Complex Numbers

GeoGebra does not support complex numbers directly, but you may use points to simulate operations with complex numbers.

If you enter the complex number 3 + 4ί into the Input Bar, you get the point (3, 4) in the Graphics View. This point’s coordinates are shown as 3 + 4ί in the Algebra View.

You can display any point as a complex number in the Algebra View. Open the Properties Dialog for the point and select Complex Number from the list of Coordinates formats on tab Algebra.

The imaginary unit ί can be chosen from the symbol box in the Input Bar or written using Alt + i. Unless you are typing the input in CAS View or you defined variable i previously, variable i is recognized as the ordered pair i = (0, 1) or the complex number 0 + 1ί. This also means, that you can use this variable i in order to type complex numbers into the Input Bar (e.g. q = 3 + 4i), but not in the CAS.

Examples: Addition and subtraction:

(2 + 1ί) + (1 – 2ί) gives you the complex number 3 – 1ί.
(2 + 1ί) - (1 – 2ί) gives you the complex number 1 + 3ί.

Examples: Multiplication and division:

(2 + 1ί) * (1 – 2ί) gives you the complex number 4 – 3ί.
(2 + 1ί) / (1 – 2ί) gives you the complex number 0 + 1ί.

The usual multiplication (2, 1)*(1, -2) gives you the scalar product of the two vectors.

The following commands and predefined operators can also be used:

x(w) or real(w) return the real part of the complex number w
y(w) or imaginary(w) return the imaginary part of the complex number w
abs(w) or Length[w] return the absolute value of the complex number w
arg(w) or Angle[w] return the argument of the complex number w

arg(w) is a number between -180° and 180°, while Angle[w] returns values between 0° and 360°.

conjugate(w) or Reflect[w,xAxis] return the conjugate of the complex number w

GeoGebra also recognizes expressions involving real and complex numbers.

Examples:

3 + (4 + 5ί) gives you the complex number 7 + 5ί.
3 - (4 + 5ί) gives you the complex number -1 - 5ί.
3 / (0 + 1ί) gives you the complex number 0 - 3ί.
3 * (1 + 2ί) gives you the complex number 3 + 6ί.