Real Number

**AlgebraicallyChallenged** · January 6th 2009, 08:15 PM

Good evening all of our class is having a online debate with the following question, any feedback would be appreciated:

In the Real Number realm,
ab = 0 è a = 0 or b = 0
Is the same theorem true in the Complex Number realm? (Why or why not?)

**Mush** · January 6th 2009, 08:24 PM

Originally Posted by AlgebraicallyChallenged

Good evening all of our class is having a online debate with the following question, any feedback would be appreciated:

In the Real Number realm,
ab = 0 è a = 0 or b = 0
Is the same theorem true in the Complex Number realm? (Why or why not?)

In the complex number 'realm', for a complex number to be considered equal to $0$ , then both it's real and imaginary parts must be equal to zero!

$z_1 = a+ib$
$z_2 = c+id$

$z_1 \times z_2 = ac+adi+cbi-bd = (ac-bd) + i(ad+cb) = 0 + 0i$

Hence $ac-bd = 0$ and $ad+cb=0$ for $z_1 \times z_2=0$

So let's look at the case where $z_1 = a+ib =0+0i$ . For this $a=0$ and $b = 0$ , which means that $ac-bd = 0(c)-(0)d = 0$ . So the first equation is satisfied. $(0)d+(0)c = 0$ . So the 2nd is satisfied.

Now look at the case where $z_2 = c+id =0+0i$ . For this c=0 and d = 0, which means that $ac-bd = 0(a)-(0)b = 0$ . So the first equation is satisfied. $(0)b+(0)b = 0$ . So the 2nd is satisfied.

**Mush** · January 6th 2009, 09:11 PM

Ahhh I see what you were asking. But the question remains, are there any combinations of $z_1$ and $z_2$ for which their product is zero, but for which neither of them are 0?!

Well. Let's try this then. For a complex number to be zero, then it's modulus must be zero, yes? So let's find the modulus of our product.

$|z_1.z_2| = |(ac-bd) + i(ad+bc)|$

$= \sqrt{(ac-bd)^2 + (ad+bc)^2}$

$= \sqrt{(ac)^2-2abcd+(bd)^2 + (ad)^2+2abcd+(bc)^2}$

$= \sqrt{(ac)^2+(bd)^2 + (ad)^2+(bc)^2} = 0$

Clearly, if this is zero, then the expression inside the square root is zero!

$= (ac)^2+(bd)^2 + (ad)^2+(bc)^2 = 0$

$= a^2(c^2+d^2)+b^2(c^2+d^2) = 0$

$= (a^2+b^2)(c^2+d^2) = 0$

Hence, by the logic of REAL numbers (a, b, c and d must all be real, remember!) either:

$a^2+b^2 = 0$

OR

$c^2+d^2 = 0$

For these to be true:

$a = \pm \sqrt{-b^2}$

or

$c = \pm \sqrt{-d^2}$

$b^2$ and $d^2$ are always positive numbers, which means that the solutions to these 2 equations for all values of b and d are not real solutions, but purely imaginary solutions. And by the definition of the complex numbers $z_1$ and $z_2$ , a, b, c and d must be REAL numbers. Hence there are no two non-zero complex numbers whose product is zero, and hence if $z_1.z_2 = 0 +0i$ then $z_1 = 0+0i$ or $z_2 = 0+0i$

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