Proof of |ab|=|a||b| in an Ordered Field

[Jamin2112]

Homework Statement

I'm supposed to prove |ab|=|a||b|. My proof is really sloppy, jumbled, at times nonsensical. Show me how to make it clear and concise.

Homework Equations

Just the ordered field properties

The Attempt at a Solution

If a, b ∈ F, where F is an ordered field, then, by the trichotomy law, one of 4 things is true: a,b≥0; a≥0, b<0; a<0, b≥0; a,b<0.

If a=0 or b=0, then ab=0. Since |0|=0 by definition, |ab|=ab. Also, since a=0 or b=0, the product |a||b| will be zero because it is the product of |0|=0 and another element of F. Therefore |a||b|=0=|ab|.

If a,b>0, then |a|=a and |b|=b. Also, by the properties of an ordered field, ab>0, and so |ab|=ab. Therefore |ab|=ab=|a||b|.

If a>0 and b<0, then |a|=a and |b|=-b. Also, ab<0 [b<0 implies –b>0. Thus a(-b) is the product of two positive numbers and –(a(-b)) is the negative of the product of two positive numbers] . Therefore |ab|=-ab=|a||b|=(-a)b. Because multiplication is commutative, making a<0 and b>0 will yield the same result.

If a,b<0, then |a|=-a and |b|=-b. Also, ab>0 [a,b<0 imply –b,-a>0. Thus (-a)(-b) is the product of two positive numbers]. Therefore |a||b|=(-a)(-b)=|ab|.

 

[JonF]

|b|=-b, |ab|=-ab, |a|=-a and |b|=-b.

all can't be true by def