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Gauss's quote for positive, negative, and imaginary number

Recently I watched the following great videos about imaginary numbers by Welch Labs.

https://youtu.be/T647CGsuOVU?list=PLiaHhY2iBX9g6KIvZ_703G3KJXapKkNaF

I like this article about naming of math by Kalid Azad.

https://betterexplained.com/articles/learning-tip-idea-name/

Both articles mentioned about Gauss, who suggested to use other names of positive, negative, and imaginary numbers. Gauss wrote these names are wrong and that is one of the reason people didn't get why negative times negative is positive, or, pure positive imaginary times pure positive imaginary is negative real number. I made a few videos about explaining why -1 * -1 = +1, too.

Explanation: why -1 * -1 = +1 by pattern
https://youtu.be/uD7JRdAzKP8

Explanation: why -1 * -1 = +1 by climbing a mountain
https://youtu.be/uD7JRdAzKP8

But actually Gauss's insight is much powerful. The original is in the GauÃŸ, Werke, Bd. 2, S. 178.
HÃ¤tte man +1, -1, √-1) nicht positiv, negative, imaginÃ¤re (oder gar ummÃ¶gliche) Einheit, sondern etwa directe, inverse, laterale Einheit gennant, so hÃ¤tte von einer solchen Dunklelheit kaum die Rede sein kÃ¶nnen.
If I translate this to English:
If we call +1, -1, and √-1 had been called direct, inverse and lateral units, instead of positive, negative, and imaginary (or impossible) units, such an obscurity would have been out of the question.
Gauss suggested negative should be coined as inverse. So inverse times inverse is direct, like Kalid coined positive and negative as forward and backward. If you do inverse and then inverse, of course it is original direction, like backward and then backward is forward.

Gauss's imaginary number name is lateral number (side number). When direct lateral times direct lateral, which is √-1 * √-1 = -1, then -1 * √-1 is inverse lateral, - √-1, then inverse lateral * direct lateral is direct, -√-1 * √-1 = +1. You can see in the following figures.

 Figure 1: Inverse and direct instead of positive and negative

 Figure 2: direct, inverse, and lateral
In this terminology, not ``negative times negative is positive.'', but ``inverse times inverse is direct.''

Why A^{T}A is invertible? (2) Linear Algebra

Why A^{T}A has the inverse Let me explain why A^{T}A has the inverse, if the columns of A are independent. First, if a matrix is n by n, and all the columns are independent, then this is a square full rank matrix. Therefore, there is the inverse. So, the problem is when A is a m by n, rectangle matrix.  Strang's explanation is based on null space. Null space and column space are the fundamental of the linear algebra. This explanation is simple and clear. However, when I was a University student, I did not recall the explanation of the null space in my linear algebra class. Maybe I was careless. I regret that... Explanation based on null space This explanation is based on Strang's book. Column space and null space are the main characters. Let's start with this explanation. Assume  x  where x is in the null space of A .  The matrices ( A^{T} A ) and A share the null space as the following: This means, if x is in the null space of A , x is also in the null spa

Why parallelogram area is |ad-bc|?

Here is my question. The area of parallelogram is the difference of these two rectangles (red rectangle - blue rectangle). This is not intuitive for me. If you also think it is not so intuitive, you might interested in my slides. I try to explain this for hight school students. Slides:  A bit intuitive (for me) explanation of area of parallelogram  (to my site, external link) .