What makes a number system?

[mklabgr]

What Makes a Number System?

Author: Lukáš Lejdar
Topic: Number systems, geometry, normed division algebras, Hurwitz’s theorem

The article asks why the familiar sequence of number systems seems to jump through dimensions

$\mathbb{R};(1),\qquad \mathbb{C};(2),\qquad \mathbb{H};(4),\qquad \mathbb{O};(8)$

rather than continuing naturally through dimensions $3,5,6,\ldots$.
Its starting point is geometric. Suppose numbers are vectors in $\mathbb R^n$, addition is ordinary vector addition, and multiplication by a fixed nonzero number must act geometrically as a rotation together with a uniform scaling.
The author shows that this apparently simple requirement forces multiplication to be linear and distributive.
In the complex plane, for example, multiplication by $i$ is a quarter-turn, giving

$i^2=-1$

and recovering ordinary complex multiplication.
Why does three-dimensional multiplication fail?
The same geometric idea explains why a three-dimensional analogue cannot exist.
A rotation in $3$-dimensional space necessarily has an axis of fixed points. Its infinitesimal velocity field therefore vanishes along that axis.
However, multiplication by a nonzero number is supposed to be an invertible rotation-and-scaling transformation. It therefore cannot send a nonzero vector to zero.
More generally, if $u$ is a unit vector perpendicular to the multiplicative identity, the geometry forces

$u(ur)=-r$

From this and related orthogonality conditions, multiplication tables can be constructed progressively.
Two dimensions give the complex numbers, four dimensions force the quaternions, and extending the construction further eventually produces the eight-dimensional octonions.
Along the way, the article derives relations closely related to Clifford algebra identities:

$u(vr)+v(ur)=-2\langle u,v\rangle r$

Why only dimensions 1, 2, 4 and 8?
The construction cannot continue indefinitely.
Attempts to extend the same type of multiplication beyond the octonions eventually lead to contradictory multiplication rules.
Therefore, the only possible finite-dimensional real systems satisfying these geometric requirements occur in dimensions

$\boxed{1,;2,;4,;8}$

These correspond precisely to

$\boxed{\mathbb R,;\mathbb C,;\mathbb H,;\mathbb O}$

that is:
1 dimension: Real numbers $\mathbb R$
2 dimensions: Complex numbers $\mathbb C$
4 dimensions: Quaternions $\mathbb H$
8 dimensions: Octonions $\mathbb O$
This result is essentially Hurwitz's theorem, which states that the only finite-dimensional real normed division algebras are the real numbers, complex numbers, quaternions and octonions.
Their fundamental algebraic property is

$|xy|=|x|,|y|$

This means that multiplication by a nonzero element uniformly scales lengths while preserving the underlying geometric structure.
Key Takeaways
• The dimensions $1,2,4,8$ are not an accident. They arise from strong geometric restrictions on multiplication.
• There is no three-dimensional number system behaving like the complex numbers under multiplication.
• The natural progression is

$\mathbb R\rightarrow\mathbb C\rightarrow\mathbb H\rightarrow\mathbb O$

• The sequence ends with the octonions.
• The article gives a particularly geometric and intuitive route toward understanding Hurwitz's theorem, instead of presenting it only as an abstract algebraic classification.

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Original article:
https://numbersystems.lejdar-lukas.workers.dev/