- The problem asks to evaluate the expression \(\frac{-2}{0}\).
- Using the definition of division, we look for a number \(z\) such that \(z \cdot 0 = -2\).
- Since any number multiplied by zero equals zero, the equation \(0 = -2\) is impossible.
- Therefore, the expression is undefined, which corresponds to choice B.
- Final Answer: \(\boxed{\text{The expression is undefined.}}\)
Explanation
- Definition of Division
The goal is to evaluate the expression \(\frac{-2}{0}\). By the definition of division for any real number \(x\) divided by a non-zero divisor \(y\), the quotient \(z\) must satisfy the relationship \(z \cdot y = x\). Applying this to the given expression, we seek a value \(z\) such that \(z \cdot 0 = -2\).
- Analysis of the Property of Zero
We know that for any real number \(z\), multiplying it by zero always results in zero, represented by the property \(z \cdot 0 = 0\). If we set \(z \cdot 0 = -2\), we obtain the equation \(0 = -2\), which is a clear contradiction because zero is not equal to negative two. Therefore, no real number \(z\) exists that satisfies the condition for this division.
- Conclusion
Since there is no real number that satisfies the definition of the quotient for \(\frac{-2}{0}\), we conclude that the expression is mathematically undefined. Thus, the correct choice is B. The expression is undefined.
Examples
Division by zero is an important concept in mathematics, appearing frequently in areas like algebra and calculus. For example, when studying the slope of a vertical line on a coordinate plane, the change in x is zero, leading to an undefined slope. Recognizing when an expression is undefined prevents errors in complex calculations and helps in understanding domains of functions and geometric properties.