Editing Brahmagupta–Fibonacci identity (section)

== Multiplication of complex numbers ==

If ''a'', ''b'', ''c'', and ''d'' are [[real number]]s, the Brahmagupta–Fibonacci identity is equivalent to the [[multiplicative function|multiplicative property]] for absolute values of [[complex numbers]]:

:<math>  | a+bi | \cdot | c+di | = | (a+bi)(c+di) | .</math>

This can be seen as follows: expanding the right side and squaring both sides, the multiplication property is equivalent to

:<math>  | a+bi |^2 \cdot | c+di |^2 = | (ac-bd)+i(ad+bc) |^2,</math>

and by the definition of absolute value this is in turn equivalent to

:<math>  (a^2+b^2)\cdot (c^2+d^2)= (ac-bd)^2+(ad+bc)^2. </math>

An equivalent calculation in the case that the variables ''a'', ''b'', ''c'', and ''d'' are [[rational number]]s shows the identity may be interpreted as the statement that the [[field norm|norm]] in the [[field (mathematics)|field]] '''Q'''(''i'') is multiplicative: the norm is given by 
: <math>N(a+bi) = a^2 + b^2,</math>
and the multiplicativity calculation is the same as the preceding one.