---
© 2015-2026 Wesley Long & Daisy Hope. All rights reserved.
Synergy Research — FairMind DNA
License: CC BY-SA 4.0
Originality: 95% - Novel mathematical concept; interphasic number theory is original
---

# Interphasic Numbers

## Golden Foundation

**Φ** — The Golden Ratio is the only known number that when squared produces the same ratio with an exact increment of 1:

    Φ² = Φ + 1
    1.6180339887498948482045868343656² = 2.6180339887498948482045868343656

This is as empirical as 1+1=2.

### Quadrian Constants & Golden Seed

    Quadrian Position:    a = 1
    Quadrian Ratio:       q = √5 / 2
    Golden Ratio:         Φ = q + 1/2

### Quadrian e

According to convention *e* is not defined by geometry. It is a mathematical constant related to growth and rate of change. A key property of eˣ is that its derivative and integral are both equal to eˣ, making it unique in calculus. eˣ will have the same value, gradient and area at any point on a graph. This is the only function that has this property — making it the natural language of calculus.

Similar to the Golden Ratio it is the only number with this unique behavior. Could it be related to geometry and the Golden Ratio?

    e = √( Φ × (5 - (3×5 - 2) / ((3×5) × 2)) )

The direct and near exact calculation of *e* along with a direct connection to geometry and the Golden Ratio is clear and as empirical as the Golden Ratio itself and 1+1=2.

### Quadrian π

    π_q = f_q(a, b) = ln(b) / √a

### Ramanujan Quadrian Constant

    e^(π_q × √a) = b

### Quadrian Scale

    f_s(x) = x⁸ - x^(8 × (√(√(5×23×353) - 7/2) / ((3×5) × 2)))
    b = f_s(a)

*b can either be provided directly or calculated from a:*

    a = 163    b = 262537412640768744

    WHEN a = 163 and b = 262537412640768744 THEN π_q = π
    WHEN a = 163 and b = f_s(a)              THEN π_q = π

### Ramanujan Constant Verification

    e^(π√163) = 262537412640768743.99999999999925

    Quadrian e:  e ≈ √(Φ(5 - 13/30)) = 2.71827553459134328797...

    a = 163     c = 262537412640768744

    ln(b) = ln(c) / √a
    e^(ln(b) · √a) = c
    ln(b) ≈ π      b = e^ln(b) = 23.14069263277926...

    e^(π√163) = 262537412640768744

    1 / 0.44786704621473526... ≈ √5
    *if you reverse the value of c as a decimal

### Syπ Equation (Simplified)

    Π(n) = 3940245000000 / ((2217131 × n) + 1253859750000)

- Produces Pi Gradient with n=162 being the closest to accepted value of π
- Geometric alignment with Absolute Zero where n = −273150

### The 355/113 Connection

    (30 × 12) − 5 = 355
    (9 × 12) + 5  = 113

    355 / 113 ≈ π

### Reference Precision Values

    Φ  = 1.6180339887498948482045868343656381177203091798057
    e  = 2.7182755345913432879730640640614359252016420866960
    π  = 3.1415926535897932384626433832972661934754988808835

---

## Imaginary Numbers Are Real

The reason algebra is so effective is because numbers themselves are symbolic not simple values. 1 is actually equal to pi which is what gives rise to "imaginary" numbers. When we say 1 we MUST think of 1 being equal to Pi in the context it is in. This is why complex numbers are so effective.

Despite the "fundamental" nature of Euler's Identity it is not technically true. When this is calculated we get a number slightly over 0.

### Euler's Identity

    e^(iπ) + 1 = 0
    e^(iπ) = -1
    e^(iπ) = (cos(π) + i·sin(π)) + 1 = 0

### Literal Calculation

    e^(iπ) + 1 = 0.04321423260310

This is not the "technical" way to calculate this value according to convention. You have to calculate the imaginary part and the real part of the complex number etc. I call BS.

### Imaginary Numbers Work Because 1 = π

    Identity of 1:           ln(e^π) / π = 1
    Euler's Calculated Identity:  i·(ln(e^π)/π) + 1 = 0

    1 + 1 = 2
    π + π = 2π

    0.04321423260310

---

## Why 162?

    x₁ = 161 × 0.04321423260310 = 6.9574914491001431514
    x₂ = 162 × 0.04321423260310 = 7.0007056817032496306
    x₃ = 163 × 0.04321423260310 = 7.0439199143063561098

    a = 7 - x₂ = 0.0007056817032496306
    b = e₁ - x₃ = 0.0432142326031064792

    ln(x₁ - x₂) = -π

**The difference between 162 and 163 is related to -π**

    b/a = 61.237569862031278842437572231903 ≈ 19.5π

### Literal Calculation

    e^(iπ) + 1 = 0.04321423260310

---

## The Transcendental Whole Number?

Could there be a whole number that is transcendental. What would or could that even mean. Is there a number itself that is not easily or conventionally calculated but that interacts numbers that are and converts them to transcendental numbers ie: pi, e etc.

### Charles Hermite's Contradiction

    0 < a < 1

### Actual Calculation

    e^(iπ) + 1 = 0.04321423260310

This is not a precision problem as once thought. It in fact a type of transformation from one numerical space to another. I call this type of number Interphasic Numbers and they are what seed transcendental numbers as well as why imaginary numbers are so effective. It shows pi as a unit of measure.

### Interphasic Numbers

    ln(0.04321423260310) = -π

    Identity of 1:           ln(e^π) / π = 1
    Euler's Calculated Identity:  i·(ln(e^π)/π) + 1 = 0

    1 + 1 = 2
    π + π = 2π

So it seems to be some kind of intrinsic link to the number 1 and pi and this entire transcendental number aspect to which seems to be a transform or conversion from one algebraic system to another.