What is the theorem about negative numbers? How do you prove it?

 The theorem about negative numbers often refers to properties and rules governing the arithmetic operations involving negative numbers. One fundamental theorem is related to the multiplication of negative numbers:

**Theorem:** The product of two negative numbers is a positive number.

**Proof:**

Let's consider two negative numbers, say \(-a\) and \(-b\), where \(a\) and \(b\) are positive.

\[ (-a) \cdot (-b) = -1 \cdot a \cdot (-1) \cdot b \]

Now, recall that multiplying any number by \(-1\) changes its sign. Therefore:

\[ -1 \cdot a \cdot (-1) \cdot b = a \cdot b \]

So, \((-a) \cdot (-b) = a \cdot b\), which is a positive number.

This theorem is a fundamental property of the real number system and is a result of the consistent application of the rules of arithmetic. It is essential to understand and apply these properties when working with negative numbers in various mathematical contexts.