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What is The Value of Sqrt (-1)?

Paul TE Cusack*

Eng, Dule 1641 Sandy Point Rd, Saint John Nb, Canada

*Corresponding Author:
Paul TE Cusack
Eng, Dule 1641 Sandy Point Rd, Saint John Nb, Canada
[email protected]

Received date: 10/05/2016 Accepted date: 24/05/2016 Published date: 28/05/2016

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The Complex number, which relies on the Imaginary Number (sqrt-1), can be determined by the use of the Golden Mean Equation. Using simple algebraic manipulation, we can show that the sqrt (-1) has a real value and it the “Conjugate of the Golden Mean”, or 0.618. This paper shows how.


Imaginary number, Complex number, Golden mean.


Since we know that, on the real number line, the point where the fraction meets the multiple occurs at the real value of 1. We reason that, since there are fractions and multiples on the Complex Plane, and then the Golden Mean equation must work there too. In fact, it does. Below, I show, in simplicity, how this works out. This fact that the sqrt (-1) opens up a new vista in Complex numbers [1,2].


The Golden Mean is where the Multiple equals the Fraction, given by the famous equation:


Or Letting n=multiple and n=fraction denominator, we have,

Multiple = Fraction

n[1- sqrt (-1)]=[1+ sqrt (-1)]/ n


Let n=sqrt (-1)

n+n [sqrt (-1)]=[1/ sqrt (-1)] +1


Let n=1

1+ sqrt (-1)=[1/ sqrt (-1)] + 1

(sqrt(-1)=1/ sqrt(-1)

Cross Multiplying,

Cancelling the 1’s on both sides, and simplifying,

[sqrt (-1)][sqrt (-1)]=1


Taking the square root of both sides, (top determine the value of the sqrt (-1), (as yet undetermined) :

Sqrt (-1) = sqrt(- 1)

Or, sqrt 1=1=sqrt (-1)

1 is where the fraction meets the multiple. So, the Golden Mean applies,

Using sqrt (-1)=1, then,

x=1/ (x-1)

x=1/ [x-sqrt(-1)]



1=1 (True)

Therefore, the fraction equals the multiple in the Complex Plane as well as the Real Plane. Thus, we can use the Golden Mean Equation there too.


Let x=1+i






0.618=1/1.618 (True)

So i=sqrt (-1)=0.618


The Imaginary number =sqrt (-1) has a real value and it is the Conjugate of the Golden Mean, i.e., 0.618.