## 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

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.''

### Random quotes from ancient time to now

The quality measure of a city (country) is what it does with its referendum.
-- a philosopher

A good decision is based on knowledge and not on numbers.
-- Plato

Dictatorship naturally arises out of democracy, and the most aggravated form of tyranny and slavery out of the most extreme liberty.
-- Plato

One of the penalties for refusing to participate in politics is that you end up being governed by your inferiors.
-- Plato

Yes, he said; but what are the characteristics of this form of government, and what are the defects of which we were speaking?
First of all, I said, consider the nature of the qualification just think what would happen if pilots were to be chosen according to their property, and a poor man were refused permission to steer, even though he were a better pilot?
-- The republic, Plato