As you may know i hold a torch for geometry. particularly synthetic geometry.
I have a sneeking suspicion i will have to study the work of Jakob Steiner, too.
Well not many are told that geometry has been extended by logarithms and by differential calculus, because these are usually hived off and hidden under the different names people want to call them. Similarly not many are told that complex "numbers" are an extension of geometry, and that is why they are used in mathematical descriptions of space-time.
Even fewer are tod that probability is an extension of geometry, and that De Moivre in particular used his extensive command of the trigonometry of the unit circle, both with rational and imaginary magnitudes to develop and extend the measurement of games of chance.
We may therefore not be so mystified to find that a notion of negative probability exists. I have bee redacting a paper by Vasil Penchev in which he goes into great dpth on the subject. The relation to classical and quantum physics is revealing. Though the maths is complex, and the argument or exposition is dealing with unfamiliar ideas, the exposition is clear.
I have just found a connection to the solving of the quadratic equation for roots.
We all have heard of roots, but probably have not heard that this is the term for the magnitude of the side of a square. In fact the word square derives from "exquadrature" which means "finding the 4 sided figure equal in area to".
Euclid provides several methods of doing this geometrically, and some chinese and indian vedic sutras rovide a method for doing this by cipherism. but which ultimately are based on geometrical procedures.
Finding the general roots of a quadratic therefore is finding the appropriate square which gives the measurement. The method that is used is called completing the square, and is due to Bhaksharia, an indian mathematician, but the full geometric representation is not often drawn.
The full geometric representation involves iteration. That means successive approximations to the correct square are drawn until they settle on the exact square. This is a curious geometrical design! The image is of a morphing shape that eventually settles, thus sometimes the shape is too large and sometimes to small. Sometimes the boundary of the shape exceeds the ideal, and sometimes the boundary lies within the ideal. This geometric pattern corresponds precisely to positive and negative area.
No one has really bothered to define negative area, but it is easy to do . When it is defined then one immediately sees the correspondence to positive and negative probaility, especially if you meditate on the unit circle definitions of the trigonometric ratios.
Thus the quadratic formula holds within it the information required to understand positive and negative probailities, and thus to give those notions there geometric interpretation.