 + 0i are identities for addition and multiplication respectively, and there are multiplicative inverses for each non zero element in C. The difference and division of complex numbers are also defined. We do not interpret the set of all real numbers R as a subset of C. From here on we do not abandon the ordered pair notation for complex numbers. For example: i 2 = (0+1i) 2 = −1+0i � = −1. We conclude this article by introducing two operations on C which are not field operations. We define the absolute value of z denoted by |z | and the conjugate of z denoted by z ∗.

### Citations

1297 | Functions and their basic properties
- Byliński
- 1990
(Show Context)
Citation Context ...ute value of z denoted by |z| and the conjugate of z denoted by z ∗ . MML Identifier:COMPLEX1. WWW:http://mizar.org/JFM/Vol2/complex1.html The articles [8], [11], [1], [5], [9], [6], [7], [10], [12], =-=[2]-=-, [3], and [4] provide the notation and terminology for this paper. In this paper a, b denote elements of R. One can prove the following proposition (2) 1 For all real numbers a, b holds a 2 + b 2 = 0... |

1292 | Tarski Grothendieck set theory
- Trybulec
- 1990
(Show Context)
Citation Context ...ch are not field operations. We define the absolute value of z denoted by |z| and the conjugate of z denoted by z ∗ . MML Identifier:COMPLEX1. WWW:http://mizar.org/JFM/Vol2/complex1.html The articles =-=[8]-=-, [11], [1], [5], [9], [6], [7], [10], [12], [2], [3], and [4] provide the notation and terminology for this paper. In this paper a, b denote elements of R. One can prove the following proposition (2)... |

1229 | Properties of subsets
- Trybulec
- 1990
(Show Context)
Citation Context ...e not field operations. We define the absolute value of z denoted by |z| and the conjugate of z denoted by z ∗ . MML Identifier:COMPLEX1. WWW:http://mizar.org/JFM/Vol2/complex1.html The articles [8], =-=[11]-=-, [1], [5], [9], [6], [7], [10], [12], [2], [3], and [4] provide the notation and terminology for this paper. In this paper a, b denote elements of R. One can prove the following proposition (2) 1 For... |

1049 | Functions from a set to a set
- Byliński
- 1990
(Show Context)
Citation Context ...alue of z denoted by |z| and the conjugate of z denoted by z ∗ . MML Identifier:COMPLEX1. WWW:http://mizar.org/JFM/Vol2/complex1.html The articles [8], [11], [1], [5], [9], [6], [7], [10], [12], [2], =-=[3]-=-, and [4] provide the notation and terminology for this paper. In this paper a, b denote elements of R. One can prove the following proposition (2) 1 For all real numbers a, b holds a 2 + b 2 = 0 iff ... |

1024 | Relations and their basic properties
- Woronowicz
- 1990
(Show Context)
Citation Context ... absolute value of z denoted by |z| and the conjugate of z denoted by z ∗ . MML Identifier:COMPLEX1. WWW:http://mizar.org/JFM/Vol2/complex1.html The articles [8], [11], [1], [5], [9], [6], [7], [10], =-=[12]-=-, [2], [3], and [4] provide the notation and terminology for this paper. In this paper a, b denote elements of R. One can prove the following proposition (2) 1 For all real numbers a, b holds a 2 + b ... |

673 | The ordinal numbers
- Bancerek
- 1990
(Show Context)
Citation Context ...field operations. We define the absolute value of z denoted by |z| and the conjugate of z denoted by z ∗ . MML Identifier:COMPLEX1. WWW:http://mizar.org/JFM/Vol2/complex1.html The articles [8], [11], =-=[1]-=-, [5], [9], [6], [7], [10], [12], [2], [3], and [4] provide the notation and terminology for this paper. In this paper a, b denote elements of R. One can prove the following proposition (2) 1 For all ... |

563 | Basic properties of real numbers
- Hryniewiecki
- 1990
(Show Context)
Citation Context ...s. We define the absolute value of z denoted by |z| and the conjugate of z denoted by z ∗ . MML Identifier:COMPLEX1. WWW:http://mizar.org/JFM/Vol2/complex1.html The articles [8], [11], [1], [5], [9], =-=[6]-=-, [7], [10], [12], [2], [3], and [4] provide the notation and terminology for this paper. In this paper a, b denote elements of R. One can prove the following proposition (2) 1 For all real numbers a,... |

370 |
Subsets of real numbers
- Trybulec
(Show Context)
Citation Context ...ations. We define the absolute value of z denoted by |z| and the conjugate of z denoted by z ∗ . MML Identifier:COMPLEX1. WWW:http://mizar.org/JFM/Vol2/complex1.html The articles [8], [11], [1], [5], =-=[9]-=-, [6], [7], [10], [12], [2], [3], and [4] provide the notation and terminology for this paper. In this paper a, b denote elements of R. One can prove the following proposition (2) 1 For all real numbe... |

252 | The modification of a function by a function and the iteration of the composition of a function
- Byliński
- 1990
(Show Context)
Citation Context ... denoted by |z| and the conjugate of z denoted by z ∗ . MML Identifier:COMPLEX1. WWW:http://mizar.org/JFM/Vol2/complex1.html The articles [8], [11], [1], [5], [9], [6], [7], [10], [12], [2], [3], and =-=[4]-=- provide the notation and terminology for this paper. In this paper a, b denote elements of R. One can prove the following proposition (2) 1 For all real numbers a, b holds a 2 + b 2 = 0 iff a = 0 and... |

90 |
Some properties of functions modul and signum
- Popiołek
- 1990
(Show Context)
Citation Context ... define the absolute value of z denoted by |z| and the conjugate of z denoted by z ∗ . MML Identifier:COMPLEX1. WWW:http://mizar.org/JFM/Vol2/complex1.html The articles [8], [11], [1], [5], [9], [6], =-=[7]-=-, [10], [12], [2], [3], and [4] provide the notation and terminology for this paper. In this paper a, b denote elements of R. One can prove the following proposition (2) 1 For all real numbers a, b ho... |

85 |
Andrzej Trybulec and Czesław Byliński. Some properties of real numbers operations
- min, square, et al.
- 1989
(Show Context)
Citation Context ...ne the absolute value of z denoted by |z| and the conjugate of z denoted by z ∗ . MML Identifier:COMPLEX1. WWW:http://mizar.org/JFM/Vol2/complex1.html The articles [8], [11], [1], [5], [9], [6], [7], =-=[10]-=-, [12], [2], [3], and [4] provide the notation and terminology for this paper. In this paper a, b denote elements of R. One can prove the following proposition (2) 1 For all real numbers a, b holds a ... |

20 | Introduction to arithmetic
- Committee
- 2003
(Show Context)
Citation Context ... operations. We define the absolute value of z denoted by |z| and the conjugate of z denoted by z ∗ . MML Identifier:COMPLEX1. WWW:http://mizar.org/JFM/Vol2/complex1.html The articles [8], [11], [1], =-=[5]-=-, [9], [6], [7], [10], [12], [2], [3], and [4] provide the notation and terminology for this paper. In this paper a, b denote elements of R. One can prove the following proposition (2) 1 For all real ... |