Thread: How do you algebraically manipulate absolute valued variables?

1. How do you algebraically manipulate absolute valued variables?

For example: 2x^2-|x|^2 and 2y^2-|y|^2 in the attached img. Does 2x^2-|x|^2 equal x^2 or |x|^2 or both?

Is there a difference between |x|^2 and x^2?

It seems the first term refers to the square of the magnitude and the second of whatever happens to be the domain.

2. Re: How do you algebraically manipulate absolute valued variables?

Hey Elusive1324.

There is no difference between |x|^2 and x^2 in terms of the map. In other words for the same values of x you get the same y for both functions. If you really want to prove it then break up the |x| into negative and positive x and show that the relationship holds since you get analytic functions (-x for x < 0 and x for x >= 0) which can be compared.

3. Re: How do you algebraically manipulate absolute valued variables?

in fact when we talk of square it does not make a difference. in that case x^2 = ( |x| )^2
As we know the modulus function is defined as
|x| = -x for x < 0 and |x| = x for x = > 0.

4. Re: How do you algebraically manipulate absolute valued variables?

If the mapping is the same, then what is the point of writing it as |x|^2 in the steps?

5. Re: How do you algebraically manipulate absolute valued variables?

Originally Posted by Elusive1324
If the mapping is the same, then what is the point of writing it as |x|^2 in the steps?
Let's just prove this. It has been several days now.

For all real numbers $a~\&~b$ it is true that $a^2+b^2\ge 2|ab|$

Suppose that $z=x+yi$, so
$\\x^2+y^2\ge 2|xy|\\2x^2+2y^2\ge x^2+ 2|xy|+y^2\\2|z|^2\ge (|x|+|y|)^2$

Thus $\sqrt2|z|\ge|\text{Re}(z)|+|\text{Im}(z)|~.$