### 3.6 Complex Numbers

#### 3.6.1 Representation as a Pair of Real Numbers

The even numbered root of a negative radicand can not be represented in the domain of real numbers, since for all real numbers $Math content$ we always have $Math content$. For a polynomial of second degree with real numbers $Math content$ the generally known solution only yields numbers when the radicand is larger or equal to zero.

$Math content$

To allow a solution for all $Math content$, i.e. the positive ones, on extends the one-dimensional space to two-dimensional number pairs of real numbers, that are called complex numbers $Math content$, and for which a special multiplication rule is agreed to.

Complex numbers were first used in the 16th century in connection with roots of negative redounds by the mathematicians Girolamo Gardano and Raffaele Bombelli.

Complex numbers satisfy the following rules:

$Math content$

The main innovation relative to the “one-dimensional” real numbers is the multiplication rule. While it yields for numbers, whose second components vanish, the rule for real numbers,

$Math content$

one obtains from the definition for the product of two numbers, whose first components vanish,

$Math content$

The product is in both cases a one-dimensional, real number. The second case is equal to the first case except for an additional sign.

The practical justification for these rules follows from their consequences, historically especially from the fact, that in the domain of number pairs defined in this way, the taking of roots with rational exponents is possible without restriction. Simplest example: one looks for the solution of $Math content$:

$Math content$

The real numbers are a one-dimensional subset of the two-dimensional complex numbers $Math content$, i.e. those with $Math content$; thus the real numbers are rare exceptions among the complex numbers. The complex numbers with $Math content$, i.e {$Math content$} are called “imaginary numbers”. Their square is negative: $Math content$

#### 3.6.2 Normal representation with the “imaginary unit i”

The usual notation, the normal representation of complex numbers distinguishes between the two components instead of their sequence in the curly brackets via a marker in front of the second component, for which since Leonhard Euler (1707 – 1783) the letter $Math content$ is used (in electrical engineering one uses instead the letter $Math content$ to distinguish the marker from the current $Math content$). The plus sign indicates, that both components belong together.

However, there is no class of “imaginary numbers”, but both components of the pair that form a complex number are real. The notation $Math content$ does not refer to a “multiplication of $Math content$ with $Math content$”, but means, that the second component of the complex number is $Math content$.

The suggestive normal representation z=a+ib simplifies the calculations, since one can use in it the usual multiplication rules for real numbers, if one takes into account the convention $Math content$. Thus the following rules for the normal representation have to interpreted accordingly

Let us consider an example:

$Math content$

$Math content$

$Math content$

$Math content$

Using the normal representation the solution of the square root problem becomes clearer.

$Math content$

$Math content$

In the set of complex numbers the square root of a real number always has two solutions. They are either both purely real or imaginary, depending on the sign of which the root has to taken.

The general solution for a quadratic polynomial with real coefficients $Math content$,$Math content$ and $Math content$, with which we started, now reads:

$Math content$

If $Math content$,$Math content$ and $Math content$ are themselves complex, the general formula is still valid, but not the distinction between two cases since the order relations $Math content$ and $Math content$ are not applicable for complex numbers.

What is the situation in the complex number space for the cube/third root, and how in general for odd root exponents? In the space of real numbers there is always one real negative solution ($Math content$ for negative radicands. In the space of complex numbers we however obtain the following:

$Math content$

$Math content$

Thus three roots $Math content$ of $Math content$ exist, of which one is real and the other two are complex conjugates of each other.

#### 3.6.3 Complex Plane

The complex numbers are mapped for visualization purposes to points in a plane, where the abscissa corresponds to the real number line and the ordinate corresponds to the complex number line, and distances on both are measured using real numbers.

The simple cubic equation $Math content$ has three solutions in the space of complex numbers, of which one is real; two are complex. As the last representation for $Math content$ shows (for $Math content$ the points are mirrored on the imaginary axis), the roots are situated symmetrically on a circle with radius $Math content$.

In the diagram the cube roots are indicated as squares and the two square roots as circles.

The general polynomial of $Math content$th degree has in the space of complex numbers $Math content$ roots Gauss’s fundamental theorem of algebra. Figure 3.6 shows that for the second and third root of $Math content$.

Taking into account the rules for additions and multiplications all usual arithmetic operations known for real numbers can also be applied to complex numbers.

The complex numbers densely cover the complex plane, as the real numbers cover the number line densely. Unlike the real numbers the complex numbers are however no ordered set, since they each consist of two real numbers, and therefore the relation $Math content$ is not defined in general. However they can be ordered according to absolute values $Math content$, which are real numbers.

The real numbers are a subset of the complex numbers, of character {a, 0}, i.e. with imaginary part zero. Real numbers are rare special cases of complex numbers. Complex

Using complex numbers (complex analysis) has many advantages in physics and engineering.

As shown for the example of the parabola every algebraic equation has solutions in the domain of complex numbers (property of algebraic closure); Gauss’s fundamental theorem.

In addition every complex function that can be differentiated once, can be differentiated an arbitrary number of times. Finally one can show with complex numbers relationships between individual functions that are independent in the domain of real numbers (exponential function and trigonometric function, see below).

#### 3.6.4 Representation in Polar Coordinates

In the representation using polar coordinates, the absolute value $Math content$ gives the distance $Math content$ from the origin and the ratio of imaginary part to real part is equal to the tangent of the angle $Math content$ to the real axis.

The following definition for the polar representation is applicable:

$Math content$

To obtain $Math content$ and $Math content$ from $Math content$ or vice versa the following relations apply:

$Math content$

$Math content$

The polar representation shows a special position of the number zero in the domain of complex numbers, which is not visible in the domain or real numbers. It is the only complex number, that does not have a direction associated with it, since $Math content$ means $Math content$, irrespective of the value of $Math content$. This is also compatible with the tangent of $Math content$ being undetermined as a ratio of two zeroes.

The number $Math content$ corresponds in polar representation to the end point of a vector that starts at the origin with length $Math content$ and makes an angle $Math content$ with the real axis.

#### 3.6.5 Simulation of Complex Addition and Subtraction

Pressing the Ctrl-key and clicking on the following pictures activates the interactive Java simulations, which visualize the complex operations addition and subtraction.

The operations are visualized in Figure 3.8 as mapping of a rectangular array of points in the $Math content$-plane to a $Math content$-plane shown on the right hand side. For the red point on the lower left corner of the arrays the position vector is indicated. On the $Math content$-plane you can change the position of the red corner of the array as well as the tip of the green vector via pulling with the mouse. The $Math content$-plane shows the result of the complex operation. Clicking on the ”Initialize“ button restores the original state. The distance between the points in the array can be adjusted with the slider. In particular you can collapse the array to a single point.

In addition to the simulation a text with several pages is shown. This text contains a description of the simulation and hints for possible experiments.

The windows can be hidden or blown up to full screen size with the usual symbols on the top right; it makes however more sense to blow up the simulation windows via pulling on one corner in order for the quadratic structure of the system of coordinates to be preserved.

If you click on a point with the mouse its coordinates appear in window with coloured background. With the right mouse button you can access further options in the context menus.

The addition of two numbers corresponds in the complex plane to the addition of both vectors (according to absolute value and direction). The subtraction shown in Figure 3.9 corresponds to a subtraction of both vectors. Considered as a mapping of the $Math content$-plane, addition and subtraction are equivalent to a translation of the plane without rotation or change of scale.

#### 3.6.6 Simulation of Complex Multiplication and Division.

The following two interactive pictures deal with the simulation of complex multiplication (Figure 3.10) and division (Figure 3.11). The presentation and handling is identical as described above for addition and subtraction.