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Helpful tricks in complex numbers

Krystian Karczyński

Founder and General Manager of eTrapez.

Graduate of Mathematics at Poznan University of Technology. Mathematics tutor with many years of experience. Creator of the first eTrapez Courses, which have gained immense popularity among students nationwide.

He lives in Szczecin, Poland. He enjoys walks in the woods, beaches and kayaking.

Complex numbers as a whole are not a complicated and difficult topic. However, things can get “heated” in less typical and less schematic situations. The key then, as always, is to understand the topic and keep a “cool head,” meaning being clear-headed and confident.


For example:

  1. When raising a complex number to some larger power, we often end up with a somewhat strange form like:

    To improve the aesthetic of the number, you can perform the following transformation:

    Improving aesthetics is often very important – especially if this number is part of an even more complicated mathematical entity.
  2. A complex number in Cartesian (or algebraic) form does not have to be nice looking, for example: . Its real and imaginary parts can be ugly, irrational messes, such as:

    But don’t panic, it’s just an ordinary complex number, where (real part not multiplied by ‘i’, imaginary part multiplied):

    – changing the sign in the imaginary part

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