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William Rowan Hamilton invented the terms Scalars and Vectors with his Quaternions. Quaternions are the sum of a scalar and three vectors. Q=s + ix +jy + kz, = s +v where s is the scalar and ix,jy and kz are the vectors. The rule for vectors is i^2=j^2=k^2= ijk = -1.

Scalar and vector quantities are the two kinds of numbers. Scalar quantities are real number quantities having a positive square. s^2>0. Scalars are commutative s1s2=s2s1.

Vector quantities v, are directional numbers and have a negative square, V^2= -1.

Typical vector denotations are i ,j and k.

Vectors are non-commutative ij=-ji

There are false vectors used in physics today ala Gibbs Heaviside "vectors" where v^2=+1. These vectors are non-associative i(jk) does not equal (ij)k when v^2=+1.

Associativity is true when v^2= -1.

Quaternions the sum of a scalar and three vectors constitutes a four dimensional division space. This is only associative division space. Quaternions contains two associative division sub spaces, the Complex space (Reals + Imaginaries)) and the Real space (Reals).

There is another division space, Octonions consisting of reals and vectors but it is not associative.

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