F4:  Unique Solution to  x·a=b

"For any  a, b  [in]  R,  [with  a  not equal to zero] the equation  xa=b  has one and only one solution."  (see ita-2-1-10-pg18)

With very minor modifications the proof and solution test of the previous case can be adapted to this one.  Note that the 'center dot' symbol  '·'  has replaced  ×  for multiplication.  This is a common notation, and more will be said on this subject in the 'convention' section below.  Furthermore, what was previously referred to as the multiplicative inverse  'a to the minus 1'  is symoblized by the letter  a  with a superscript of  -1.  For now this notation should not be viewed as having any significance other than representing the multiplicative inverse.


Similarly to what was said in the previous case, the following points are made:  (1) If we let  b=a  in  F4.1,  then we get  x=1,  which shows that the neutral element for multiplication is unique.  (2)  If we let  b=1  in  F4.1,  then we get  x  equal to the multiplicative inverse of  a.  This shows that, for any nonzero real number  a,  the multiplicative inverse is unique.

Definition of Division



Converted [ in part ] by Mathematica      December 4, 2007