Geometry

Core Theoretical Principles

This subject revisits standard algebraic equations and trigonometric angle systems from fresh, intuitive angles.

1. Inverting the Quadratic Formula

The classic quadratic equation $ax^2 + bx + c = 0$ has roots:

$$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$

However, when $a \to 0$ (nearly linear equations), numerical cancellation error occurs in the numerator. By dividing the equation by $x^2$ and substituting $u = 1/x$, we obtain $c u^2 + b u + a = 0$. Solving for $u$ and inverting gives:

$$ x = \frac{2c}{-b \pm \sqrt{b^2 - 4ac}} $$

This dual formulation is numerically stable for small $a$ and reveals a direct relationship with Vieta’s formulas:

$$ x_1 + x_2 = -\frac{b}{a}, \qquad x_1 x_2 = \frac{c}{a} $$

2. Dimensional Radians vs Arbitrary Degrees

A degree ($1^\circ$) is defined arbitrarily as $1/360$ of a full rotation, rooted in ancient Babylonian sexagesimal calendars.

In contrast, the radian is the mathematically pure ratio of arc length $s$ to radius $r$:

$$ \theta = \frac{s}{r} \quad [\text{dimensionless ratio}] $$

Because of this definition, calculus derivatives take their purest form without scaling factors:

$$ \frac{d}{d\theta} \sin(\theta) = \cos(\theta) \quad (\text{only valid when } \theta \text{ is in radians}) $$