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How can you calculate with them and what can you actually make up from the calculations? And what is exactly meant by normed division-algebras?

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As to your second question, a normed division algebra is $A$ is an algebra over a field which has a sub-multiplicative norm: $$\forall x,y\in A\qquad \|xy\|\le\|x\|\|y\|$$

The interesting thing about octonions is that they form the highest dimensional normed division algebra over the reals (up to an isomorphism): see Hurwitz's theorem.

References: "Normed Algebra". Encyclopaedia of Mathematics. Retrieved 6 June 2013.

Edit: For your first question, I suggest you check out What are some real-world uses of Octonions?.

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If you're interested at all in representation theory, the octonion algebra is the setting for a representation of the smallest exceptional Lie group.

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