# 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 + 4i into the Input Bar, you get the point (3, 4) in the Graphics View. This point’s coordinates are shown as 3 + 4i 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.

If the variable i has not already been defined, it is recognized as the ordered pair i = (0, 1) or the complex number 0 + 1i. 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).

Example: Addition and subtraction:
• (2 + 1i) + (1 – 2i) gives you the complex number 3 – 1i.
• (2 + 1i) - (1 – 2i) gives you the complex number 1 + 3i.
Example: Multiplication and division:
• (2 + 1i) * (1 – 2i) gives you the complex number 4 – 3i.
• (2 + 1i) / (1 – 2i) gives you the complex number 0 + 1i.
Note: The usual multiplication (2, 1)*(1, -2) gives you the scalar product of the two vectors.

GeoGebra also recognizes expressions involving real and complex numbers.

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

## 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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