Insert two irrational numbers between `(2)/(3)` and `(3)/(2)`

Answer

To insert two irrational numbers between \( \frac{2}{3} \) and \( \frac{3}{2} \), we can follow these steps:

### Step 1: Convert the fractions to decimal form
First, we convert \( \frac{2}{3} \) and \( \frac{3}{2} \) into decimal form for easier comparison.

\[ \frac{2}{3} = 0.6666... \quad (\text{which is } 0.\overline{6}) \]
\[ \frac{3}{2} = 1.5 \]

### Step 2: Identify the range
Now we have the decimal equivalents:
- Lower bound: \( 0.6666... \)
- Upper bound: \( 1.5 \)

### Step 3: Choose irrational numbers
We need to find two irrational numbers that lie between \( 0.6666... \) and \( 1.5 \).

One common way to generate irrational numbers is to use square roots.

1. **First irrational number**: We can take \( \sqrt{2} \) and convert it to decimal form:
\[ \sqrt{2} \approx 1.4142... \]
This number lies between \( 0.6666... \) and \( 1.5 \).

2. **Second irrational number**: We can take \( \sqrt{3} \) and convert it to decimal form:
\[ \sqrt{3} \approx 1.7320... \]
However, this number is greater than \( 1.5 \). So, we can instead take \( 1 + \frac{\sqrt{2}}{10} \):
\[ 1 + \frac{\sqrt{2}}{10} \approx 1 + 0.1414... \approx 1.1414... \]
This number also lies between \( 0.6666... \) and \( 1.5 \).

### Conclusion
Thus, the two irrational numbers we can insert between \( \frac{2}{3} \) and \( \frac{3}{2} \) are:
- \( \sqrt{2} \approx 1.4142... \)
- \( 1 + \frac{\sqrt{2}}{10} \approx 1.1414... \)