Category: Complex numbers

There are days when nothing seems to work. And there are examples with complex numbers where nothing goes right. The known and memorized methods don’t help.

Take, for instance, this innocent-looking exponentiation: (1+2i)^8. Following the well-trodden path you’ve used in many examples, you want to write the number 1+2i in trigonometric form and then raise it to the eighth power using the appropriate formula. But you encounter complications along the way… See the trick I use.

Many fourth-degree polynomial equations can be transformed into quadratic equations using a well-known high school trick of substitution. This also works perfectly for polynomials in complex numbers.

See how to turn these equations into lower-degree equations.

When calculating square roots in Cartesian (or: algebraic) form in my Complex Numbers Course, I showed a method that involves adding a third equation to the existing system of two equations, which greatly simplifies and shortens the further calculations.

I showed this method, but didn’t justify it in any way. Recently, I received an email regarding this:

“Could you explain why we can use the method of adding a third equation when calculating the square root of a complex number?” So here’s the explanation.

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.

When solving problems with complex numbers, it’s important to keep in mind that a complex number in trigonometric form has its typical form. And only that form. No more, no less. Therefore, you need to pay attention to when a complex number is in trigonometric form and when it is not.