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PostPosted: Thu, 7 Oct 2010 16:50:55 UTC 
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How do we know that the decimal number system can represent all numbers ?


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PostPosted: Thu, 7 Oct 2010 16:54:56 UTC 
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wajed wrote:
How do we know that the decimal number system can represent all numbers ?


No you can't. They don't represent the p-adics, or the surreals, for example.

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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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PostPosted: Thu, 7 Oct 2010 21:08:04 UTC 
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Thanks, that adds to my info.

But I had "integers" in my mind when I asked the question, how do we know that the decimal number system represents / can represent all integer numbers?


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PostPosted: Fri, 8 Oct 2010 00:49:41 UTC 
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wajed wrote:
Thanks, that adds to my info.

But I had "integers" in my mind when I asked the question, how do we know that the decimal number system represents / can represent all integer numbers?


Um...just write out the decimal for each integer....

1.0000..., 0.00000..., -1.00000..., et cetera

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