# Absolute value no longer sqrt(x^2) for complex numbers

• Jun 23rd 2011, 04:34 PM
lamp23
Absolute value no longer sqrt(x^2) for complex numbers
Why is |a+bi| defined as: $\displaystyle \sqrt{a^2+b^2$

Why is it not defined as $\displaystyle \sqrt{(a+bi)^2$

This would yield $\displaystyle \sqrt{a^2-b^2+2abi$

I understand that it comes from applying pythagorean's theorem to the complex plane but since proofs of pythagorean's theorem obviously involve only real numbers I guess it's just a convenient definition so that other results come out the way we want? Is that the idea of even defining the imaginary plane to begin with?
• Jun 23rd 2011, 04:44 PM
HallsofIvy
Re: Absolute value no longer sqrt(x^2) for complex numbers
The point of an absolute value is that |x| is the distance from x to 0. Of course, a "distance" must be a non-negative real number. For a complex number, we can represent the number x+ iy by the point (x,y) in the "complex plane". The distance form (x, y) to (0, 0), by the Pythagorean theorem: $\displaystyle \sqrt{x^2+ y^2}$.

That is not $\displaystyle \sqrt{x^2}$ but it is $\displaystyle \sqrt{z\overline{z}}$ where $\displaystyle \overline{z}$ is the "complex conjugate" of z: the complex conjugate of z= x+ iy is $\displaystyle \overline{z}= x- iy$ which, in the case that z is real, z= x+ 0i, reduces to x so that $\displaystyle \sqrt{z\overline{z}}= \sqrt{x^2}$.
• Jun 23rd 2011, 05:10 PM
lamp23
Re: Absolute value no longer sqrt(x^2) for complex numbers
Well it's our choice which interpretation of absolute value we want to stay with us when we expand to the complex numbers. Why do we choose that particular interpretation?
• Jun 23rd 2011, 05:28 PM
Plato
Re: Absolute value no longer sqrt(x^2) for complex numbers
Quote:

Originally Posted by lamp23
Well it's our choice which interpretation of absolute value we want to stay with us when we expand to the complex numbers. Why do we choose that particular interpretation?

No, sorry in this case it is not your choice.
In mathematics absolute value is a metric (i.e. a distance).
You may chose to redefine a distance function but it must conform with the axioms of a metric. If it does not then it is a new definition and therefore needs a new name.
• Jun 23rd 2011, 06:52 PM
ojones
Re: Absolute value no longer sqrt(x^2) for complex numbers
Quote:

Originally Posted by lamp23
Well it's our choice which interpretation of absolute value we want to stay with us when we expand to the complex numbers. Why do we choose that particular interpretation?

You're right in that we're certainly free to define modulus anyway we like. The problem is, you're definition has no use!