Complex Plane
Complex Numbers
Definition: The "number" :
To solve the equation we create a symbol and declare
that so solves the equation and we write .
Definitions: A complex number is a number that can be expressed in the form where and are real numbers.
If then is called the real part of and (and sometimes ) is called the imaginary part of .
Examples: and .
Complex Geometry- The Complex Plane
Complex numbers are identified with points in a cartesian plane by having identified with the point with coordinates or with position vectors by identifying
with the vector .
With this identification is identified with the point or the vector and is identified with or the vector .
Complex Numbers in GeoGebra
From the Toolbar:
From the point dropdown, select 
complex number. Then click on any point in the Graphics Frame to create a complex number.
Comments: The point will appear in the frame labeled z1.
In the Algebra Frame will appear z1 = ... + ....i.
This number/point is "free" and can be changed with the mouse (reset to the pointer 
) by moving it, or by entry of a new value by double clicking on current value in the Algebra Frame.
From the Input Bar:
Enter the complex number with or without a name: e.g., z1= 2+3i or just 2+3i.
Comments: A point will appear in the Graphics Frame labeled z1.
In the Algebra Frame will appear z1 = 2 + 3i.
This number/point is "free" and can be changed with the mouse (reset to the pointer 
) by moving it, or by entry of a new value by double clicking on current value in the Algebra Frame.
The Norm of a Complex Number
From the Input Bar:
As a function: Enter abs(...) and hit return. A real number will appear in the Algebra Frame determined as the norm of the complex number. E.g. abs(3+4i) gives ...=5. This real number is dependent on the argument of abs.
With bar notation: Enter |...| and hit return. A real number will appear in the Algebra Frame determined as the norm of the complex number. E.g. |3+4i| gives ...=5. This real number is dependent on the number between the bars.
Polar Representation of a Complex Number
From the Input Bar:
As a function: Enter ToPolar(...) and hit return. A point will appear in the Algebra Frame determined with the polar coordinates of the complex number. E.g. ToPolar(3+4i) gives ...=(5; 53.13). This point is
dependent on the on the argument of ToPolar.
Comment: The Argument of a complex number is the second coordinate of the Polar Form.
This can be found directly by using the "Angle" function applied to the complex number. E.g. Angle(3 + 4ί) gives ...=53.13.
complex number. Then click on any point in the Graphics Frame to create a complex number.
Comments: The point will appear in the frame labeled z1.
In the Algebra Frame will appear z1 = ... + ....i.
This number/point is "free" and can be changed with the mouse (reset to the pointer ) by moving it, or by entry of a new value by double clicking on current value in the Algebra Frame.
From the Input Bar:
Enter the complex number with or without a name: e.g., z1= 2+3i or just 2+3i.
Comments: A point will appear in the Graphics Frame labeled z1.
In the Algebra Frame will appear z1 = 2 + 3i.
This number/point is "free" and can be changed with the mouse (reset to the pointer ) by moving it, or by entry of a new value by double clicking on current value in the Algebra Frame.
The Norm of a Complex Number
From the Input Bar:
As a function: Enter abs(...) and hit return. A real number will appear in the Algebra Frame determined as the norm of the complex number. E.g. abs(3+4i) gives ...=5. This real number is dependent on the argument of abs.
With bar notation: Enter |...| and hit return. A real number will appear in the Algebra Frame determined as the norm of the complex number. E.g. |3+4i| gives ...=5. This real number is dependent on the number between the bars.
Polar Representation of a Complex Number
From the Input Bar:
As a function: Enter ToPolar(...) and hit return. A point will appear in the Algebra Frame determined with the polar coordinates of the complex number. E.g. ToPolar(3+4i) gives ...=(5; 53.13). This point is
dependent on the on the argument of ToPolar.
Comment: The Argument of a complex number is the second coordinate of the Polar Form.
This can be found directly by using the "Angle" function applied to the complex number. E.g. Angle(3 + 4ί) gives ...=53.13.