# Rational-Addition Numbers: A Theory of Delayed Collapse ## Preface This is a working theory of a three-parameter number representation. A **Rational-Addition Number** (RAN) is written as \[ \mathrm{RAN}(a,b,c) = \frac{a}{b} + c \] where \(a,b,c\) are chosen from some number system. The novelty is not the set of values that RANs can represent—often that set is just the rationals or the reals—but the computational structure that delays the final evaluation. The purpose of this text is to make that structure explicit and to explore its consequences for numerical accuracy, symbolic computation, and the philosophy of measurement. --- ## Chapter 1: Introduction and Motivation A rational number is usually presented as a ratio of two integers: \[ \frac{p}{q} \] A mixed fraction separates the whole part from the fractional part: \[ c + \frac{a}{b} \] The Rational-Addition Number makes that separation a first-class object. We write \[ \mathrm{RAN}(a,b,c) = \frac{a}{b} + c \] and treat the triple \((a,b,c)\) as the *state* of the number. The actual value is produced only when we force an evaluation, which we call **collapse**. The motivation comes from two sources: 1. **Numerical accuracy.** When a large number is added to a small number inside a fixed-precision format, the small number can be lost. By keeping the small term as a separate numerator and denominator, we prevent premature rounding. 2. **The delayed-collapse paradigm.** In quantum mechanics, the wavefunction is evolved in a richer parameter space and only collapsed when a measurement is performed. RANs apply the same philosophy to arithmetic: calculate in parameter space, and collapse only at the boundary. --- ## Chapter 2: Formal Definition ### 2.1 The parameter space Let \(S\) be a set of numbers. The most common choices are: - \(S = \mathbb{Z}\), the integers - \(S = \mathbb{Q}\), the rationals - \(S = \mathbb{R}\), the reals - \(S = \mathbb{C}\), the complex numbers The **RAN space** over \(S\) is the set of triples \[ \mathcal{R}(S) = \{(a,b,c) : a,b,c \in S, b \neq 0\} \] An element of \(\mathcal{R}(S)\) is called a **raw RAN**. ### 2.2 The collapse map The **collapse map** \(\Phi : \mathcal{R}(S) \to S\) is defined by \[ \Phi(a,b,c) = \frac{a}{b} + c \] This is the numerical value of the RAN. Collapse is the analogue of measurement: it produces a single number and discards the structural information in the triple. ### 2.3 Equivalence Two raw RANs are **equivalent** if they collapse to the same value: \[ (a,b,c) \sim (d,e,f) \iff \frac{a}{b} + c = \frac{d}{e} + f \] This is an equivalence relation. The **value** of a RAN is its equivalence class. A RAN is therefore a *representation* of a value, not the value itself. Examples over the reals: \[ \mathrm{RAN}(1,2,0) \sim \mathrm{RAN}(1,4,0.25) \sim \mathrm{RAN}(2,4,0) \sim \mathrm{RAN}(0,1,0.5) \] All collapse to \(0.5\). --- ## Chapter 3: Algebraic Structure ### 3.1 Operations on raw RANs We define addition, subtraction, multiplication, and division on raw RANs by the following algebraic formulas. **Addition:** \[ \mathrm{RAN}(a,b,c) + \mathrm{RAN}(d,e,f) = \mathrm{RAN}(ae + bd,\; be,\; c + f) \] **Subtraction:** \[ \mathrm{RAN}(a,b,c) - \mathrm{RAN}(d,e,f) = \mathrm{RAN}(ae - bd,\; be,\; c - f) \] **Multiplication:** \[ \mathrm{RAN}(a,b,c) \cdot \mathrm{RAN}(d,e,f) = \mathrm{RAN}(ad + afe + cbd,\; be,\; cf) \] **Division:** \[ \frac{\mathrm{RAN}(a,b,c)}{\mathrm{RAN}(d,e,f)} = \mathrm{RAN}\big(e(a + bc),\; b(d + ef),\; 0\big) \] These formulas are chosen so that they match the corresponding operations on the collapsed values. ### 3.2 The fundamental theorem **Theorem (Consistency of operations).** For any two raw RANs \(x,y \in \mathcal{R}(S)\) and any operation \(\star \in \{+, -, \cdot, /\}\), \[ \Phi(x \star y) = \Phi(x) \star \Phi(y) \] provided the right-hand side is defined. **Proof.** The proof is a direct calculation. For addition: \[ \Phi\big(\mathrm{RAN}(ae+bd,\; be,\; c+f)\big) = \frac{ae+bd}{be} + c + f = \frac{a}{b} + \frac{d}{e} + c + f = \Phi(x) + \Phi(y) \] The other operations follow similarly. ∎ This theorem is the heart of the theory. It says that operations performed in parameter space are guaranteed to agree with the corresponding operations on values. Therefore, we may postpone collapse indefinitely. ### 3.3 The quotient structure Because the operations are compatible with collapse, they descend to the quotient space \[ \mathcal{R}(S)/\sim \] which is the set of RAN values. **Theorem.** If \(S\) is a field, then \(\mathcal{R}(S)/\sim\) is a field isomorphic to \(S\). **Proof.** The collapse map \(\Phi\) is a surjective homomorphism from the quotient to \(S\). Its kernel consists of the RANs equivalent to zero, i.e., those with \(a/b + c = 0\). The first isomorphism theorem gives the result. ∎ So the RAN system is not a new algebraic number system in the strict sense. It is a *redundant* representation of an existing number system. The redundancy is the point. --- ## Chapter 4: Canonical Forms A single value has many RAN representations. It is useful to choose a standard form. ### 4.1 Integer canonical form For \(S = \mathbb{Z}\), we define the **canonical form** of a RAN by the following rules: 1. \(b > 0\) 2. \(0 \le a < b\) 3. \(\gcd(a,b) = 1\) **Theorem (Uniqueness of canonical form).** Every integer RAN is equivalent to exactly one canonical RAN. **Proof.** This is the standard mixed-fraction theorem. Given \(p/q\), divide \(p\) by \(q\) to obtain quotient \(c\) and remainder \(a\) with \(0 \le a < b\), then reduce the fractional part. ∎ This canonical form is unique, but it is a *partial collapse*: it moves the whole part into \(c\) and reduces the fraction, but it does not produce a single decimal or binary value. ### 4.2 Float canonical form For \(S = \mathbb{R}\) or \(S = \mathbb{Q}\), no exact canonical form exists because many equivalent representations differ only by floating-point rounding. In this case we usually do not normalize; instead, we keep the raw form and delay collapse as long as possible. --- ## Chapter 5: Delayed Collapse and Accuracy ### 5.1 Information loss at collapse When a RAN \(\mathrm{RAN}(a,b,c)\) is collapsed to a single floating-point number, two things happen: 1. The division \(a/b\) is rounded to a single number. 2. The addition \(a/b + c\) is rounded, which may cause the smaller term to be absorbed into the larger. If \(c\) is much larger than \(a/b\), the term \(a/b\) can be lost entirely. This is the numerical analogue of losing phase information upon measurement. ### 5.2 Catastrophic cancellation Consider two nearly equal numbers: \[ x = 1{,}000{,}000 + \frac{1}{1{,}000{,}000} \] \[ y = 1{,}000{,}000 \] In ordinary floating-point arithmetic, the small term is below the spacing of representable numbers near \(1{,}000{,}000\). So \(x\) rounds to \(1{,}000{,}000\), and \(x - y = 0\). In RAN form: \[ x = \mathrm{RAN}(1,\; 1{,}000{,}000,\; 1{,}000{,}000) \] \[ y = \mathrm{RAN}(0,\; 1,\; 1{,}000{,}000) \] Subtraction gives \[ x - y = \mathrm{RAN}(1,\; 1{,}000{,}000,\; 0) \] The large offsets cancel exactly in parameter space, and the small term survives. ### 5.3 Accumulation of small terms Adding three copies of \(1/10\) in floating-point arithmetic gives \[ 0.1 + 0.1 + 0.1 = 0.30000000000000004 \] In RAN form: \[ \mathrm{RAN}(1,10,0) + \mathrm{RAN}(1,10,0) + \mathrm{RAN}(1,10,0) = \mathrm{RAN}(3,10,0) \] The intermediate arithmetic is exact. Collapse occurs only once, at the end, producing the closest floating-point approximation to \(3/10\). --- ## Chapter 6: The Quantum Analogy The parallel between RANs and quantum mechanics is conceptual, not physical. | Quantum mechanics | RAN theory | |---|---| | Wavefunction \(|\psi\rangle\) | Raw RAN \((a,b,c)\) | | Hilbert space | RAN space \(\mathcal{R}(S)\) | | Unitary evolution | Algebraic RAN operations | | Superposition | Unevaluated form \(a/b + c\) | | Measurement / collapse | Evaluation \(\Phi(a,b,c)\) | | Observable | The value \(a/b + c\) | | Interference | Cancellation of large offsets in parameter space | In both cases, the richest information exists *before* collapse. The act of producing a single value discards structure that could have been exploited by further calculation. --- ## Chapter 7: Generalizations ### 7.1 Multi-term RANs A natural extension is to allow several rational terms: \[ \mathrm{RAN}_k(a_1,b_1,\dots,a_k,b_k,c) = \sum_{i=1}^k \frac{a_i}{b_i} + c \] This is useful when a number is composed of many corrections of different magnitudes. It corresponds to a floating-point expansion or a sum of compensated corrections. ### 7.2 Complex RANs If \(a,b,c \in \mathbb{C}\), then the RAN represents a complex number. The same algebraic rules apply, with complex division and multiplication. The real and imaginary parts can be kept as separate RANs if desired. ### 7.3 Probabilistic RANs One can imagine RANs whose parameters are random variables. Collapse then produces a distribution over values, not a single number. This is a speculative direction that connects RANs to interval arithmetic and uncertainty quantification. ### 7.4 Matrix RANs If \(a,b,c\) are matrices, then \(\mathrm{RAN}(a,b,c)\) represents a matrix expression. The collapse is \(b^{-1}a + c\). Delayed collapse might preserve matrix structure that is destroyed when the inverse is computed early. --- ## Chapter 8: Implementation A minimal RAN implementation stores the triple and overloads the arithmetic operators. The Python class given in the companion note provides this behavior. The key design choices are: - Keep \((a,b,c)\) as separate fields. - Perform operations using the algebraic formulas above. - Provide a `.collapse()` method that evaluates the number only when requested. - Optionally provide a `.simplify()` method for partial collapse. The implementation is small enough to be a teaching example and rich enough to show real accuracy improvements in cancellation-prone calculations. --- ## Chapter 9: Applications ### 9.1 Numerical computing RANs are useful whenever a computation mixes large offsets with small corrections. Examples include: - N-body simulations with widely separated scales - Financial calculations with large principal and small interest - Geometric computations with large coordinates and small displacements ### 9.2 Symbolic algebra Computer algebra systems already keep expressions unevaluated. RANs provide a particularly simple normal form for expressions of the form rational + constant. ### 9.3 Quantum-inspired algorithms The delayed-collapse paradigm is not merely an analogy. It suggests a design principle for algorithms: represent intermediate states in the richest possible parameter space, and collapse only at the output boundary. This principle appears in quantum computing, symbolic computation, and automatic differentiation. --- ## Chapter 10: Conclusion and Open Problems Rational-Addition Numbers are a simple idea with several layers: 1. As a set of values, they are equivalent to the rationals or reals, depending on the parameter set. 2. As a representation, they introduce redundancy that can be exploited for numerical accuracy. 3. As a computational paradigm, they embody the principle of delayed collapse. The theory raises several questions: - What is the optimal canonical form for mixed integer-float RANs? - Can RANs be extended to include irrational offsets in a computationally useful way? - What is the complexity of RAN arithmetic compared to exact rational arithmetic? - How can RANs be integrated with floating-point expansion libraries? The most important lesson is that a number is not only its value. The *way* a number is represented can determine whether a calculation succeeds or fails. In that sense, RANs are a study in the ontology of numerical representation: a number is its value plus the structure that survives collapse. --- ## References - Higham, N. J. *Accuracy and Stability of Numerical Algorithms*. (Numerical accuracy) - Feynman, R. P., Leighton, R. B., & Sands, M. *The Feynman Lectures on Physics, Vol. 3*. (Quantum measurement and collapse) - Knuth, D. E. *The Art of Computer Programming, Vol. 2*. (Seminumerical algorithms) - The Python `fractions` module and SymPy symbolic library. (Exact and symbolic arithmetic)