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Commutativity of multiplication is not “lost”. It’s something you shouldn’t generally expect to have. It would be good if people would spend more time learning

Commutativity of multiplication is not “lost”. It’s something you shouldn’t generally expect to have. It would be good if people would spend more time learning about non-commutative multiplications at a younger age, which should be expected in general. Mathematicians’ bias towards commutativity is often so culturally ingrained from a lifetime of commutative algebraic structures that it’s very hard for them to break away from. If they spent more time on the geometry earlier on it could help quite a bit. That multiplication of a vector by a scalar + bivector oriented in a plane containing the vector on the left happens to have the same result as multiplication by the reverse on the right Zv = vZ† comes down to the identity a(bc) = (bc)a for any three coplanar vectors a, b, c, which happens because vectors commute with both scalars and coplanar bivectors. This doesn’t work more generally because the trivector part of the product doesn’t vanish when the vectors are not coplanar. This convenient feature results in many nice results for plane geometry / complex numbers which don’t hold in more general settings. It would be good if students would start with the general version (in high school or early in undergraduate science/engineering programs) so they can separate out the special features in their minds and not take them for granted.

jacobolus 6y ago
P.S. you can “graphically” demonstrate the non-commutativity of 3-dimensional rotation to a layperson (including children) in a few minutes by turning around an arbitrary 3-dimensional object in your hands. P.P.S. The hypervolume of an n-dimensional ball of unit radius only tends to zero as n tends to infinity if you use an n-dimensional hypercube with unit side length as your unit for hypervolume. If you use a standard simplex as your unit for hypervolume, then the volume of the ball tends to infinity. If you use the volume of the ball as your unit, then the volume of the ball tends to 1 by definition. Using a unit-length n-cube as your unit hypervolume is a convenient but arbitrary cultural convention. One main reason this ratio is ”counterintuitive” is because we conventionally use a unit-side-length square or cube but a unit-radius disk or ball, and the extraneous factors of 2 caused by the discrepancy cause confusion in low dimensions; people weirdly think of disks as being bigger than squares. If we used a unit-diameter disk/ball instead, we would see how the ratio of hypervolume of ball/cube goes strictly down as dimension goes up: dimension 1: 1 dimension 2: π/4 ≈ 0.79 dimension 3: π/6 ≈ 0.52 dimension 4: π²/32 ≈ 0.31 dimension 5: π²/60 ≈ 0.16 dimension 6: π³/384 ≈ 0.08 dimension 7: π³/840 ≈ 0.04 ...

jacobolus 6y ago

P.S. you can “graphically” demonstrate the non-commutativity of 3-dimensional rotation to a layperson (including children) in a few minutes by turning around an arbitrary 3-dimensional object in your hands. P.P.S. The hypervolume of an n-dimensional ball of unit radius only tends to zero as n tends to infinity if you use an n-dimensional hypercube with unit side length as your unit for hypervolume. If you use a standard simplex as your unit for hypervolume, then the volume of the ball tends to infinity. If you use the volume of the ball as your unit, then the volume of the ball tends to 1 by definition. Using a unit-length n-cube as your unit hypervolume is a convenient but arbitrary cultural convention. One main reason this ratio is ”counterintuitive” is because we conventionally use a unit-side-length square or cube but a unit-radius disk or ball, and the extraneous factors of 2 caused by the discrepancy cause confusion in low dimensions; people weirdly think of disks as being bigger than squares. If we used a unit-diameter disk/ball instead, we would see how the ratio of hypervolume of ball/cube goes strictly down as dimension goes up: dimension 1: 1 dimension 2: π/4 ≈ 0.79 dimension 3: π/6 ≈ 0.52 dimension 4: π²/32 ≈ 0.31 dimension 5: π²/60 ≈ 0.16 dimension 6: π³/384 ≈ 0.08 dimension 7: π³/840 ≈ 0.04 ...