Are Complex Numbers Just 2D Vectors? Here’s Why Not! (with Visual Examples)
I suggest you to watch this video to get a better understanding of the relationship between the complex multiplication and the dot and cross product:
• Complex Multiplication in Terms of Dot Pro...
In this video, I explore the fascinating differences between complex numbers and vectors. I’ll start by graphing some complex numbers and adding them on the 2D plane, showing how they can look and behave similarly to vectors. However, when it comes to multiplication, complex numbers reveal their true nature—they behave quite differently from vectors!
The key difference lies in their underlying structures: complex numbers belong to a field, whereas vectors exist in a vector space. I’ll explain why complex numbers can be multiplied directly within their field, while vector multiplication requires additional structures, like an inner product space. Join me for a visual breakdown that clarifies why complex numbers are not just 2D vectors!
PS: Complex numbers form a field which is more than just a vector space. Actually complex numbers form a vector space over the real numbers and are not JUST like vectors, but more. In a future video, I'll talk more about fields, groups, sets, maps and also vector spaces.
00:00 How to graph complex numbers in a 2D plane
02:27 Are complex numbers just 2D vectors?
05:02 Fields Vs. Vector Spaces
06:37 Multiplication of complex numbers: Scaling and Rotation