This exercise indicates one of the reasons why multiplication of complex numbers is not...

This exercise indicates one of the reasons why multiplication of complex numbers is not carried out simply by multiplying the corresponding real and imaginary parts of the numbers. (Recall that addition and subtraction are carried out in this manner.) Suppose for the moment that we were to define multiplication in this seemingly less complicated way:

(a) Compute (2 + 3i)(5 + 4i), assuming that multiplication is defined by

(b) Still assuming that multiplication is defined by (*), find two complex numbers z and w such that z ≠ 0, w ≠ 0,but zw = 0 (where 0 denotes the complex number 0 + 0i).

Now notice that the result in part (b) is contrary to our expectation or desire that the product of two nonzero numbers be nonzero, as is the case for real numbers. On the other hand, it can be shown that when multiplication is carried out as described in the text, then the product of two complex numbers is nonzero if and only if both factors are nonzero.