Find an irrational number between `1/5` and `2/5`

Answer

To find an irrational number between \( \frac{1}{5} \) and \( \frac{2}{5} \), we can follow these steps:

### Step 1: Convert the fractions to decimal form
First, we need to convert \( \frac{1}{5} \) and \( \frac{2}{5} \) into decimal form.

\[ \frac{1}{5} = 0.2 \]
\[ \frac{2}{5} = 0.4 \]

### Step 2: Identify the range
Now we know that we are looking for an irrational number between \( 0.2 \) and \( 0.4 \).

### Step 3: Choose a decimal number in the range
A simple approach to find an irrational number is to choose a decimal that is non-terminating and non-repeating. One common example of such a number is \( \sqrt{2} - 1 \).

### Step 4: Calculate \( \sqrt{2} - 1 \)
We know that \( \sqrt{2} \) is approximately \( 1.414 \). Therefore:

\[ \sqrt{2} - 1 \approx 1.414 - 1 = 0.414 \]

### Step 5: Check if it lies between \( 0.2 \) and \( 0.4 \)
Now we check if \( 0.414 \) lies between \( 0.2 \) and \( 0.4 \):

\[ 0.2 < 0.414 < 0.4 \]

### Step 6: Conclusion
Thus, \( \sqrt{2} - 1 \) is an irrational number that lies between \( \frac{1}{5} \) and \( \frac{2}{5} \).

### Summary
An irrational number between \( \frac{1}{5} \) and \( \frac{2}{5} \) is \( \sqrt{2} - 1 \approx 0.414 \).

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