Question

Prove that if a is positive and b is negative, then ab is negative

Answer 1

What do we get to assume is true? Can we assume the product of two positive numbers are positive?

Answer 2

If we let \(c=-b\), we know \(c\) is positive. Then \(ac\) is the product of two positive numbers, thus positive. \(ab=-ac\)

Answer 3

Proof by false assumption:
assume "a" is +ve and "b" is -ve and their product is >0
\[ab>0\qquad\text{and}\qquad a(-b)<0\\
\implies ab+a(-b)=0\\
\implies a(b-b)=0\\
\implies a=0
\]
so, there can be no such number except for "0"