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In [[mathematics]], some thinkers<ref>Hamilton, Tait, Cayley, Hardy, and all the modern thinkers agree there is some type of relationship, but they differ on the nature of this relationship.</ref> believe there is a relationship between the norm of a [[Classical Hamiltonian quaternions|quaternion]] and the [[tensor]] of a quaternion. Some writers<ref>[[Quaternion#Conjugation.2C_the_norm.2C_and_division|modern thinkers, see proper section of main article]]</ref> define the norm of a quaternion as having the same formula as the tensor of a quaternion, while other writers<ref>Hamilton, Tait, Cayley</ref> define the norm of a quaternion as the square of the tensor. Hamilton uses the term tensor in two different sences as a [[Classical_Hamiltonian_quaternions#Tensor|positive numerical quantity]] and as an operator that operates on other mathematical entities extracting a tensor quantity from them.
HamiltonA defined the new word ''tensor'' as a positive or more properly signless number.{{Failed verification|date=March 2009}} <ref>[http://books.google.com/books?id=TCwPAAAAIAAJ&printsec=frontcover&dq=tensor+definition+positive+properly+new+word#PRA1-PA57,M1 Hamilton 1853 pg 57 ]</ref><ref>[http://books.google.com/books?id=YNE2AAAAMAAJ&printsec=frontcover&dq=positive+tensor+strictly+speaking+number+without+sign&as_brr=1#PPA5,M1 Hardy 1881 pg 5]</ref><ref>[http://books.google.com/books?id=CGZLAAAAMAAJ&pg=PA146&dq=Hamilton+positive+signless+quotients#PPA31,M1 Tait 1890 pg.31 explains Hamilton's older definition of a tensor as a positive number]</ref>The [[tensor of a quaternion]] is a number which represents its magnitude,<ref>Tait</ref> the "stretching factor"<ref>[http://books.google.com/books?id=CGZLAAAAMAAJ&pg=PA146&dq=Hamilton+positive+signless#PPA31,M1 Tait (1890), pg 32]</ref>, the amount by which the application of the quaternion lengthens a quantity; specifically, the tensor is defined{{Fact|date=March 2009}} as the square root of the [[#Common norm|norm]] <ref>[http://books.google.com/books?id=CGZLAAAAMAAJ&pg=PA146&dq=tensor+called+norm#PPA146,M1 Cayley (1890), pg 146], </ref> — this is a one-dimensional quantity, quite distinct from the modern sense of [[tensor]], coined by [[Woldemar Voigt]] in 1898 to express the work of [[Bernhard Riemann|Riemann]] and [[Gregorio Ricci-Curbastro|Ricci]].<ref>''OED'', "Tensor", def. 2b, and citations.</ref> As a square root, tensors cannot be negative{{Fact|date=March 2009}}, and the only quaternion to have a zero tensor is the zero quaternion{{Fact|date=March 2009}}. Since tensors are numbers, they can be added, multiplied, and divided. The tensor of the product of two quaternions is the product of their tensors; the tensor of a quotient (of non-zero quaternions) is the quotient of their tensors; but the tensor of the sum of two quaternions ranges between the sum of their tensors (for parallel quaternions) and the difference (for anti-parallel ones) .
The tensor of the quaternion ''q'' is denoted '''''T'''q''.
==References==
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