How to Prove Zero Product Property for Rational Numbers?

[MathematicalMatt]

Howdy, I just stumbled on this forum and was hoping someone could help with this proof:

If a and b are elements of the Rational Number set, then a*b=0, if and only if a=0 or b=0

With that in mind, I need to prove that:

• Prove that if a=0 or b=0, then a*b=0
• Prove that if a*b=0, then a=0 or b=0

Any help is appreciated, cheers.

 

[Muzza]

a) One can assume, without loss of generality, that a = 0. Then for any b, we have that ab = 0b = (0 + 0)b = 0b + 0b. Subtract 0b from both sides and you'll find that 0b - 0b = 0b, or equivalently 0b = 0.

b) Can we assume that every non-zero rational has an inverse? If both a and b are zero, then we are done. Suppose a is non-zero. Then b = 0*a^-1 = 0. The same argument can be repeated if b is non-zero.

 

[HallsofIvy]

One of the "axioms" or defining properties of the rational numbers is that every rational number, except 0, has a multiplicative inverse.

Given that ab= 0, either
1) a= 0 in which case we are done, or

2) a is not 0, in which case a^-1(ab)= a^-10 or b= 0.

 

[MathematicalMatt]

Thanks for the help!