Expert: Sherry Wallin - 1/17/2009

Question
Equality of complex number
two complex numbers z_1=a+ib  and z_2=c+id  are said to be equal if and only if a=c and b=d.
proof:

a+ib=c+id
⟹  a-c=i(d-b)
⟹    (a-c)^2=-(d-b)^2
⟹ (a-c)^2+(d-b)^2=0
⟹  a-c=0  and d-b=0       [∴  x^2+y^2=0  is possible iff x=0 and y=0]

Sherry please explain avobe example


Answer
Hi Pratap~
    Just out of curiosity, where did this proof come from? I ask because it is fallacious (wrong).
⟹ (a-c)^2+(d-b)^2=0 this step is true, however the next step is not ⟹  a-c=0 and d-b=0. If the first step was ⟹ (a-c)^2*(d-b)^2=0 [note the multiplication * in place of the addition +],then the next step would follow but in order to solve for something by using the zero product rule you need a product, not a sum. The idea behind something using the zero product rule is this:
an example:  (x-2)(x+1) = 0 what you have is two numbers being multiplied together equaling zero and the only way that can happen is if one or both numbers equal zero so either x-2=0 or x+1=0 [or both]. Ok so back to this proof. First off all it is really saying is that the 'real parts' (the part without the i) must be the same and the 'imaginary parts' must be the same, so that is what you are trying to show or prove.
By the way in an if and only if proof you are required to show both directions, what I mean to say is that you really need to prove both
if a+bi = c+di then a=c and b=d AND if a=c and b=d then a+bi = c+di. [in a complex number you usually put the i after the coefficient, standard form for a complex number is a+bi not a+ib although they are equivalent]. In this proof the attempt is only one direction. It is not stated but should be 'if a+bi=c+di, then...'
in the second step the only possible result is that a=b=c=d=0 since the only way a-c can equal (b-d)i is if both sides equal 0, since a real number (a-c) cannot equal an imaginary number (b-d)i.

Please tell me where this proof came from. I need to know if this is your attempt, something you found on the web, or your teachers attempt or from a book. Also I need you to make absolutely sure you have typed it in correctly.

Until I hear back from you,
Math Prof