Find irrational number between a and b .

Answer

To find an irrational number between two given numbers \( a \) and \( b \), we can follow these steps:

### Step 1: Identify the Numbers
Let’s say we are given two numbers \( a \) and \( b \). For example, let’s take \( a = 2 \) and \( b = 3 \).

### Step 2: Calculate the Product
Next, we need to calculate the product of \( a \) and \( b \):
\[ \text{Product} = a \times b = 2 \times 3 = 6 \]

### Step 3: Check for Perfect Square
We need to ensure that the product \( ab \) is not a perfect square. In this case, \( 6 \) is not a perfect square (since \( \sqrt{6} \) is not an integer).

### Step 4: Find the Irrational Number
To find an irrational number between \( a \) and \( b \), we can use the formula:
\[ \text{Irrational Number} = \sqrt{a \times b} \]
Substituting the values we have:
\[ \text{Irrational Number} = \sqrt{6} \]

### Step 5: Verify the Range
Now, we need to check if \( \sqrt{6} \) lies between \( 2 \) and \( 3 \). We know that:
\[ \sqrt{4} = 2 \quad \text{and} \quad \sqrt{9} = 3 \]
Since \( 4 < 6 < 9 \), it follows that:
\[ 2 < \sqrt{6} < 3 \]

### Conclusion
Thus, the irrational number between \( 2 \) and \( 3 \) is \( \sqrt{6} \).

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