Answer :
Sure, let's insert a rational number and an irrational number between each pair of given numbers:
1. **Between 2 and 3:**
- Rational: \( \frac{5}{2} = 2.5 \)
- Irrational: \( \sqrt{2} \approx 1.414 \)
2. **Between 0 and 0.1:**
- Rational: \( \frac{1}{10} = 0.1 \)
- Irrational: \( \pi \approx 3.14159 \)
3. **Between \( \frac{1}{3} \) and \( \frac{1}{2} \):**
- Rational: \( \frac{1}{2} = 0.5 \)
- Irrational: \( e \approx 2.71828 \)
4. **Between \( -\frac{2}{5} \) and \( \frac{1}{2} \):**
- Rational: \( 0 \)
- Irrational: \( \sqrt{2} \approx 1.414 \)
5. **Between 0.15 and 0.16:**
- Rational: \( 0.155 \)
- Irrational: \( \sqrt{2} \approx 1.414 \)
6. **Between \( \sqrt{2} \) and \( 2 \):**
- Rational: \( 1.5 \)
- Irrational: \( e \approx 2.71828 \)
7. **Between 2.357 and 3.121:**
- Rational: \( 2.5 \)
- Irrational: \( \pi \approx 3.14159 \)
8. **Between 0.0001 and 0.001:**
- Rational: \( 0.0005 \)
- Irrational: \( \sqrt{2} \approx 1.414 \)
9. **Between 3.623623 and 3.623624:**
- Rational: \( 3.6236235 \)
- Irrational: \( \pi \approx 3.14159 \)
10. **Between 6.375289 and 6.375738:**
- Rational: \( 6.3755 \)
- Irrational: \( \sqrt{2} \approx 1.414 \)
These examples provide one rational number and one irrational number inserted between each given pair of numbers. Rational numbers are those that can be expressed as a fraction, and irrational numbers are those that cannot be expressed as a fraction and have non-terminating and non-repeating decimal expansions.
1. **Between 2 and 3:**
- Rational: \( \frac{5}{2} = 2.5 \)
- Irrational: \( \sqrt{2} \approx 1.414 \)
2. **Between 0 and 0.1:**
- Rational: \( \frac{1}{10} = 0.1 \)
- Irrational: \( \pi \approx 3.14159 \)
3. **Between \( \frac{1}{3} \) and \( \frac{1}{2} \):**
- Rational: \( \frac{1}{2} = 0.5 \)
- Irrational: \( e \approx 2.71828 \)
4. **Between \( -\frac{2}{5} \) and \( \frac{1}{2} \):**
- Rational: \( 0 \)
- Irrational: \( \sqrt{2} \approx 1.414 \)
5. **Between 0.15 and 0.16:**
- Rational: \( 0.155 \)
- Irrational: \( \sqrt{2} \approx 1.414 \)
6. **Between \( \sqrt{2} \) and \( 2 \):**
- Rational: \( 1.5 \)
- Irrational: \( e \approx 2.71828 \)
7. **Between 2.357 and 3.121:**
- Rational: \( 2.5 \)
- Irrational: \( \pi \approx 3.14159 \)
8. **Between 0.0001 and 0.001:**
- Rational: \( 0.0005 \)
- Irrational: \( \sqrt{2} \approx 1.414 \)
9. **Between 3.623623 and 3.623624:**
- Rational: \( 3.6236235 \)
- Irrational: \( \pi \approx 3.14159 \)
10. **Between 6.375289 and 6.375738:**
- Rational: \( 6.3755 \)
- Irrational: \( \sqrt{2} \approx 1.414 \)
These examples provide one rational number and one irrational number inserted between each given pair of numbers. Rational numbers are those that can be expressed as a fraction, and irrational numbers are those that cannot be expressed as a fraction and have non-terminating and non-repeating decimal expansions.