For example, 1/3 is the inverse of 3, because 3(1/3) = 1.
The complex number (3 - 4 i)/25 is the inverse of 3 + 4 i because (3 + 4 i)( (3 - 4 i)/25 ) = 1.
There is no inverse of 0 because no number inserted in 0( ) = 1 can ever make the equation true (zero times anything is zero).
If any quaternion is placed inside the parentheses of the equation
(*) (q)( ) = 1
and the equation remains true, then by definition the content of ( ) is the multiplicative inverse of quaternion q. If q*/|q|² is inserted inside the parentheses in (*), then the left side of the equation can be evaluated:
q(q*/|q²) = qq*/|q|² = |q|²/|q|² = 1. (See [5] in the main text for chapter 3.) Therefore q-1 = q*/|q|²
Since a quaternion and its conjugate commute (see [7] in the main text for chapter 3.), it is easily shown that (q-1)q = 1. This proves that any non-zero quaternion and its inverse commute.
It is easier to show the equivalent statement:
|qr|² = |q|²|r|²
|qr|² = |(a + bi + cj + dk)(e + fi + gj + hk)| =
(ae - bf - cg - dh)² + (af + be + ch - dq)² + (ag - bh + ce +df)² + (ah + bg - cf + de)² =
a²e² + b²f² + c²g² + d²h² - 2abef - 2aceg - 2adeh + 2bcfg + 2bdfh + 2cdgh +
a²f² + b²e² + c²h² - d²q² + 2abef + 2acfh - 2adfg +2bceh - 2bdeg - 2cdgh +
a²g² + b²h² + c²e² + d²f² - 2abgh + 2aceg + 2adfg - 2bceh - 2bdfh + 2cdef +
a²h² + b²g² + c²f² + d²e² + 2abgh -2acfh + 2adeh - 2bcfg + 2bdeg - 2cdef =
a²e² + b²f² + c²g² + d²h² +
a²f² + b²e² + c²h² - d²q² +
a²g² + b²h² + c²e² + d²f² +
a²h² + b²g² + c²f² + d²e² =
(a² + b² + c² + d²)(e² + f² + g² + h²) = |q|²|r|²
For this theorm it is necessary that ji = - ij, ik = -ki, kj = -jk. In general, multiplication of quaternions is not commutative.