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jackie.abb
 Post subject: Complex numbers Q
PostPosted: Wed, 23 Nov 2011 05:46:05 UTC 
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What happens when a complex number is raised to a power?

For example

(-1-i)^10

....
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Shadow
 Post subject: Re: Complex numbers Q
PostPosted: Wed, 23 Nov 2011 06:02:12 UTC 
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jackie.abb wrote:
What happens when a complex number is raised to a power?

For example

(-1-i)^10

....


If it's a normal power like 1, 2, 3, 4, . . . et cetera, you just compute it by multiplying by itself that many times, just like you do with all other numbers.

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outermeasure
 Post subject: Re: Complex numbers Q
PostPosted: Thu, 24 Nov 2011 15:02:55 UTC 
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\begin{aligned} Spin(1)&=O(1)=\mathbb{Z}/2&\quad&\text{and}\\ Spin(2)&=U(1)=SO(2)&&\text{are obvious}\\ Spin(3)&=Sp(1)=SU(2)&&\text{by }q\mapsto(\mathop{\mathrm{Im}}\mathbb{H}\ni p\mapsto qp\bar{q})\\ Spin(4)&=Sp(1)\times Sp(1)&&\text{by }(q_1,q_2)\mapsto(\mathbb{H}\ni p\mapsto q_1p\bar{q_2})\\ Spin(5)&=Sp(2)&&\text{by }\mathbb{HP}^1\cong S^4_{round}\hookrightarrow\mathbb{R}^5\\ Spin(6)&=SU(4)&&\text{by the irrep }\Lambda_+\mathbb{C}^4 \end{aligned}
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