How do I prove |xy| = |x| * |y|?

[D H]

This massive off-topic misunderstanding over the meaning of √x was a bit unwarranted.

As an aside, proving the conjecture with |x| defined as √(x²) is rather trivial.

 

[Mark44]

Surely not true. √9 = ± 3. √-9 = ± 3 i

D H already dealt with this, but I'll say a bit more. It's true that 9 has two square roots, but the symbol √9 is generally recognized to mean the positive square root, or 3.

It would be just like the one already given for |x|. Instead of we'd have something like |x|= √(x²) if √(x²)> 0, |x|= -√(x²) if √(x²)<0, and |x|=0 if x=0, introducing and then inverting an operation to no purpose.

For any real number x, √(x²) is always nonnegative.

 

[epenguin]

1MileCrash is correct. This is not. √9 is 3, never -3. √x is the function that uniquely maps the non-negative reals to the non-negative reals such that (√x)²=x. This function can be extended to the negative reals and to all of the complex numbers by defining a a branch cut. There are some problems with this extension (e.g., (a^b)^c = a^bc is no longer universally true), but √z still a function, not a multifunction.

OK I guess you're right, now I think of it otherwise we wouldn't write ± in the quadratic solution formula.

However I maintain the appearance of greater simplicity is merely verbal or notational and apparent - you happen to have a convention for something a bit elaborate, you have this convention because you do these operations often. But it is not simpler than Avodyne's definition. It represents an operation on x which is then inverted with a condition. Two unnecessary operations which no one in their senses would ever do. Of course anyone presented with calculating √(x²), say √(-237124.629²) can do it in their heads, write the answer immediately and what do they actually do? They write |x| as calculated according to the Avodyne's definition!

 

[1MileCrash]

OK I guess you're right, now I think of it otherwise we wouldn't write ± in the quadratic solution formula.

However I maintain the appearance of greater simplicity is merely verbal or notational and apparent - you happen to have a convention for something a bit elaborate, you have this convention because you do these operations often. But it is not simpler than Avodyne's definition. It represents an operation on x which is then inverted with a condition. Two unnecessary operations which no one in their senses would ever do. Of course anyone presented with calculating √(x²), say √(-237124.629²) can do it in their heads, write the answer immediately and what do they actually do? They write |x| as calculated according to the Avodyne's definition!

It's not just an appearance of greater simplicity. It is simpler in practice. This proof is one example, just as any proof involving the absolute value of a real number. Other examples aren't hard to find. What is the derivative of |x|? That any person would use Avodyne's definition for a quick computation means it is a useful way to quickly compute something, not that it is a better working definition in proofs or studying qualities of the function.

And another thing I want to mention is that √(x²) isn't just chosen because it's a "function and inverse" that works out like we want it to. Consider that the absolute value for a complex number (the set of which is a superset of the reals, so this MUST hold in the reals too) is |a + bi| = √(a² + b²). That's what it is, there is no way to define it with another function and its inverse. |x| = √(x²) is the exact same thing, the imaginary component is 0. That's why we say that |x| = √(x²) and not some other "function and inverse." It does not require any conditional at all (I still don't see what you mean by "inverted with a condition"), and is guaranteed to work every time.