# Complex Numbers

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

**Example:**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.

**Note:**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.

**Note:**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*

**Note:**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.

## Comments

## Workaround: IsComplex[][edit | edit source]

Sometimes you may want to check if a number is treated as complex number in GeoGebra, as function such as `x()`

and `y()`

do not work with real numbers. As there is no such command as `IsComplex`

you currently have to employ a small trick to check if the number `a`

is complex: `complex = IsDefined[sqrt(a) + sqrt(-a)] ∧ (a ≠ 0)`

.

**Note:**Complex with imaginary part 0, like

`a = 2 + 0i`

, also pass this test. If you just want to check if the imaginary part of a complex number `a`

is not 0 you can use `y(a) != 0`

.