Prove why the product of two negative real numbers is a positive real number.
Solution
Let \(A\) and \(B\) be two positive real numbers. Subsequently, \( –A\) and \(–B\) are the respective additive inverses.
\[-A \cdot -B+ -A\cdot B = -A \cdot (-B+B)\] \[-A \cdot (-B+B) = 0\]
We then add \(A \cdot B\) to both sides of the equation:
\[-A \cdot -B+ -A \cdot B+A \cdot B = 0+A \cdot B\] \[-A \cdot -B+(-A + A) \cdot B = 0+A \cdot B\]
which simplifies to \[-A \cdot -B = A \cdot B.\]
To prove why the quotient of two negative real numbers is a positive real number, just treat either \(A\) or \(B\) as the multiplicative inverse (division is the multiplication of reciprocals).
Check out my other notes at Proof, Disproof, and Derivation
Note by
Steven Zheng
3 years, 8 months ago
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Top NewestYou should include the definition of a positive number, and explain why the product of 2 positive numbers is also positive.
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This problem assumes that you only know that the product of two positive reals is a positive real, and that there exists an additive inverse element for every real number. Maybe I should include that, but I assume the reader knows algebra. They just have to prove why this is so.
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