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"Proof surfaced in 1898 that the reals, complex numbers, quaternions and octonions are the only kinds of numbers that can be added, subtracted, multiplied and d
by ChrisLomont 8y ago
"Proof surfaced in 1898 that the reals, complex numbers, quaternions and octonions are the only kinds of numbers that can be added, subtracted, multiplied and divided. "
Not true. Any field (in the algebra sense) has these properties. [1]
Quaternions and octnonions also have weirder properties: quaternions are non-commutatve (jk=-kj) and octnonions are non-associative: a(bc) != (ab)c.
I think the article meant these are the only Euclidean Hurwitz algebras [2], which is a far cry from the claim.
[1] https://en.wikipedia.org/wiki/Field_(mathematics) https://en.wikipedia.org/wiki/Field_(mathematics)
[2] https://en.wikipedia.org/wiki/Hurwitz%27s_theorem_(composition_algebras) https://en.wikipedia.org/wiki/Hurwitz%27s_theorem_(compositi...
- paulddraper 8y agoExample: rational numbers
- throwawaymath 8y agoThe rationals are a subfield of the reals.
- Sharlin 8y agoYeah, but they are still a counterexample to the stated claim. It could be that only ℝ as a whole forms a field.
- pishpash 8y agoIt's in the article towards the end, but these are the only kinds of ... "over real numbers," which constraint Furey believes may be only an approximation.
- HerrMonnezza 8y agoEven when adding "over the real number", the claim in the article is wrong: R(x), the algebra of rational functions of one variable with real coefficients is a field, and a module over R containing a isomorphic image of R as a subfield. You really do need the additional constraint of the Hurwitz form to restrict the possibilities to R, C, H, O ... (Of course this has nothing to do with the work of Furey - I'm just seconding that the claim in the article is incomplete and inexact as worded.)