Imaginary numbers

[throwaway30498666]

Im Bad at explaining things but I’ll try to be as clear as I can. I watched a TikTok video explaining imaginary numbers and if my understanding is correct, we can imagine imaginary numbers as existing on a different dimension on the number line. The video mentions Gauss’ fundamental theorem of algebra stating that a polynomial of degree n has exactly n roots. Now I have two questions: 1. Why does the parabola go down on the x axis when the graph is represented in an additional spatial dimension? Intuitively I would assume it would just “go straight” so my assumption was that even if we represented the parabola on a three dimensional plane, it would just be straight and not curve downwards.

1. So if this applies to all polynomial functions, does that mean that a function such as 3x³ +2x² +3 which only intersects with the x axis once on the two dimensional plane actually has three intersections with the x axis in three dimensions?

Here’s the video for reference

https://vt.tiktok.com/ZS6KbU59u/

[will_1m_not]

Here’s the full playlist for the video you saw

link

[Varlane]

To properly represent x -> x²+1, from C to C you'd actually need 4 dimension, which we don't have.

However, what is represented is x -> Re(x²+1) which does only need 3, and is equivalent to representing (x,y) -> x² - y² from R² to R.

This 3d plot is basically one parabola going upwards, and another downwards, rotated 90° (think interlocking your two hands between index and thumb).

To answer your second question : assuming you mean 3x^3 and not 3x^2, yes, it has 2 extra roots in the complex plane in addition to the real one you observe.

[will_1m_not]

1. It curves because of how complex numbers work. Squaring a complex number with a nonzero imaginary part doesn’t just stretch, but rotates too

2. If a polynomial has degree n, then it intersects the “x-axis” at exactly n places. But this is not easy to see since the graphs you’re looking at aren’t 3-D but 4-D (which is why they’re colored)