Understanding Absolute Value
The absolute value of an integer represents its distance from zero on the number line, regardless of direction. It is denoted by two vertical bars surrounding the number, e.g., |x|.
Key Points about Absolute Value:
- Definition: For any integer \( x \):
- If \( x \geq 0 \), then \( |x| = x \).
- If \( x < 0="" \),="" then="" \(="" |x|="-x" />
- Comparison with the Integer:
- If \( x \) is a positive integer, then \( |x| = x \), meaning they are equal.
- If \( x \) is zero, \( |0| = 0 \), thus equal as well.
- If \( x \) is a negative integer, then \( |x| = -x \), which is a positive value, thus greater than \( x \).
Conclusion:
The statement "the absolute value of an integer is always greater than the integer" is false. The absolute value can be equal to the integer when the integer is non-negative (zero or positive).
Summary:
- True for negative integers: \( |x| > x \)
- True for zero: \( |0| = 0 \)
- True for positive integers: \( |x| = x \)
Understanding these distinctions is crucial for evaluating the characteristics of integers in mathematical contexts, including competitive exams like UPSC.