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complex equation

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  complex equation Antonio Mezzacapo <ant.mezzacapo@gmail.com> - 2011-04-27 09:38 +0000
    Re: complex equation Gary Wardall <gwardall@gmail.com> - 2011-04-28 10:36 +0000

#1898 — complex equation

Hi everyone,
I have a question. Do you know why Mathematica finds solution to this equation
Solve[Sqrt[1 - z^2] == 2, z]
{{z -> -I Sqrt[3]}, {z -> I Sqrt[3]}}
while if I change sign of the right part it doesn't find solution anymore?
Solve[Sqrt[1 - z^2] == -2, z]
{}

Is this related to the phase specification for complex numbers?

Thank you
Antonio Mezzacapo

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#1964

On Apr 27, 4:38 am, Antonio Mezzacapo <ant.mezzac...@gmail.com> wrote:
> Hi everyone,
> I have a question. Do you know why Mathematica finds solution to this equation
> Solve[Sqrt[1 - z^2] = 2, z]
> {{z -> -I Sqrt[3]}, {z -> I Sqrt[3]}}
> while if I change sign of the right part it doesn't find solution anymore?
> Solve[Sqrt[1 - z^2] = -2, z]
> {}
>
> Is this related to the phase specification for complex numbers?
>
> Thank you
> Antonio Mezzacapo


Antonio,

Sqrt[1 - z^2] = -2

has no solutions, real or complex. That is the solution set is empty.
Mathematica is correct when it yields {}.

Note:
Sqrt[1 - z^2] = -2

(Sqrt[1 - z^2] )^2 =  (-2)^2

1 - z^2 = 4

 - z^2 = 3

 z^2 =- 3

z = -i Sqrt[3]  or z= i Sqrt[3]

Checking/Proving:

Sqrt[1 - (-i Sqrt[3])^2] = -2

2 = -2  NO!


Sqrt[1 - (i Sqrt[3])^2] == -2

2 = -2  NO!


Gary Wardall

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