One, two, four, eightNumbers

Each algebra is the one before it, doubled. Take a pair of quaternions and call the pair an octonion; the rule for multiplying pairs is one line, and it is the same line at every rung:

Start with the reals, turn the crank, and you get the complex numbers. Turn it again: quaternions. Again: octonions. Again: sedenions, sixteen of them, and the bottom falls out.

What each turn costs

• 1 → 2. You lose the ordering. There is no saying whether i is bigger or smaller than zero.
• 2 → 4. You lose commuting. e1 e2 = e3, but e2 e1 = −e3. Order of operations now matters, which is exactly why quaternions describe rotations — rotations do not commute either.
• 4 → 8. You lose associating. (ab)c and a(bc) part company. What survives is the weaker alternative law: any two octonions still generate an associative patch, so (aa)b = a(ab) always. That is enough structure to build on, and it is the last rung where you get it.
• 8 → 16. You lose the norm law, |ab| = |a||b| — and with it, division. Two nonzero sedenions can multiply to zero. There are 84 such pairs of basis planes; the first one the search finds is (e1 + e10)(e5 + e14) = 0.

So the norm law, |ab| = |a||b|, is the thing that picks out 1, 2, 4 and 8 and nothing else. It is also, read a different way, the triality form that runs the rest of this site.

Beside each division algebra sits a split one of the same size, written ℂ′, ℍ′, 𝕆′. Same construction, one sign flipped, so the length of a thing can come out negative. They are not division algebras — some nonzero elements have length zero and cannot be divided by — but they multiply the same way, and they are how the same Lie algebra shows up in several real forms Lisi 2026 §2. Physics cares about that, because spacetime has a minus sign in it too.

Switch the multiplier above to 𝕆′ and watch |ab| = |a||b| keep holding while the signs on the diagonal go the other way.