To find the modulus (or absolute value) of a real number [tex]\( n \)[/tex], let's start by understanding what modulus means.
The modulus, also known as the absolute value, of a number [tex]\( n \)[/tex] is denoted by [tex]\( |n| \)[/tex]. It represents the distance of that number from zero on the number line, without considering the direction. Therefore, the modulus of [tex]\( n \)[/tex] is always a non-negative value.
Given the options:
1. [tex]\( \sqrt{n} \)[/tex]
2. [tex]\( 1 \pi \)[/tex]
3. [tex]\( n \)[/tex]
4. [tex]\( \pi^2 \)[/tex]
Let's analyze each one:
1. [tex]\( \sqrt{n} \)[/tex]: The square root of [tex]\( n \)[/tex] is not necessarily the modulus of [tex]\( n \)[/tex]. For example, if [tex]\( n = -4 \)[/tex], [tex]\( \sqrt{n} \)[/tex] is not defined for real numbers.
2. [tex]\( 1 \pi \)[/tex]: The expression [tex]\( 1 \pi \)[/tex] represents a fixed value (approximately 3.14), which is unrelated to the actual value of [tex]\( n \)[/tex].
3. [tex]\( n \)[/tex]: If [tex]\( n \)[/tex] is a real number, its modulus is indeed [tex]\( |n| \)[/tex]. For positive [tex]\( n \)[/tex], [tex]\( |n| = n \)[/tex]. For negative [tex]\( n \)[/tex], [tex]\( |n| = -n \)[/tex], which can be viewed as representing [tex]\( n \)[/tex] as the non-negative value of itself in the context of this question.
4. [tex]\( \pi^2 \)[/tex]: This is a constant (approximately 9.87) and does not represent the modulus of [tex]\( n \)[/tex] since it has nothing to do with the value of [tex]\( n \)[/tex].
From our analysis, the correct option that represents the modulus of a real number [tex]\( n \)[/tex] is:
[tex]\[
n
\][/tex]
So, the modulus of a real number [tex]\( n \)[/tex] is equal to [tex]\( \boxed{n} \)[/tex].
