If $n$ is a negative number, then which of the following is the least?

### Concept & Logic When working with negative numbers, multiplying by a positive constant scales the number further into the negatives, making its value "lesser" (further left on the number line). Squaring a negative number yields a positive result. ### Step-by-Step Solution * **Given:** $n$ is a negative number ($n < 0$). * **Deduction:** Let's substitute a simple negative integer, like $n = -2$, to evaluate the magnitude of each option. * **Calculation:** * Option (a): $0$ * Option (b): $-(-2) = 2$ (Positive) * Option (c): $2(-2) = -4$ (Negative) * Option (d): $(-2)^2 = 4$ (Positive) * Comparing the results: $-4 < 0 < 2 < 4$. The smallest value is $-4$, which corresponds to the expression $2n$. ### Exam Strategy & Shortcut **Conceptual Elimination:** A negative number ($n$) multiplied by a positive integer ($2$) yields a negative number ($2n$). Option (b) negates a negative, yielding a positive. Option (d) squares a negative, yielding a positive. Since $2n$ is the only expression that remains strictly negative, it must be less than $0$ and the positive options. ### Common Pitfall Confusing "least" with "closest to zero". Students sometimes pick $0$ or $-n$ thinking about absolute value. Remember that on a number line, $-4$ is smaller (less) than $-2$. ### Final Answer Therefore, the correct answer is **$2n$**.