Complex Numbers

GeoGebra does not support complex numbers directly, but you can use points and vectors to display complex numbers in the plane and perform algebraic operations with complex numbers.

When entering the complex number 3 + 4ί into the Input Bar, the point (3, 4) is displayed in the Graphics View. The complex number is shown in rectangular form 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.

Unless you are in the CAS View or you have previously defined a variable named i, entering i is recognized as the ordered pair i = (0, 1) or the complex number 0 + 1ί. This means that you can use i to enter complex numbers into the Input Bar (e.g. w = 3 + 4 i) of the Algebra View. In the CAS View use the shortcut Alt + i. to enter the imaginary unit ί.

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.

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.

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.