Complexe getallen

Some equations have no real solution because they involve the square roots of negative numbers.

To address this, complex numbers are introduced, defining the square root of −1 as the imaginary unit i.

This can be visualized on the complex plane, where the real and imaginary parts form perpendicular axes, placing each complex number as a point.

The addition of complex numbers involves separately adding their real and imaginary parts.

Multiplication of complex numbers follows the distributive property. Since i²=−1, any occurrence of i² is replaced with −1 during simplification.

Dividing complex numbers involves multiplying both the numerator and denominator by the denominator’s conjugate—which has the same real part and the opposite imaginary part—to eliminate the imaginary part.

Just as every positive real number has two square roots, every negative real number also has two complex square roots, which are complex conjugates.

Complex numbers are used in Magnetic Resonance Imaging, where the scanner collects complex signal data called k-space. This data is converted into spatial images using inverse Fourier transforms.