Are the complex numbers well ordered?

Mathematics 1 pp 169-198 | Cite as

Part of the Springer textbook book series (SLB)


If complex numbers are used to represent alternating current circuits, these can be calculated according to the same laws as those for direct current circuits. Calculating with complex numbers is therefore an important part of electrical engineering.

However, the trigger for the definition of the complex numbers is the desire that every quadratic equation can be solved. This is also used as a motivation for introducing the complex numbers. It should be noted that the symbol \ (\ sqrt {- 1} \) does not make any sense in the real-number domain. Therefore, the complex numbers are first introduced as ordered pairs of numbers, provided with the two links of addition and multiplication. This gives the definition for the imaginary unit i = (0, 1), the notation in mathematics. As in electrical engineering, however, j is used here, da i stands for the amperage.

The terms real part, imaginary part and the absolute value of a complex number as well as the to z conjugate complex number z* are defined and the most important calculation rules are summarized. The Gaussian number plane and the representation of a complex number as a pointer in the Gaussian number plane follow. This means that a complex number can also be introduced into the trigonometric representation. The terms are further explained using examples.

How complex numbers can be raised to the power, square rooted and logarithmized is shown in detail. The Fundamental Theorem of Algebra is presented (without proof) to recall the motivation. The notation of a complex number with the help of the exponential function is finally mentioned and deepened with a few examples.

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© Springer-Verlag Berlin Heidelberg 2008

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