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 x we always have x2 > 0. For a polynomial of second degree with real numbers x the generally known solution only yields numbers when the radicand is larger or equal to zero.

ax2 + bx + c = 0 has the two solutions x1,2 = -bb2-4ac2 2a b2 4ac realnumberassolution b2 < 4ac no solution in the domain of the real numbers in the simplest case x2 + c = 0we would have x = -c; then there exists for positive c, i.e. c > 0,no solution in the domain of the real numbers. 

To allow a solution for all c, i.e. the positive ones, on extends the one-dimensional space to two-dimensional number pairs of real numbers, that are called complex numbers C, 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:

 General definition of the complex number z as ordered pair of numbers z = a,ba,b real numbers  addition rule z1 + z2 = a1,b1 + a2,b2 = a1 + a2,b1 + b2 multiplication rulez1 z2 = a1,b1 a2,b2 = a1a2 - b1b2,a1b2 + a2b1 conjugatecomplexnumberdefinition: z̄ = a,-b;this leads to  zz̄ = a,b a,-b = a2 + b2,0 a2 + b2 division:z1 z2 = a1,b1 a2,b2 = a1,b1 a2,-b2 a2,b2 a2,-b2 = a1a2+b1b2,-a1b2+a2b1 a22+b22 = z1z2 z2z2

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,

a1,0 a2,0 = a1a2 = sign(a1)sign(a2) a1 a2 ; withsign(a1) : sign of a1 a1 : absolute value of a1

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

0,b1 0,b2 = -b1b2 = -sign(b1)sign(b2) b1 b2 .

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 z2 = -1:

z2 = -1;the approach : z = a,bleads to z2 = a,b ×a,b = a2 - b2,2ab} = -1 = -1,0 comparison of coefficients yields: a2 -b2 = - 1and2ab = 0 the second equation gives a = 0 or b = 0 the latter possibility is excluded, since we must have a2 - 1 therefore a = 0 and thus b 1 thus z1 = 0,1;z2 = 0,-1

The real numbers are a one-dimensional subset of the two-dimensional complex numbers {a,b}, i.e. those with b = 0; thus the real numbers are rare exceptions among the complex numbers. The complex numbers with a = 0, i.e {0,b} are called “imaginary numbers”. Their square is negative: 0,b 0,b = -b2 < 0

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 i is used (in electrical engineering one uses instead the letter j to distinguish the marker from the current i). 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 5i does not refer to a “multiplication of 5 with i”, but means, that the second component of the complex number is 5.

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 i2 = -1. Thus the following rules for the normal representation have to interpreted accordingly

Let us consider an example:

z1z2 = (a1 + ib1)(a2 + ib2) = a1a2 + i2b1b2 + i(a1b2 + a2b1) = a1a2 - b1b2 + i(a1b2 + a2b1)

complex numberz = a,b real number:a = a,0 definitionimaginary numberib = 0,b definition:real part(z) = Re(z) = a definition:imaginary part (z) = Im(z) = b definition:imaginaryunit 0,1 = i definition:z = Re(z) + iIm(z) = a + ib definition:conjugatecomplexnumberz̄ = Re(z) - iIm(z) = a - ib consequencezz̄ = a2 + b2 definition: absolute value z = zz ̄ 0

computation rules in normal representation: z1 + z2 = a1 + a2 + i(b1 + b2) z1z2 = (a1 + ib1)(a2 + ib2) = (a1a2 - b1b2) + i(a1b2 + a2b1)

z1 z2 = z1 z2 z2 z2 = a1a2+b1b2 a22+b22 - ia1b2-a2b1 a22+b22 = a1a2+b1b2 zz̄ - ia1b2-a2b1 zz̄ i2 = ii = 0,1 0,1 = -1,0 = -1 In this specific sense i is the square root of (-1) 

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

crealnumber z2 = (a + ib)(a + ib) = a2 - b2 + i2ab = c creell 2ab = 0  a product vanishes if and only if one of the factors vanishes

Therefore:1.solutiona = 0 -b2 = c b = -c = c-1 = ic z = 0 ic forc < 0 or:2.solution b = 0 a2 = c a = c z = c + i 0 forc 0

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 a,b and c, with which we started, now reads:

z1,2 = - b 2ab2 - 4ac 2a = -b 2a b2 -4ac 2a forb2 > 4ac = -b 2a ib2 -4ac 2a forb2 < 4ac

If a,b and c are themselves complex, the general formula is still valid, but not the distinction between two cases since the order relations > and < 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 ( - c3 = -c3) for negative radicands. In the space of complex numbers we however obtain the following:

z3 = c;creal (a + ib)(a + ib)(a + ib) = (a2 - b2 + i2ab)(a + ib) = a3 - 3ab2 + i(3a2b - b3) = c sincecisreal b(3a2 - b2) = 0 eitherb = 0 or (3a2 - b2) = 0 1.solutionb = 0 a3 = c,a = c3 z = a = c3 always  present real solution

2.solution 3a2 - b2 = 0 b2 = 3a2 a(a2 - 3b2) = c a(a2 - 9a2) = -8a3 = c a = -c 8 3 = -1 2c3 b2 = 3 4(c3)2 b = 3 2 c3 z2 = -1 2c3 + i3 2 c3 = c3(-1 2 + i3 2 ) z3 = -1 2c3 - i3 2 c3 = c3(-1 2 - i3 2 ) two conjugate complex solutions 3 2 = sin120 = -sin240 -1 2 = cos120 = cos240 z1 = c3cos0 z2 = c3(cos120 + i3 2 sin120) z3 = c3(cos240 + i3 2 sin240) = z 2, sincesin240 = -sin120

Thus three roots z1,z2,z3 of z3 = c 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 z3 = c has three solutions in the space of complex numbers, of which one is real; two are complex. As the last representation for c > 0 shows (for c > 0 the points are mirrored on the imaginary axis), the roots are situated symmetrically on a circle with radius 1.

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

The general polynomial of n-th degree has in the space of complex numbers n roots Gauss’s fundamental theorem of algebra. Figure 3.6 shows that for the second and third root of 1.


Figure 3.6: Roots in the complex plane: the blue round points show 1, the red round points show -1 and the squares show 13

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 z1 > z2 is not defined in general. However they can be ordered according to absolute values |zi|, 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


Figure 3.7: Complex numbers in representation via polar coordinates

In the representation using polar coordinates, the absolute value |z| gives the distance r from the origin and the ratio of imaginary part to real part is equal to the tangent of the angle ϕ to the real axis.

The following definition for the polar representation is applicable:

z = r(cosφ + isinφ)

To obtain r and Φ from z or vice versa the following relations apply:

r = z = +zz ̄ = +Re2 (z) + Im2 (z) tanφ = Im(z) Re(z)

z1z2 = r1r2 (cosφ1 cosφ2 - sinφ1 sinφ2) + i(cosφ1 sinφ2 + cosφ2 sinφ1) z1z2 = r1r2[cos(φ1 + φ2) + isin(φ1 + φ2)] z1 z2 = r1 r2 [cos(φ1 - φ2) + isin(φ1 - φ2)]

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 z = 0 means r = 0, irrespective of the value of ϕ. This is also compatible with the tangent of ϕ being undetermined as a ratio of two zeroes.

The number z corresponds in polar representation to the end point of a vector that starts at the origin with length r and makes an angle ϕ 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 z-plane to a u-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 z-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 u-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.


Figure 3.8: Addition of a complex number z2 to all numbers z1 of a point grid. This grid is moved with the tip of the red arrow, that leads to the lower left corner of the array in z-plane; in the same way z1 + z2 moves for all complex z1 the whole complex plane. The supplementary sides of the parallelogram for the vector construction are drawn on the right hand side

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 z-plane, addition and subtraction are equivalent to a translation of the plane without rotation or change of scale.


Figure 3.9: Subtraction of a complex number z2 from all points of a grid. In the left window the point array and the tip of the vector to be subtracted can be pulled with the mouse. The supplementary sides of the parallelogram of the vector construction are shown on the right hand side.

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.


Figure 3.10: The multiplication of z1 with z2 corresponds to a rotation of the vector z1 by the angle of the vector z2 in the mathematical positive sense (anticlockwise), while at the same time being stretched by the absolute value of z2 (compressed, if the absolute value is smaller than 1). The point array and the complete plane is rotated via the angle of z2 while being stretched by the absolute value of z2.


Figure 3.11: Division corresponds to rotation of z1 in the mathematical negative sense (clockwise) by the angle of the vector z2 while undergoing compression by its absolute value (stretching , if the absolute value is smaller than 1).