2 Equals the Sum of All the Unit Fractions ≡ Sum of All the Integers
Year: 2026
Type: article
Abstract: In Quaternionic Eulerian Calculus (QEC) a tangent is a derivative and the area it encloses is an integral. On 𝑦 = 1/𝑥 every tangent cuts from the axes a triangle of area exactly 2, and the QEC sum, or Johnson sum, of all the unit fractions, 1/1 included, is that 2. This paper carries the result to the integers. The unit fraction 1/1 sits at the vertex (1, 1) of the hyperbola and carries the unit square beneath it, of area 1. All the unit fractions except 1/1 therefore carry the remaining 2 − 1 = 1. Flipping that set, {½, ⅓, …}⁻¹ ≡ (−1) · {½, ⅓, …} ≡ {2, 3, …}, is the reflection (𝑥, 𝑦) ↦ (𝑦, 𝑥), which preserves area, so the integers from 2 on also carry 1. Restoring 1/1, which the flip leaves fixed, gives the QEC sum of all the positive integers as 1 + 1 = 2. Negating the set carries everything onto the opposite branch of the hyperbola, where the signed area is again 2: the QEC sum of the negative unit fractions has absolute value 2, equal to that of the positive integers. Every equation is dimensionally homogeneous.
Author: Willie Johnson Jr.
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225817995
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