Let's solve the given problem step-by-step to determine the smallest set of real numbers that contains [tex]\(-\frac{18}{6}\)[/tex].
1. Evaluate [tex]\(-\frac{18}{6}\)[/tex]:
- We start with the fraction [tex]\(-\frac{18}{6}\)[/tex].
- Divide the numerator by the denominator: [tex]\(-\frac{18}{6} = -3.0\)[/tex].
2. Simplify the result:
- The result of the division [tex]\(-\frac{18}{6}\)[/tex] simplifies to [tex]\(-3.0\)[/tex].
- While the result is a floating-point number, it is essentially [tex]\(-3\)[/tex], since [tex]\(-3.0\)[/tex] and [tex]\(-3\)[/tex] are equivalent.
3. Determine the smallest set of real numbers:
- Let's examine each option given:
- Option A: Irrational Numbers - These include numbers that cannot be expressed as simple fractions (e.g., [tex]\(\sqrt{2}\)[/tex], [tex]\(\pi\)[/tex]). Since [tex]\(-3\)[/tex] can be expressed as a simple fraction, it is not irrational.
- Option B: Rational Numbers - These include numbers that can be written as fractions, including integers (e.g., [tex]\(\frac{1}{2}\)[/tex], 3, [tex]\(-3\)[/tex]). [tex]\(-3\)[/tex] fits into this set.
- Option C: Natural Numbers - These are the positive integers starting from 1 (e.g., 1, 2, 3). [tex]\(-3\)[/tex] is not a natural number because it is negative.
- Option D: Integers - These include all positive and negative whole numbers, including zero (e.g., [tex]\(..., -2, -1, 0, 1, 2, ...\)[/tex]). [tex]\(-3\)[/tex] fits into this set.
4. Identify the smallest set:
- While [tex]\(-3\)[/tex] is indeed a rational number, the smallest set that includes [tex]\(-3\)[/tex] among the given options is the set of integers.
Therefore, the smallest set of real numbers that contains [tex]\(-\frac{18}{6}\)[/tex] is:
D. Integers
