The Complex Numbers

Czeslaw Bylinski

Warsaw University, Bialystok

Supported by RPBP.III-24.C1.

Summary.

We define the set $\Bbb C$ of complex numbers as the set of all ordered pairs $z =\langle a,b\rangle$ 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 $a = \Re(z)$, $b = \Im(z)$. These definitions satisfy all the axioms for a field. $0_{\Bbb C} = 0+0i$ and $1_{\Bbb C} = 1+0i$ are identities for addition and multiplication respectively, and there are multiplicative inverses for each non zero element in $\Bbb C$. The difference and division of complex numbers are also defined. We do not interpret the set of all real numbers $\Bbb R$ as a subset of $\Bbb C$. From here on we do not abandon the ordered pair notation for complex numbers. For example: $i^2 = (0+1i)^2 = -1+0i \neq -1$. We conclude this article by introducing two operations on $\Bbb C$ which are not field operations. We define the absolute value of $z$ denoted by $|z|$ and the conjugate of $z$ denoted by $z^\ast$.

MML Identifier: COMPLEX1

Contents (PDF format)

Bibliography

[1] Grzegorz Bancerek. The ordinal numbers. Journal of Formalized Mathematics, 1, 1989.

[2] Czeslaw Bylinski. Functions and their basic properties. Journal of Formalized Mathematics, 1, 1989.

[3] Czeslaw Bylinski. Functions from a set to a set. Journal of Formalized Mathematics, 1, 1989.

[4] Czeslaw Bylinski. The modification of a function by a function and the iteration of the composition of a function. Journal of Formalized Mathematics, 2, 1990.

[5] Library Committee. Introduction to arithmetic. Journal of Formalized Mathematics, Addenda, 2003.

[6] Krzysztof Hryniewiecki. Basic properties of real numbers. Journal of Formalized Mathematics, 1, 1989.

[7] Jan Popiolek. Some properties of functions modul and signum. Journal of Formalized Mathematics, 1, 1989.

[8] Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 1989.

[9] Andrzej Trybulec. Subsets of real numbers. Journal of Formalized Mathematics, Addenda, 2003.

[10] Andrzej Trybulec and Czeslaw Bylinski. Some properties of real numbers operations: min, max, square, and square root. Journal of Formalized Mathematics, 1, 1989.

[11] Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, 1989.

[12] Edmund Woronowicz. Relations and their basic properties. Journal of Formalized Mathematics, 1, 1989.