Complex numbers

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What to Do When Exponentiation Just Doesn’t Work (Complex Numbers)

06 August 2024 / Krystian Karczyński

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.

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Complex Polynomial Equations Reducible to Quadratic Equations

05 August 2024 / Krystian Karczyński

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.

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Where did this patent in complex roots come from? Where did the third equation come from when calculating the square root of a complex number?

24 July 2024 / Krystian Karczyński

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.

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Helpful tricks in complex numbers

02 July 2024 / Krystian Karczyński

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.

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The (Almost) Trigonometric Form of a Complex Number

02 July 2024 / Krystian Karczyński

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.

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Fourth Degree Polynomial Equations in Complex Numbers

18 April 2024 / Krystian Karczyński

When solving complex polynomial equations, we generally use the same methods as for solving real polynomial equations in high school. How to handle fourth-degree polynomials?

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Trigonometric form of a complex number – different angles, same solution?

18 April 2024 / Krystian Karczyński

Trigonometric functions (used in the trigonometric form of a complex number) are 2π-periodic, which means that for any arguments differing by e.g., 2π, 4π, 6π, they assume the same value.

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