Become Hyper-rational Numbers.
by Bruce E. Camber, working draft, https://81018.com/irrational-numbers-2/

Introduction: The four irrational numbers— π (pi), e (Euler’s number), φ (golden ratio or phi), and √2 (square root of 2)—are commonly highlighted in mathematical and scientific contexts as some of the most fundamental or well-known examples of irrational numbers. While there isn’t a single, universally standardized collective term like “primary” in mainstream scientific literature, they are frequently grouped and referred to as “famous irrational numbers” or “prominent irrational constants” due to their historical significance, frequent appearance in nature, geometry, physics, and calculus, and their role as the earliest or most commonly introduced irrationals in education and research.

Few suggestions for renaming: 1) Stabilizers, 2) stabilizing numbers, 3) hyper-rationals, 4).foundational numbers, 5) hypostatic numbers, 6) infinite numbers, 7) (your idea)

Here’s how the scientific and mathematical community typically refers to and describes each:

• Pi (π): Universally referred to as “pi” or the “circle constant”. It’s the ratio of a circle’s circumference to its diameter and is transcendental (not algebraic, proven in 1882). It’s foundational in fields like physics (e.g., wave equations, quantum mechanics) and engineering, with its irrationality proven in the 18th century. In 2020 we added the qualitative, qualities of infinity, continuity, symmetry and harmony.
• Golden ratio (φ or phi): Referred to as “the golden ratio,” “phi,” or “the golden mean”. It’s an algebraic irrational, defined as (1 + √5)/2 ≈ 1.618, and appears in geometry (e.g., pentagons), nature (e.g., phyllotaxis in plants), art, and architecture. It’s tied to the Fibonacci sequence, where the ratio of consecutive terms approaches φ.
• Euler’s number (e): Commonly called “Euler’s number,” “the base of the natural logarithm,” or simply “e”. It’s transcendental (proven in 1873) and central to exponential growth, calculus, probability, and physics (e.g., in radioactive decay or compound interest models). It’s often introduced as the limit of (1 + 1/n)^n as n approaches infinity.
• Square root of 2 (√2): Often called “the square root of two” or simply “sqrt(2)”. It’s recognized as the first historically proven irrational number (attributed to ancient Greek mathematicians like the Pythagoreans around the 5th century BCE) and appears in basic geometry, such as the diagonal of a unit square. It’s an algebraic irrational, meaning it’s a root of a polynomial equation with integer coefficients.

These are indeed among the first irrationals that anybody mentions in educational settings because π is encountered in elementary geometry, e arises in high school calculus, φ in algebra or art-math crossovers, and √2 demonstrates basic irrationality proofs. In scientific bodies like the American Mathematical Society or physics textbooks (e.g., those discussing constants in quantum or classical mechanics), they are discussed individually for their properties rather than as a fixed quartet, but popular math resources and tutoring programs (such as Mathnasium) often list them together as “famous” or “coolest” irrationals to illustrate the concept.

We can say that these four irrational numbers—π, e, φ, √2—are “derivative of infinity,” though this phrasing isn’t standard mathematical terminology. It captures the idea that their precise definitions and representations inherently rely on infinite processes, sequences, or expansions, which distinguish them from rational numbers (which can be exactly expressed with finite information). Below, I’ll explain this for each, grounding it in mathematical reasoning without implying a literal “derivation” from infinity (as infinity isn’t a number but a concept in limits, sets, or cardinality).

1. π (Pi)

• Defined as the ratio of a circle’s circumference to its diameter, π is transcendental (not the root of any finite polynomial with rational coefficients).
• Its “connection to infinity”: The decimal (3.1415926535…) is infinite and non-repeating. Classically, it’s computed via infinite series, such as the Leibniz formula: π/4 = 1 – 1/3 + 1/5 – 1/7 + …, or Archimedes’ method of inscribed polygons, where accuracy improves as the number of sides approaches infinity. In physics (e.g., Fourier series or quantum wave functions), π emerges from limits involving infinite sums or integrals.
• Thus, π is fundamentally tied to infinity; finite approximations (like 22/7) are useful but inexact, requiring infinite terms for the true value.

2. e (Euler’s Number)

• The base of the natural logarithm, e arises in growth processes, like continuous compounding interest or exponential functions.
• Its “connection to infinity”: Explicitly defined as the limit e = lim (n→∞) (1 + 1/n)^n, which directly invokes infinity. Alternatively, it’s the sum of an infinite series: e = Σ (1/n!) from n=0 to ∞ (1 + 1/1! + 1/2! + 1/3! + …). This series converges, but only the infinite sum yields the exact irrational value. In calculus and physics (e.g., Schrödinger’s equation), e appears in solutions to differential equations that model infinite-dimensional spaces or limits.
• e is perhaps the most direct example of being “derivative of infinity,” as its core definitions are limits or sums extending to infinity.

3. φ (Golden Ratio)

• Defined as φ = (1 + √5)/2, it’s the limit of ratios in the Fibonacci sequence (F_{n+1}/F_n → φ as n → ∞).
• Its “connection to infinity”: Though algebraic (root of x² – x – 1 = 0), its decimal (1.6180339887…) is infinite and non-repeating. The continued fraction form is φ = 1 + 1/(1 + 1/(1 + 1/(1 + …))), the simplest infinite continued fraction (all 1’s repeating infinitely). In nature and geometry (e.g., pentagons or spirals), φ emerges from self-similar patterns that theoretically extend infinitely.
• Here, infinity is key to its convergence in sequences; finite Fibonacci ratios approximate it, but exactness requires an infinite sequence.

4. √2 (Square Root of 2)

• This is the length of the diagonal of a square with side length 1, proven irrational by the ancient Greeks via reductio ad absurdum (assuming it’s rational leads to a contradiction).
• Its “connection to infinity”: The decimal expansion (1.41421356237…) is infinite and non-repeating. More profoundly, it can be represented as an infinite continued fraction: √2 = 1 + 1/(2 + 1/(2 + 1/(2 + …))), which requires an infinite nesting. Alternatively, it’s the limit of infinite sequences, like Newton’s method for root approximation, converging only after infinitely many steps for exactness.
• In this way, √2 “derives” from an infinite process; without invoking infinity, we can’t express it exactly beyond its radical form.

Broader Context

• Why “derivative of infinity”? All irrationals share this trait: They can’t be captured by finite decimal expansions or repeating patterns, implying they embody “infinite complexity” in their representation. In set theory, the set of irrationals has the cardinality of the continuum (uncountably infinite), far larger than the countable rationals. Philosophically, thinkers like Georg Cantor (who developed transfinite numbers) highlighted how irrationals reveal the richness of the infinite real line. However, in “rigorous math” we don’t say they “derive from infinity” instead infinity is a tool to define or approximate them.

With this treatment of these numbers, we expect to make a few changes.

• All the mathematics can be represented both as equations and
  

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