2016-10-12

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 Gauß, Werke, Bd. 2, S. 178.
Hätte man +1, -1, √-1) nicht positiv, negative, imaginäre (oder gar ummögliche) Einheit, sondern etwa directe, inverse, laterale Einheit gennant, so hätte von einer solchen Dunklelheit kaum die Rede sein können.
 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.''

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