In schaumm's series book for complex variables, I found this question

Why this fallany happens

-1=\sqrt(-1)*\sqrt(-1)=\sqrt(-1*-1)=\sqrt(1)=1

Could someone help me find the error?

Thank you so much.

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- Jun 13th 2013, 01:50 AMerich22Complex Variables cocerning i
In schaumm's series book for complex variables, I found this question

Why this fallany happens

-1=\sqrt(-1)*\sqrt(-1)=\sqrt(-1*-1)=\sqrt(1)=1

Could someone help me find the error?

Thank you so much. - Jun 13th 2013, 03:50 AMPlatoRe: Complex Variables cocerning i
- Jun 13th 2013, 05:19 AMHallsofIvyRe: Complex Variables cocerning i
Specifically, is not true for complex numbers.

- Jun 13th 2013, 12:23 PMerich22Re: Complex Variables cocerning i
I started to think, Plato....

I think it's maybe the \sqrt(1) is not just 1 depends on which branch are you on, isn't it?

What do you think...? - Jun 13th 2013, 12:27 PMerich22Re: Complex Variables cocerning i
Ok, HallsofIvy, if it is not true, so we can not calculate \sqrt(-4)=\sqrt(4*-1)=\sqrt(4)*\sqrt(-1)=2i, can't we?

- Jun 13th 2013, 12:40 PMPlatoRe: Complex Variables cocerning i
- Jun 13th 2013, 12:53 PMPlatoRe: Complex Variables cocerning i
- Jun 13th 2013, 02:17 PMerich22Re: Complex Variables cocerning i
I'm rather confused, Plato.

In the complex system, is \sqrt(1) means we want to find complex numbers z such that z^2=1?

That means in complex system, the \sqrt(1) = +1 and -1, doesn't it?

It differs from the real system that defines \sqrt(1) as a positive number x such that x^2=1. Thus the solution is just x=1, without x=-1?

CMIW - Jun 13th 2013, 02:24 PMerich22Re: Complex Variables cocerning i
Plato, concerning \sqrt{-4}=\sqrt{4\cdot-1}=\sqrt{4}*\sqrt{-1}=2i.

But is it true when we want to find for example the solution of z^2-i*z+2=0, then we could use

z_12=(i \pm \sqrt((-i)^2-4*1*2))/2=(i \pm \sqrt(-9))/2=(i \pm \sqrt(9*(-1)))/2=(i \pm \sqrt(9)*\sqrt(-1))/2=2i or -i? - Jun 13th 2013, 02:49 PMPlatoRe: Complex Variables cocerning i
Absolutely NOT! Nowhere, is that true.

The fact is: the use of a radical is for real numbers.

Now it is true that for historical reasons we*wave our hands*and allow .

BUT**the rules for radicals apply only to real numbers**and do not carry over to complex.

Example: ,*(hand waving)*.

Because is not real, we do not apply rules for radicals. - Jun 13th 2013, 10:53 PMerich22Re: Complex Variables cocerning i
Thank you so much, Plato. Now I understand.

One more question, Plato. But what about the function f(z)=\sqrt(z)? Does it exist? - Jun 14th 2013, 03:44 AMProve ItRe: Complex Variables cocerning i
- Jun 14th 2013, 08:23 AMerich22Re: Complex Variables cocerning i
hmmm, but then 'Prove It', it means that f(1)=\sqrt(1)=1 or -1, doesn't it?