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I'm trying to answer a question that asks me to regard $S^3$ as the set of all quaternions of modulus $1$. What does this actually mean?

Thanks

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    Multiplication by any quaternion is continuous; in fact it can be described by polynomial functions, which are clearly continuous.2012-05-24

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This question was answered in the comments:

A quaternion $a+bi+cj+dk$ has norm $a^2+b^2+c^2+d^2$. The quaternions of norm $1$ can therefore be identified with $S^3$.

Multiplication by any quaternion is continuous; in fact it can be described by polynomial functions, which are clearly continuous.

– Qiaochu Yuan May 24 '12


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