# Complex Numbers

#### GeoGebra Objects

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.

• `(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.

## Workaround: IsComplex[]

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`.
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