I am looking for a convincing argument to show that if $w$ is a complex number, $w^{\frac 12}$ has 2 different roots.
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The adjectives "positive" and "negative" carry little meaning when talking about complex numbers. – Arthur Jan 19 '20 at 22:53
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What do you mean by a "positive" complex number? – Brian61354270 Jan 19 '20 at 22:53
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my apologies. what i meant was some similarity with the root of real numbers – math_student Jan 19 '20 at 22:56
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Welcome to Mathematics Stack Exchange. Note that $0$ has only one square root – J. W. Tanner Jan 19 '20 at 23:13
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$w=0$ is an exception. You don't have two different roots in this case. – Kavi Rama Murthy Jan 19 '20 at 23:20
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Consider the polar representation $w=r e^{i(\phi+2k\pi)}$.
Its square root yields 2 distinct roots that repeat ad infinitum.
That is: $$w^{1/2}=\sqrt r e^{i(\phi/2+k\pi)}$$ If $r=0$ we have only 1 root, which is 0. Otherwise we get the 2 distinct roots $\sqrt r e^{i\phi/2}$ and $\sqrt r e^{i(\phi/2+\pi)}$. All other values of $k$ map to these 2 roots due to the $2\pi$ periodicity.
Klaas van Aarsen
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this doesn't answer the question. we are talking about 2 distinct (non-trivial) answers – math_student Jan 20 '20 at 12:08
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@math_student, I have extended my answer to clarify how we do get 2 distinct roots. – Klaas van Aarsen Jan 20 '20 at 18:06