Question

Which set includes rational numbers but not natural numbers?
Negative 6 and two-thirds, negative 2, StartFraction 5 Over 11 EndFraction, StartFraction 7 Over 8 EndFraction
Negative 5, negative 4, 4, 5
Three-fourths, 4 and StartFraction 5 Over 6 EndFraction, StartFraction 18 Over 5 EndFraction, 8
0, 1, 2, 3

Answer 1

To identify which set includes rational numbers but not natural numbers, let's first clarify the definitions:

- **Natural numbers** are the positive integers starting from 1 onwards: {1, 2, 3, 4, ...}.
- **Rational numbers** are numbers that can be expressed as a fraction \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b \neq 0 \). This category includes positive and negative fractions, integers, and zero.

Now, let's analyze each set:

1. **Set 1:** Negative 6 and two-thirds, negative 2, \( \frac{5}{11} \), \( \frac{7}{8} \)
- Contains: -6 (integer), -2 (integer), \( \frac{5}{11} \) (rational), \( \frac{7}{8} \) (rational).
- **Does not contain natural numbers.** Contains rational numbers.

2. **Set 2:** Negative 5, negative 4, 4, 5
- Contains: -5 (integer), -4 (integer), 4 (natural number), 5 (natural number).
- **Contains natural numbers.**

3. **Set 3:** Three-fourths, 4 and \( \frac{5}{6} \), \( \frac{18}{5} \), 8
- Contains: \( \frac{3}{4} \) (rational), \( \frac{5}{6} \) (rational), \( \frac{18}{5} \) (rational), 4 (natural number), 8 (natural number).
- **Contains natural numbers.**

4. **Set 4:** 0, 1, 2, 3
- Contains: 0 (rational), 1 (natural number), 2 (natural number), 3 (natural number).
- **Contains natural numbers.**

Now, we can conclude:

The **only set** that includes rational numbers but does not include natural numbers is **Set 1 (Negative 6 and two-thirds, negative 2, \( \frac{5}{11} \), \( \frac{7}{8} \)).**