Question
True or false. Two numbers have the same absolute value when the product of the two numbers is 1
Answers
GPT-4o mini
This statement is false.
Two numbers can have the same absolute value without their product being 1. For example, the numbers \(3\) and \(-3\) have the same absolute value (which is \(3\)), but their product is \(3 \times (-3) = -9\), not \(1\).
For two numbers \(a\) and \(b\) to have the same absolute value, it suffices that \(a = b\) or \(a = -b\). The product of the two numbers could be \(1\) if one of them is the reciprocal of the other, specifically in the case when one number is positive and the other is its reciprocal. For instance, the numbers \(1\) and \(-1\) have the same absolute value, but their product is \(-1\).
Thus, the assertion "Two numbers have the same absolute value when the product of the two numbers is 1" is false.
Two numbers can have the same absolute value without their product being 1. For example, the numbers \(3\) and \(-3\) have the same absolute value (which is \(3\)), but their product is \(3 \times (-3) = -9\), not \(1\).
For two numbers \(a\) and \(b\) to have the same absolute value, it suffices that \(a = b\) or \(a = -b\). The product of the two numbers could be \(1\) if one of them is the reciprocal of the other, specifically in the case when one number is positive and the other is its reciprocal. For instance, the numbers \(1\) and \(-1\) have the same absolute value, but their product is \(-1\).
Thus, the assertion "Two numbers have the same absolute value when the product of the two numbers is 1" is false.