RedCrab Math Tutorial

Geometric representation of complex numbers

With complex numbers, operations can also be represented geometrically. With the geometric representation of the complex numbers we can recognize new connections, which make it possible to solve further questions.

The complex plane (Gaussian plane)

Complex numbers are defined as numbers in the form z = a + bi; where i is the imaginary part and a and b are real numbers. A complex number z is thus uniquely determined by the numbers (a, b).

The geometric representation of complex numbers is defined as follows:

  • A complex number z = a + bi is assigned the point (a, b) in the complex plane. The complex plane is similar to the Cartesian coordinate system, it differs from that in the name of the axes.

  • The x-axis represents the real part of the complex number. This axis is called real axis and is labelled asRor Re.

  • The y-axis represents the imaginary part of the complex number. This axis is called imaginary axis and is labelled withiR or Im.

  • The origin of the coordinates is called zero point.

 The figure below shows the number 4 + 3i.

complex number on Gaussian plane

As another example, the next figure shows the complex plane with the complex numbers 
 5i, -4 – 5i, 5 + 3i und 5 - 3i

Gaussian plane

To a complex number z we can build the number -z opposite to it, or the complex number konjugierte komplexe Zahlconjugated to it.

The next figure shows the complex numbers w and z and their opposite numbers -w and -z, as well as the conjugate complex numbers  konjugiert komplexe Zahl wund konjugierte komplexe Zahl

visual complexe plane


The position of an opposite number in the Gaussian plane corresponds to a point reflection around the zero point. Following applies:

-z = - (a + bi) = -a + (-b) i.

The position of the conjugate complex number corresponds to an axis mirror on the real axis in the Gaussian plane.  Following applies:

entgegengesetzte komplexe Zahl= a - bi = a + (- b) i.

The opposite number -ω to ω, or the conjugate complex number  konjugierte komplexe Zahl to z plays an important role in solving quadratic equations. With ω and -ω is a solution of ω2 = D, even if the discriminant D is not real. Because it is (-ω)2 = ω2 = D.

If z is a non-real solution of the quadratic equation az2 + bz + c = 0 with real coefficients a, b, c, then z is always a solution of this equation. This is evident from the solution formula. Non-real solutions of a quadratic equation with real coefficients are symmetric in the Gaussian plane of the real axis.


Forming the opposite number corresponds in the complex plane to a reflection around the zero point.

Forming the conjugate complex number corresponds to an axis reflection around the real axis in the complex plane.



Complex numbers

Basics of complex numbers
Computing with complex numbers
Addition and subtraction
Multiplication of complex numbers
Conjugate complex numbers
Division of complex numbers
Quadratic equations, complex numbers
Geometric representation
Geometric addition and subtraction
Absolute value of a complex number
The polar form of complex numbers
Calculate polar form
Convert polar form to normal
Multiplication in polar form
Division in polar form

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